text:
\section{Introduction} \label{sec:intro}

Model reduction constructs computationally efficient reduced models of large-scale dynamical systems by approximating the high-dimensional states in low-dimensional subspaces of the state space. Reduced models are especially beneficial in the many-query and outer-loop setting, such as in optimization, design, uncertainty quantification, and control, where models are evaluated many times for different inputs. An overview of the field of model reduction can be found in, e.g., \cite{BCOW2017morBook,quarteroni2014reduced}.
Most traditional model reduction techniques are of intrusive nature, which means that full knowledge of the governing equations and the discretization of the full models of the dynamical systems of interest are required. The intrusive nature limits the scope of traditional model reduction methods. For example, consider proprietary software that implements the simulation of a physical process. Details, and access to, the governing equations, discretization, and solver typically are unavailable when working with proprietary software and thus traditional, intrusive model reduction methods are not applicable.
In contrast, non-intrusive model reduction techniques aim to learn reduced models from snapshots, i.e., either numerical approximations or measurements of the states and the outputs of the dynamical systems.
There is a variety of non-intrusive model reduction techniques for linear systems, yet methods for learning reduced-order nonlinear systems---which is the focus of this paper---are more scarce.

For linear systems, the Loewner framework constructs reduced models from input-output measurements by fitting a linear-time invariant model to frequency or time-domain data~\cite{morMayA07,antoulas14parametricLoewner,PSW16TLoewner,schulze2018data}. System identification techniques such as the matrix pencil approach~\cite{antoulas12pencilSYSID}, eigensystem realization algorithm~\cite{kung78new,rowley_ma11BTera,KG16TangentialInterpolationERA,KG18ERA_CUR_Hankel}, and vector fitting~\cite{gustavsen99VF,drmac15VF} can accurately approximate dynamical systems and in some cases even retain structure. For nonlinear systems, non-intrusive model reduction generally requires a choice of parametrization of the nonlinear term. For example, the Loewner approach has been extended from linear to bilinear \cite{doi:10.1137/15M1041432} and quadratic-bilinear systems \cite{gosea2018LoewnerQB}. Techniques based on libraries select the nonlinear functions that compose the right-hand-side of a dynamical system from candidate functions that are chosen either by expert knowledge~\cite{brunton2016discovering} or through sparse approximation techniques~\cite{rudy2017data,schaeffer2018extracting}. Dynamic mode decomposition~\cite{rowley09spectralDMD,schmid2010dynamic} learns reduced models for nonlinear dynamical systems, yet its success depends on knowledge of suitable observables that make the input-output map low-dimensional~\cite{williams2015data}.
A promising approach to consider DMD modes together with the fluctuation-dissipation theorem allows the authors in~\cite{khodkar_hassanzadeh_2018} to learn a linear ROM for nonlinear turbulent flows.
There are also hybrid approaches to learn only part of the nonlinear system, e.g., in data-driven closure modeling, where a projection-based framework is pursued, but correction terms are learned from data~\cite{xie2018data}, or the Galerkin-ROMs can be re-calibrated \cite{couplet2005calibration,cordier2010calibration} to achieve greater accuracy.
Another approach to learning nonlinear reduced models is operator inference~\cite{Peherstorfer16DataDriven} that is applicable if the nonlinear terms are polynomials in the state.  The operator inference framework has been extended to general nonlinear systems by utilizing variable transformations~\cite{QKMW_2019_Transform_and_Learn}.

In practice, one is often confronted with a mixture of the intrusive and non-intrusive setting, such as when the governing equations of the system of interest are known yet details about the discretization are unavailable. For example, if one considers a reaction-diffusion process (such as the tubular reactor example in \Cref{sec:numerics_tubular}), then the reaction term might be given analytically, whereas the discretization of the terms corresponding to the diffusion are unavailable.
We thus propose a novel non-intrusive model reduction method that incorporates terms that are available analytically and that learns reduced operators for all other terms from snapshot data.
To this end, we build on the operator inference method and extend it in multiple directions. First, we explicitly incorporate  knowledge about non-polynomial terms that are given analytically. Second, we combine operator inference with empirical interpolation \cite{morBarMNetal04,grepl2007efficient,deim2010} to make the learned models online efficient in case of non-polynomial nonlinear terms.
Third, we demonstrate that structure in coupled (partial) differential equations can be retained in the learned reduced models, which increases the predictive capabilities of the reduced models.
We show that under certain conditions the learned model converges to the same model that is derived with intrusive model reduction methods that require full knowledge of the governing equations and the full-model solvers. Numerical results demonstrate that our approach learns predictive reduced models for  diffusion-reaction systems and simulations of chemical processes.

The manuscript is structured as follows. \Cref{sec:problemformulation} describes the problem setting and presents background material on model reduction. \Cref{sec:use_nonlin} presents the proposed operator inference problem for nonlinear non-polynomial systems. \Cref{sec:numerics} presents numerical results that provide evidence of the wide scope of our approach. Conclusions are drawn in \Cref{sec:conclusion}.

\section{Problem setting and preliminaries} \label{sec:problemformulation}
In \Cref{sec:PDE}, we introduce the partial differential equation (PDE) model considered herein, followed by its discretized version in \Cref{sec:ODE}. In \Cref{sec:traditional}, we describe the standard projection-based model reduction technique and in \Cref{sec:problemsetting} we state the problem setting of this paper.

\subsection{Partial differential equation model} \label{sec:PDE}
Let $0 < T \in \mathbb{R}$ and consider a PDE of the form
\begin{equation}\label{eq:PDE}
	\frac{\partial \cstate}{\partial t } = \cA(\cstate) + \cH(\cstate) + f(t,\cstate) + \cB(u)
\end{equation}
for time $t \in (0, T]$, input $u(t)$, state $\cstate(x,t)$ with $x \in \Omega \subseteq \R^d, \ d=1,2,3$, and $\Omega$ a bounded open domain with boundary $\partial \Omega$. We assume boundary conditions $\cstate_{| \partial \Omega}$ and initial conditions $\cstate(x,0)$ are given such that the corresponding initial boundary value problem with governing PDE~\eqref{eq:PDE} is well posed up to time $T$.
In equation~\eqref{eq:PDE}, $\cA$ is a linear operator (satisfying $\cA(a\cstate) = a\cA(\cstate)$ for all $a\in \R$ and $\cA (\cstate_1 + \cstate_2) = \cA(\cstate_1) + \cA(\cstate_2)$, e.g., a difference or integral operator), the operator $\mathcal{H}$ is quadratic in $\cstate$ (satisfying $\cH(a\cstate) = a^2 \cH(\cstate)$ for all $\cstate$, e.g., $\mathcal{H}(\cstate) = \cstate \cdot \nabla \cstate$), the nonlinear function is $f(t,\cstate)$, and $\mathcal{B}$ is a linear input operator.

\subsection{Discretized model} \label{sec:ODE}
A semi-discrete numerical model for the PDE~\eqref{eq:PDE} is given by
\begin{align}\label{eq:FOM}
	\dot \dstate(t)& = \bA\dstate(t) + \bH(\dstate(t)\otimes \dstate(t)) + \boldf(t,\dstate(t)) +  \bB\bu(t),
\end{align}
with finite-dimensional state $\dstate(t)\in \R^n$, input $\bu(t)\in \R^m$, linear operator  $\bA\in \R^{n\times n}$, quadratic operator $\bH \in \R^{n\times n^2}$, input operator $\bB \in \R^{n \times m}$.
The initial condition for the semi-discrete model~\eqref{eq:FOM} is $\dstate(0) = \dstate_0$ and the input at time $t = 0$ is $\bu(0)=\bu_0$.  Throughout this paper, we call \eqref{eq:FOM} the full-order model (FOM).
Here, $\otimes$ denotes the column-wise Kronecker product, which for a column vector $\dstate = [\cstate_1, \cstate_2, \ldots, \cstate_n]^\top$ is given by
$$
\dstate \otimes \dstate = [\cstate_1^2 \ \ \cstate_1 \cstate_2 \ \ldots \cstate_1 \cstate_n \ \ \cstate_2 \cstate_1 \   \ \cstate_2^2 \ \ldots \ \cstate_2\cstate_m \ \ldots \cstate_m^2] \in \R^{n^2},
$$
and for a matrix $\Mstate = [\dstate_1, \dstate_2, \ldots, \dstate_k] \in \R^{n\times k}$ is given by
$$
\Mstate \otimes \Mstate = [\dstate_1 \otimes \dstate_1 \ \ \dstate_2 \otimes \dstate_2  \ \ \ldots \ \ \dstate_k \otimes \dstate_k ] \in \R^{n^2 \times k}.
$$

In the following, we assume that each component function $f_1, \dots, f_n: [0, T] \times \mathbb{R} \to \mathbb{R}$ of $\boldf$ requires evaluating $f$ from the continuous model \eqref{eq:PDE} at a single component of the state vector $\dstate(t) = [\cstate_1(t), \dots, \cstate_n(t)]^\top$. Formally, this means that $\boldf$ is
\begin{equation} \label{eq:def_f}
	\boldf(t, \dstate) = \begin{bmatrix}
		f(t, \cstate_1)\\
		\vdots\\
		f(t, \cstate_n)
	\end{bmatrix}\,.
\end{equation}
We introduced the PDE model~\eqref{eq:PDE} and the definition of $\boldf$ in~\eqref{eq:def_f} for a scalar model (describing the evolution of one physical quantity, e.g., velocity) for ease of notation.  For PDEs with $\ell$ physical variables, that means that each component function in~\eqref{eq:def_f} can depend on other physical variables at the same spatial location.
Next, we give a concrete example of a PDE and its discretization with non-polynomial nonlinear terms as in~\eqref{eq:def_f}.

\begin{example}
	Consider the PDE
	$$
	\frac{\partial}{\partial t} \cstate(x,t) = \frac{\partial^2}{\partial x^2} \cstate(x,t) + e^{-\beta t}  \cstate(x,t)^{-\alpha} + b(x) u(t)
	$$
	with state $\cstate(x,t)$, diffusion operator $\frac{\partial}{\partial x^2}$, time-dependent reaction term $f(t,\cstate) = e^{-\beta t}  \cstate^{-\alpha}$, one-dimensional input $u(t)$, input function $b(x)$ and constants $\alpha, \beta \in \R_+$. Let $\dstate$ denote the discretized, finite-dimensional state vector obtained through a finite difference approximation. A discretization of the diffusion operator results in a matrix representation $\bA$, whose structure depends on the discretization scheme, approximation order, and boundary conditions. Likewise, the discretized version of $b(x)$ is $\bB$. For a vector $\dstate$, we define $\dstate^{-\alpha}:= [s_i]^{-\alpha}, \ i=1,2, \ldots, n$ as the componentwise power of the vector entries. The discretized system reads as
	$$
	\dot{\dstate}(t) = \bA \dstate(t) + e^{-\beta t} \dstate(t)^{-\alpha} + \bB u(t).
	$$
	Note that the nonlinear function $\boldf(t,\dstate(t)) = e^{-\beta t}  \dstate(t)^{-\alpha}$ does not require approximation of spatial derivatives.  In particular, evaluating the semi-discrete nonlinear function $\boldf(t,\dstate)$ only requires application of $f(t,\cstate)$ at every component of $\dstate$.
\end{example}

In this work, we consider functions $f(t,\cstate)$ that are continuous in $\cstate$ and that are \textit{spatially local}, which means they involve nonlinear expressions in the state $\cstate$; the class of problems not considered in this paper are functions $f$ that include differentials or integrals of $\cstate$.
Other examples of spatially local nonlinear terms are the Arrhenius reaction model $\exp{(\gamma - \frac{\gamma}{\cstate} )}$ in the tubular reactor model in \Cref{sec:numerics_tubular} and the function $\dfrac{\cstate}{\alpha+\cstate}$ in the Batch Chromatography example in \Cref{sec:numerics_batchChrom}; see also \cite{morBarMNetal04,deim2010}.

\subsection{Traditional projection-based model reduction for dynamical systems} \label{sec:traditional}
In projection-based model reduction, the semi-discrete model~\eqref{eq:FOM} is projected onto a low-dimensional subspace. Let $r \ll n$ and let $\bV = [\bv_1, \dots, \bv_{r}] \in \mathbb{R}^{n \times r}$ be an orthonormal matrix, where the columns of $\bV$ span an $r$-dimensional subspace of $\mathbb{R}^{n}$. A basis matrix $\bV$ of such a subspace can be computed with, e.g., proper orthogonal decomposition (POD)~\cite{lumley1967structure,sirovich87turbulence,holmes_lumley_berkooz_1996}.  Let $\Mstate = [\dstate_1, \ldots, \dstate_k]$ be snapshot matrix containing solution snapshots $\dstate_i \approx \dstate(t_i)$, where the $\dstate_i$ are solutions to~\eqref{eq:FOM} computed with a time-stepping scheme. POD computes the basis matrix $\bV$ from $\Mstate$ via singular value decomposition (SVD).

Consider now the reduced operators
\begin{equation}
	\widetilde{\bA} = \bV^\top\bA\bV \in \R^{r \times r}\,,\quad \check{\bH} = \bV^\top \bH (\bV \otimes' \bV) \in \R^{r \times r^2}\,,\quad \widetilde{\bB} = \bV^\top\bB \in \R^{r \times m}\, ,
	\label{eq:Prelim:RedOperators}
\end{equation}
which are the restrictions of the matrices of the FOM from~\eqref{eq:FOM} to the image of $\bV$.
To remove redundant terms in the original Kronecker product, we use throughout the manuscript the product $\dstate \otimes' \dstate$ that contains as components $\dstate_i \dstate_j, i,j = 1,2, \ldots, n, j\geq i$, which is the Kronecker product yet the duplicate terms are removed. For instance, for $\dstate = [\dstate_1, \dstate_2]^\top$, the standard Kronecker product yields $\dstate\otimes\dstate = [\dstate_1^2 \ \dstate_1 \dstate_2 \ \dstate_2 \dstate_1 \ \dstate_2^2]^\top $ and without redundant terms we have $\dstate\otimes'\dstate = [\dstate_1^2 \ \dstate_1 \dstate_2 \ \dstate_2^2]^\top$.  To this end, we define the matrix $\widetilde{\bH} \in \R^{r\times (r^2+r)/2}$ (without redundant terms) as
$$
\widetilde{\bH} = \check{\bH}_{:,r(i-1) + j}, \qquad i=1, \ldots r, \  i \leq j \leq r,
$$
where $\check{\bH}_{:,j}$ denotes $j$th column of the matrix $\check{\bH}$.
Moreover, we define the function $\widetilde{\boldf}: [0, T] \times \R^r \to \R^r$ as
\[
\widetilde{\boldf}(t, \widetilde{\dstate}(t)) = \bV^\top\boldf(t, \bV\widetilde{\dstate}(t))\,,
\]
where $\widetilde\dstate \in \mathbb{R}^{r}$. Note that there are efficient reduction strategies that approximately compute $\widetilde{\boldf}$ so that $\boldf$ does not have to be evaluated at all $n$ components of $\bV\widetilde{\dstate}$ \cite{astrid2008missing,morBarMNetal04,carlberg2013gnat,deim2010,nguyen2008best}; we will return to such reduction strategies of $\widetilde{\boldf}$ in \Cref{sec:DEIM}. The reduced operators and $\hat{\boldf}$ define the reduced-order model (ROM)
\begin{align}\label{eq:ROM}
	\dot{\widetilde{\dstate}}(t)& = \widetilde \bA\widetilde \dstate(t) + \widetilde \bH(\widetilde \dstate(t) \otimes' \widetilde \dstate(t)) + \widetilde\boldf(t,\widetilde\dstate(t)) +  \widetilde \bB\bu(t)
\end{align}
with the reduced state $\widetilde{\dstate}(t) \in \mathbb{R}^r$ at time $t$ and with initial condition $\widetilde \dstate(0) = \widetilde\dstate_0$.
Constructing reduced operators \eqref{eq:Prelim:RedOperators} via projection requires that the operators $\bA, \bB, \bH$ of the system \eqref{eq:FOM} are available in assembled form or as matrix-vector products. Furthermore, each evaluation of $\widetilde{\boldf}$ requires evaluating components of $\boldf$.

\subsection{Problem setting} \label{sec:problemsetting}
In this work, we consider the situation that we have a PDE model~\eqref{eq:PDE} with spatially local nonlinear term $f$ as described in \Cref{sec:PDE}, and that we can obtain snapshot data $\dstate(t_i),\bu(t_i)$ of the model~\eqref{eq:FOM} at times $0=t_0< t_1 <\cdots < t_k$.
However, the discretized operators $\bA, \bH, \bB$ are unavailable.
The goal of this work is to learn a nonlinear ROM of the form~\eqref{eq:ROM}.

\section{Operator inference for nonlinear systems} \label{sec:use_nonlin}
In \Cref{sec:OpInf_knownF}, we extend the operator inference framework for low-order polynomial systems from~\cite{Peherstorfer16DataDriven} to the setting described in \Cref{sec:PDE} with general non-polynomial nonlinear function $f$. In \Cref{sec:OPINF_analysis}, we present a result that shows that under certain conditions, the learned operators converge to their projection-based counterparts. In \Cref{sec:DEIM}, we discuss empirical interpolation \cite{morBarMNetal04,deim2010} to efficiently reduce the non-polynomial nonlinear terms in the learned ROMs.

\subsection{Operator inference method} \label{sec:OpInf_knownF}
We now extend the operator inference method introduced in~\cite{Peherstorfer16DataDriven} for polynomially nonlinear systems to systems with non-polynomial nonlinear terms that are spatially local. Let $\dstate_1, \ldots, \dstate_k$ be the solutions of the FOM~\eqref{eq:FOM} at time steps $t_1, \ldots, t_k$ computed with a time stepping scheme and initial condition $\dstate_0$.  Let the inputs $\bu(t_0), \ldots, \bu(t_k)$ be sampled at the same time steps. Then, the snapshot and input trajectory are defined as
\begin{equation} \label{eq:Xdot_X}
	{\Mstate}  : =
	\left[
	\begin{array}{llll}
		~ \rule[-1ex]{0.5pt}{2.5ex} & ~\rule[-1ex]{0.5pt}{2.5ex} &        & ~\rule[-1ex]{0.5pt}{2.5ex} \\
		\dstate_0    & \dstate_1    & \cdots & \dstate_k   \\
		~\rule[-1ex]{0.5pt}{2.5ex} & ~\rule[-1ex]{0.5pt}{2.5ex} &        & ~\rule[-1ex]{0.5pt}{2.5ex}
	\end{array}
	\right], \qquad
	{\bU}  : =
	\left[
	\begin{array}{llll}
		~ \rule[-1ex]{0.5pt}{2.5ex} & ~\rule[-1ex]{0.5pt}{2.5ex} &        & ~\rule[-1ex]{0.5pt}{2.5ex} \\
		\bu({t_0})    & \bu({t_1})    & \cdots & \bu({t_k})    \\
		~\rule[-1ex]{0.5pt}{2.5ex} & ~\rule[-1ex]{0.5pt}{2.5ex} &        & ~\rule[-1ex]{0.5pt}{2.5ex}
	\end{array}
	\right].
\end{equation}
Note that we can evaluate $\boldf(t_i,\dstate(t_i))$ using the solution $\dstate(t)$ and the analytical form of the spatially local nonlinear term $f$. That is, we can post-process the snapshots to obtain
\begin{equation}\label{eq:F}
	{\bF}  =
	\left[
	\begin{array}{cccc}
		\rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} &        & \rule[-1ex]{0.5pt}{2.5ex} \\
		\boldf(t_0,\dstate(t_0))    & \boldf(t_1,\dstate(t_1))    & \cdots & \boldf(t_k,\dstate(t_k))    \\
		\rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} &        & \rule[-1ex]{0.5pt}{2.5ex}
	\end{array}
	\right].
\end{equation}
In the operator inference framework, the goal is to learn a ROM of the form~\eqref{eq:ROM} directly from data. One starts with the projected  trajectories of $\dstate(t)$, $\dot \dstate(t)$ and $\boldf(t,\dstate)$, which are obtained via the following projections onto the basis matrix $\bV\in \R^{n\times r}$:
\begin{equation} \label{eq:Shat}
	\hat{\Mstate} = \bV^\top{\Mstate},\qquad
	{\hat{\bF}} = \bV^\top {\bF}.
\end{equation}
We denote with $\dot{\widehat{\dstate}}_k$ the time derivative approximation of $\frac{\text{d}}{\text{d}t} \hat{\dstate}(t_k)$, which can be computed from $\hat{\dstate}$ using a time derivative approximation (see, e.g.,~\cite{martins2013review,knowles2014methodsDifferentiation,chartrand2017numericalDifferentiation}). We store the time-derivative approximations in the matrix
\begin{equation} \label{Xdot}
	\dot{\widehat{\Mstate} } : =
	\left[
	\begin{array}{llll}
		~ \rule[-1ex]{0.5pt}{2.5ex} & ~\rule[-1ex]{0.5pt}{2.5ex} &        & ~\rule[-1ex]{0.5pt}{2.5ex} \\
		\dot{\widehat{\dstate}}({t_0})    & \dot{\widehat{\dstate}}({t_1})    & \cdots & \dot{\widehat{\dstate}}({t_k})    \\
		~\rule[-1ex]{0.5pt}{2.5ex} & ~\rule[-1ex]{0.5pt}{2.5ex} &        & ~\rule[-1ex]{0.5pt}{2.5ex}
	\end{array}
	\right].
\end{equation}
The optimization problem to compute $\hat \bA$, $\hat \bB$ and $\hat \bH$ from the above projected data is:
\begin{equation} \label{eqn:optimizationProb}
	\min_{\hat \bA,\hat \bB,\hat \bH} \  \| \underbrace{\dot{\hat{\Mstate}} - \hat{\bF}}_{:=\hat{\bR}}- \hat \bA \hat{\Mstate} - \hat \bB \bU -\hat \bH (\hat{\Mstate}\otimes' \hat{\Mstate})\|_F
\end{equation}
where $\hat{\bR}$ contains only known terms.
Taken together, the solution of the operator inference problem for $\hat \bA,\hat \bB,\hat \bH$ yields a learned ROM of the form
\begin{align}  \label{eq:ROM_OPINF}
	\dot{\hat{\dstate}}(t)& = \hat \bA\hat \dstate(t) + \hat \bH(\hat \dstate (t )\otimes' \hat \dstate (t)) + \bV^\top \boldf(t,\bV\hat \dstate(t)) +  \hat \bB\bu(t),
\end{align}
which can then be used for predictive simulations. \Cref{algo:OpInfNL} summarizes the algorithmic steps of this section.
\begin{algorithm}[!tb]
	\caption{Operator inference for nonlinear systems.}
	\begin{algorithmic}[1]
		\State {\bf Input:} $\dstate_0,\Mstate,\bU,\bF$ and a user-specified tolerance $\texttt{tol}$.
		\State Collect derivative data $\dot{\Mstate}$ by numerical approximation or evaluation of a right-hand side residual on the snapshot matrix $\Mstate$.
		\State Compute the $r$  dominant POD basis vectors of $\Mstate$, resulting in $\bV$ such that
		$$\dfrac{\|\Mstate - \bV\bV^\top\Mstate\|}{\|\Mstate\|} \leq \texttt{tol}.$$
		\State Determine projected initial condition $\hat{\dstate}_0$, reduced states $\hat{\Mstate}$, reduced derivative $\dot{\hat{\Mstate}}$, and reduced nonlinear terms $\hat{\bF}$:
		$$\hat{\dstate}_0 := \bV^\top\dstate_0, \quad\hat{\Mstate} := \bV^\top\Mstate,\quad \dot{\hat{\Mstate}} = \bV^\top\dot{{\Mstate}},\quad  \hat{\bF} := \bV^\top\bF.$$

		\State Solve the optimization problem \eqref{eqn:optimizationProb}, yielding $\hat{\bA}$, $\hat{\bB}$, $\hat{\bH}_\text{sq}$.

		\State  {\bf Output:} $\hat{\dstate}_0$, $\hat{\bA}$, $\hat{\bB}$, $\hat{\bH}$.
	\end{algorithmic}\label{algo:OpInfNL}
\end{algorithm}
In \Cref{fig:compare_PODDEIM_NonIntru}, we illustrate the differences of the  proposed approach of using operator inference together with knowledge of spatially nonlinear functions to learn ROMs as compared to an intrusive projection-based ROM.
\begin{figure}[!tb]
	\centering
	\inputtikz{FlowChart}
	\caption{A comparison of a fully intrusive ROM approach with the proposed learning approach.}
	\label{fig:compare_PODDEIM_NonIntru}
\end{figure}

\begin{remark}
	We introduced the operator inference framework in the context of continuous-time nonlinear systems. However, the framework carries over directly to discrete time systems of the form
	\begin{equation}
		\dstate_{k+1} = \bA \dstate_k + \bH (\dstate_k \otimes' \dstate_k) + \boldf(t_k, \dstate_k) + \bB \bu_k.
	\end{equation}
	In this case, we replace $\Mstate$ with $\Mstate_0 = [\dstate_0, \dstate_1, \ldots, \dstate_{k-1}]$, set $\bU = [\bu_0, \bu_1, \ldots, \bu_{k-1}]$, and replace the time-derivative matrix $\dot{\Mstate}$ with $\Mstate_1 = [\dstate_1, \dstate_2, \ldots, \dstate_k]$.
\end{remark}

\subsection{Analysis of operator inference framework} \label{sec:OPINF_analysis}
We extend the results from \cite[Sec~3.2]{Peherstorfer16DataDriven} to show that under certain conditions on the time discretization, the learned operators converge to the intrusively obtained reduced operators. For this, we make two assumptions on the time discretization of the FOM and ROM.

\begin{assumption} \label{ass1}
	The time stepping scheme for the FOM~\eqref{eq:FOM} is convergent, i.e.,
	$$
	\max_{i \in \{ 1,\ldots T/\Delta t \}} \Vert \dstate_i - \dstate(t_i) \Vert_2 \rightarrow 0 \quad \text{as} \quad \Delta t\rightarrow 0.
	$$
\end{assumption}

\begin{assumption} \label{ass2}
	The derivatives approximated from projected states, $\dot{\hat{\dstate}}_k$, converge to $\frac{\text{d}}{\text{d}t} \hat{\dstate}(t_k)$ as the discretization time step $\Delta t \rightarrow 0$, i.e.,
	$$
	\max_{i \in \{ 1,\ldots T/\Delta t \}} \Vert \dot{\hat{\dstate}}_i - \frac{\text{d}}{\text{d}t} \hat{\dstate}(t_i) \Vert_2 \rightarrow 0 \quad \text{as} \quad \Delta t\rightarrow 0.
	$$
\end{assumption}
Assumption~\ref{ass2} relates to the specific derivative approximation scheme that is used, its approximation order and the time-step size. In this work, we employ a fourth-order time-stepping scheme, see \Cref{sec:numerics} for details. Other choices of time discretizations are possible, see e.g.,~\cite{martins2013review,knowles2014methodsDifferentiation,chartrand2017numericalDifferentiation} and the references therein.

\begin{theorem}
	Let Assumptions~\ref{ass1}--\ref{ass2} hold and let a basis matrix $\bV = [ \bv_1, \bv_2, \ldots, \bv_r] \in \R^{n\times r}$ be given. Let $\widetilde{\bA}, \widetilde{\bB}, \widetilde{\bH}$ be the intrusively projected ROM operators from \eqref{eq:Prelim:RedOperators}. If the data matrix $\hat{\bD} = [\hat{\Mstate}, \ \bU, \ \hat{\Mstate} \otimes' \hat{\Mstate}]$ has full column rank, then for every $\varepsilon>0$, there exists $r\leq n$ and a time step size $\Delta t>0$ such that for the difference between the learned operators $\hat{\bA}, \hat{\bH}, \hat{\bB}$ and the (intrusive) projection-based $\widetilde{\bA}, \widetilde{\bH}, \widetilde{\bB}$, we have

	\begin{equation*}
		\Vert \hat{\bA} - \widetilde{\bA} \Vert_F \leq \varepsilon, \quad \Vert \hat{\bB} - \widetilde{\bB} \Vert_F \leq \varepsilon, \qquad \Vert \hat{\bH} - \widetilde{\bH} \Vert_F \leq \varepsilon.
	\end{equation*}
\end{theorem}

\begin{proof}
	The proof follows the ideas of the proof of \cite[Theorem~1]{Peherstorfer16DataDriven}. With the data matrices from \eqref{eq:Shat}--\eqref{Xdot}, we set $\hat{\bD} = [\widehat{\Mstate}, \ \bU, \ \widehat{\Mstate} \otimes' \widehat{\Mstate}]$ as the data matrix, $\widehat{\bR} = \dot{\widehat{\bS}} - \widehat{\bF}$ as the residual, and $\hat{\bO} = [ \widehat{\bA}, \widehat{\bB}, \widehat{\bH}]$ are the operators to be inferred, so the operator inference problem~\eqref{eqn:optimizationProb} can be formulated as
	\begin{equation} \label{eqn:optimizationProbDO}
		\min_{\hat \bO} \  \| \widehat{\bD} \widehat{\bO}^\top - \widehat{\bR}^\top \|_F.
	\end{equation}
	%
	Denote by $\widetilde{\Mstate} = [ \widetilde{\dstate}_1,\widetilde{\dstate}_2, \ldots, \widetilde{\dstate}_k  ]$ the data matrix that is collected from taking $k$ snapshots of the intrusive ROM~\eqref{eq:ROM}, and let $\widetilde{\bD} = [\widetilde{\Mstate}, \ \bU, \ \widetilde{\Mstate} \otimes' \widetilde{\Mstate}]$ and $\widetilde{\bR} = \dot{\widetilde{\bS}} - \widetilde{\bF}$. With $\widetilde{\bD},\widetilde{\bR}$ as data, the projected operators $\widetilde{\bA}, \widetilde{\bB}, \widetilde{\bH}$ are a solution of the minimization problem~\eqref{eqn:optimizationProbDO}, and if $\widetilde{\bD}$ has full column rank, then the solution is also unique.
	%
	Next, we observe that $\widehat{\bS} = \widetilde{\bS} + \Delta\widetilde{\bS}$, i.e., the projected FOM data can be interpreted as a perturbation of the ROM-simulated data, and as the dimension $r$ increases, we have $\Vert \Delta \widetilde{\bS}  \Vert_F \rightarrow 0$ because of Assumption~\ref{ass1}, and the fact that for the limiting case $r=n$ the FOM~\eqref{eq:FOM} and ROM~\eqref{eq:ROM} are equivalent up to a change of coordinates.
	%
	Since $\widehat{\bS} \rightarrow \widetilde{\bS}$ we also have by the continuity of $\boldf_r$ that $\widetilde{\bF} \rightarrow \widehat{\bF}$.
	%
	Moreover, Assumption~\ref{ass2} ensures convergence of $\dot{\hat{\dstate}}_k \rightarrow \frac{\text{d}}{\text{d}t} \widehat{\dstate}(t_k)$ and using again that for the limiting case $r=n$ the FOM~\eqref{eq:FOM} and ROM~\eqref{eq:ROM} are equivalent up to a change of coordinates, we have that $\frac{\text{d}}{\text{d}t} \widehat{\dstate}(t_k) \rightarrow \frac{\text{d}}{\text{d}t} \widetilde{\dstate}(t_k)$ as $r\rightarrow n$ so $\Vert \Delta \widetilde{\bR} \Vert_F \rightarrow 0$.  Taken together we have $\Vert \widetilde{\bR} + \Delta \widetilde{\bR} \Vert_F \rightarrow \Vert \widetilde{\bR} \Vert_F$.
	%
	Therefore, we have as a limiting result that
	\begin{equation}
		\min_{\widehat \bO} \left [ \lim_{\substack{r\rightarrow n  \\ \Delta t \rightarrow 0}}    \  \| \widehat{\bD} \widehat{\bO}^\top - \widehat{\bR}^\top \|_F \right ]
		=
		\min_{\widehat \bO} \left [ \lim_{\substack{r\rightarrow n  \\ \Delta t \rightarrow 0}}    \  \| [\widetilde{\bD} + \Delta \widetilde{\bD} ]  \widehat{\bO}^\top - [\widetilde{\bR} + \Delta \widetilde{\bR}]^\top \|_F \right ]
		=
		\min_{\widehat \bO}  \| \widetilde{\bD} \widehat{\bO}^\top - \widetilde{\bR}^\top \|_F,
	\end{equation}
	so the learned operators $\widehat{\bO}$ converge to the intrusively projected operators $\widetilde{\bO}$. In the pre-asymptotic case, we then get the stated result in the theorem, which also uses the full-rank assumption on $\widehat{\bD}$ to deduce $\Vert \hat{\bA} - \widetilde{\bA} \Vert_F \leq \varepsilon, \ \Vert \hat{\bB} - \widetilde{\bB} \Vert_F \leq \varepsilon, \ \Vert \hat{\bH} - \widetilde{\bH} \Vert_F \leq \varepsilon$.
\end{proof}

The theorem shows that the learned ROM operators converge to the projected ROM operators as $r\rightarrow n$, in the coordinate system spanned by the columns of $\bV$ as $\widehat{\bA} \rightarrow \bV^\top \bA \bV$. However, we do not obtain a convergence rate, and in our practical experience there are examples where the difference in operators might not monotonically decrease for low orders of the ROM, such as the tubular reactor example in Section~\ref{sec:numerics_tubular}. Even for $r=50$ the relative error in the inferred linear system operators is $1.66\times 10^{-2}$ for the tubular reactor example. For the other examples in Sections~\ref{sec:numerics_Chaffee} and \ref{sec:numerics_batchChrom}, the norm did decrease for low orders of the ROM. Most importantly, however, we see in the numerical results in Section~\ref{sec:numerics} that in all cases we can obtain accurate learned ROMs with convergent state-errors for $r\ll n$ via the operator inference framework.

The cost of the proposed framework of operator inference for nonlinear systems is similar to a standard projection-based POD framework when $k<n$, i.e., we have more states than snapshots. Both the projection-based ROM and the learned ROM require computing the POD basis from the snapshot set as well as time integration of $r$-dimensional ROMs. The methods differ in that the operator inference requires projecting the data $\bS, \dot{\bS}$ onto the subspace $\bV$ (see step~4 of Algorithm~\ref{algo:OpInfNL}), forming $\bS\otimes'\bS$ and then solving a least-squares problem in reduced dimension. The cost scales as $\mathcal{O}(rn)$. In contrast, the projection-based ROM requires projecting the model matrices onto the subspace $\bV$, see \eqref{eq:Prelim:RedOperators}, which also scales as $\mathcal{O}(rn)$. Note, that in that last step we need to form $\bV^\top \bH (\bV\otimes' \bV)$ which is potentially expensive.

\subsection{Empirical interpolation} \label{sec:DEIM}
In this section, we discuss empirical interpolation \cite{morBarMNetal04,deim2010} to accelerate the evaluation of $\bV^\top \boldf (t, \bV \hat{\dstate})$ in the learned ROM. We employ the discrete empirical interpolation method (DEIM) from \cite{deim2010}. To this end, we approximate
\begin{equation} \label{eq:DEIMapprox}
	\hat{\boldf}(t,\bV \hat{\dstate}(t)) \approx \hat{\boldf}_r (t,\bV \hat{\dstate}(t)) = \bV^\top \bW (\mathbb{S}^\top \bW)^{-1} \mathbb{S}^\top \boldf(t,\bV \hat{\dstate}(t)).
\end{equation}
The matrix $\bW$ is computed by taking the SVD of the nonlinear snapshot matrix $\bF$ in equation~\eqref{eq:F}, and setting $\bW$ to the leading $m$ left singular vectors of $\bF$. Here, $\mathbb{S}$ is an $n\times m$ matrix obtained by selecting certain columns of the $n\times n$ identity matrix. The choice of the selected entries in $\mathbb{S}$ follows \Cref{algo:DEIM}, which enforces interpolation $\mathbb{S}^\top \boldf(t,\cdot)  = \mathbb{S}^\top [\bW (\mathbb{S}^\top \bW)^{-1} \mathbb{S}^\top \boldf(t,\cdot) ]$ and simultaneously limits the local growth of the spectral norm $\Vert(\mathbb{S}^\top \bW)^{-1}\Vert_2$, as this term occurs in the DEIM error bound \cite{deim2010}.
\begin{algorithm}[!tb]
	\caption{Discrete empirical interpolation method \cite[Algorithm 1]{deim2010}}
	\begin{algorithmic}[1]
		\State {\bf Input:} $\bW = [\bw_1, \ldots, \bw_m]$.
		\State $p_1 = \arg \max_i( | \bw_1(i) | ); \ \bW_1 = [\bw_1]; \ \mathbb{S}_1 = [e_{p_1}]; \ \rho_1= [p_1]$
		\For{$j=2:m$}
		\State Solve $\mathbb{S}^\top_{j-1} \bW_{j-1} z = \mathbb{S}^\top _{j-1} \bw_j$ for $z$;
		\State  $r_j = \bw_j - \bW_{j-1} z;  \ p_j = \arg \max_i ( | r_j (i)| );$
		\State $U_j = [\bW_{j-1}, \bw_{j}]; \ \mathbb{S}_j = [\mathbb{S}_{j-1}, e_{p_j}]; \ \rho_j = (\rho_{j-1}, p_j)$;
		\EndFor
		\State  {\bf Output:}  Selection operator $\mathbb{S} = \mathbb{S}_m$
	\end{algorithmic}\label{algo:DEIM}
\end{algorithm}
The learned ROM with DEIM interpolation is written as
\begin{align}\label{eq:ROMDEIM}
	\dot{\hat{\dstate}}(t)& = \hat \bA\hat \dstate(t) + \hat \bH(\hat \dstate(t) \otimes' \hat \dstate(t)) + \hat \boldf_r(t,\hat \dstate(t)) +  \hat \bB\bu(t),
\end{align}
where only $m\ll n$ entries of $\boldf$ are evaluated, thus achieving computational speedup.

\section{Numerical results} \label{sec:numerics}
In this section, we test the efficiency of the proposed operator inference approach by means of several problems, arising in different areas of science and engineering. We compare the quality of the learned ROM with an intrusive ROM. Here, we use a projection-based POD method. Below we give some information that applies to all our numerical simulations:
\begin{itemize}
\item All models are simulated using the routine \texttt{ode15s} in \matlab ~with relative error and absolute error tolerances of $10^{-10}$.
\item In the operator inference framework, we approximate the time derivative $\dot{\widehat{\dstate}}_i$ with a five-point stencil $\dot{\widehat{\dstate}}_i \approx (-{\widehat{\dstate}}_{i+2} + 8 {\widehat{\dstate}}_{i+1} - 8{\widehat{\dstate}}_{i-1} +{\widehat{\dstate}}_{i-2} )/(12\Delta t)$, which has fourth-order accuracy. The first two and last two time derivatives are computed using first-order forward and backward Euler approximations, respectively.
\end{itemize}

We present three examples with increasing complexity to illustrate our approach.

\subsection{Chafee-Infante model} \label{sec:numerics_Chaffee}
We consider the one-dimensional Chafee-Infante equation \cite{chafee1974bifurcation}, which is closely related to the Cahn-Hillard equation \cite{chen2004generation}.  The Chafee-Infante model has been studied extensively as a benchmark problem for nonlinear MOR techniques, and has been reduced by means of various methods, see, e.g.~\cite{morBenB15,morBenGG18}.
The Chafee-Infante equation is a  diffusion-reaction model with the  governing equation, boundary and initial conditions as follows:
\begin{equation}\label{eq:ChafeeGovEq}
	\begin{aligned}
		\dfrac{dv}{dt} + v^3 &= \frac{\partial^2 v}{\partial x^2} +  u, \quad  &&(x,t) \in (0,1)\times (0,T], &\quad v(0,t) &= u(t), \quad &&t \in (0,T),\\
		\frac{\partial v}{\partial x}(1,t) &= 0, \quad  &&t \in (0,T), &\quad v(x,0) &= 0, \quad &&x \in (0,1).
	\end{aligned}
\end{equation}
As a quantity of interest, we observe the output at the right boundary, i.e., $y(t) = v(L,t)$. The model~\eqref{eq:ChafeeGovEq} contains a cubic nonlinear term, which fits our discussion of spatially local nonlinear terms of \Cref{sec:PDE}. Hence, we can use our proposed non-intrusive approach to construct a learned ROM.

The governing equations are spatially discretized via a finite difference scheme using $500$ grid points. We generate snapshots of the discretized state $\dstate(t)$ via numerical simulations of the semi-discrete system in the time interval $T = \left[0,10\right]$s with time step $10^{-3}$.  We also collect snapshot data for the input $u(t) = 10(\sin(\pi t)+1)$. This allows us to build a snapshot matrix $\Mstate$ and input matrix $\bU$. Furthermore, we construct a nonlinear snapshot matrix $\bF$ from the snapshot matrix $\Mstate$ by applying the nonlinear function $\boldf$ from equation~\eqref{eq:def_f}, i.e., $\bF(:,i) = \boldf(\Mstate(:,i),t_i)$ where $(:,i)$ denotes the $i$th column of a matrix.

In \Cref{fig:chafee_decaySV}, we show the decay of the singular values of the snapshot matrix and nonlinear snapshot matrix, indicating exponential decay for both. By taking the dominant $12$  POD basis vectors, we construct an intrusive ROM by explicitly computing the projected matrices, and also employ our non-intrusive framework from \Cref{algo:OpInfNL} to obtain a learned ROM.
For the operator inference method, we leave out the first $10$ recorded snapshots, as the transient response in the first 10 state vectors is very fast. A much smaller time step could solve this issue, but we are often faced with a situation where a data set with a fixed time step is given to us, and we have little control over the time step selection, which is the situation we consider here.
We employ DEIM approximation for the reduced nonlinear terms as discussed in \Cref{sec:DEIM}. We select DEIM basis vectors for the nonlinear term corresponding to relative singular values up to $10^{-8}$.

In \Cref{fig:chafee_response}, we compare the output of interest for the FOM and both ROMs for the same input. Note that the system is simulated for $T=20$s, thus predicting the output for 10s, i.e., 100\% past the training interval. We observe that the learned ROM is more accurate at most time steps. While this appears counter-intuitive at first, it might be possible that unresolved features (sometimes called closure terms) are learned by the ROM. This remains a topic of further investigation.

In \Cref{fig:chafee_stateOpComp}, we compare the state approximation error of both ROMs. Both POD and our learned ROM are accurate, with the learned ROM being slightly more accurate. We found that after $r = 14$, the non-intrusive approach starts yielding unstable ROMs. This could be due to the fact that the singular values corresponding to higher basis vectors are relatively small (order $10^{-10}$ in this example). Moreover, since the training data remains the same as $r$ is increased, there can be a situation where there is not enough information in the training data to learn a more accurate ROM that inherits the stability from the FOM. We do not observe instability in non-intrusive ROMs in general.
Nevertheless, at this point the ROMs already predict the output with accuracy $10^{-7}$, which is sufficient in most practical applications.

\begin{figure}[!tb]
	\centering
	\setlength\fheight{3.0cm}
	\setlength\fwidth{.4\textwidth}
	{\footnotesize
		\inputtikz{Chafee_DecaySV}
	}
	\caption{Chafee-Infante example: Decay of singular values of the snapshot matrix and the nonlinear snapshot matrix.}
	\label{fig:chafee_decaySV}
\end{figure}

\begin{figure}[!tb]
	\centering
	\setlength\fheight{3.0cm}
	\setlength\fwidth{.4\textwidth}
	{\footnotesize
		\inputtikz{Chafee_Response}\hfill
		\inputtikz{Chafee_Error}
	}
	\caption{Chafee-Infante example: Output (left) and output errors (right) of the POD intrusive and the learned ROMs of order $r = 12$.}
	\label{fig:chafee_response}
\end{figure}

\begin{figure}[!tb]
	\centering
	\setlength\fheight{3.0cm}
	\setlength\fwidth{.4\textwidth}
	{\footnotesize
		\inputtikz{Chafee_DiffFullState}
		\caption{Chafee-Infante example: Convergence of relative state errors with increasing order of the POD intrusive ROMs and the learned ROMs.}
		\label{fig:chafee_stateOpComp}
	}
\end{figure}

\subsection{Tubular reactor model} \label{sec:numerics_tubular}
We consider a one-dimensional non-adiabatic tubular reactor model with a single reaction, describing the evolution of the species concentration $\psi(x,t)$ and temperature $\theta(x,t)$. The dynamics of the model are given by the PDEs~\cite{heinemann1981tubular} as follows:
\begin{subequations}
	\begin{align*}
		\frac{\partial \psi}{\partial t}   &= \dfrac{1}{\ensuremath{\mathop{\mathrm{Pe}}}} \frac{\partial^2 \psi}{\partial x^2} - \frac{\partial \psi}{\partial x} - \mathcal D \mathcal{F}(\psi,\theta;\gamma),\\
		\frac{\partial \theta}{\partial t } &= \dfrac{1}{\ensuremath{\mathop{\mathrm{Pe}}}}\frac{\partial^2 \theta}{\partial x^2} - \frac{\partial \theta}{\partial x}  -\beta(\theta-\theta_{\text{ref}}) + \mathcal B\mathcal D \mathcal{F}(\psi,\theta;\gamma),
	\end{align*}
\end{subequations}
with spatial variable $x\in (0,1)$, time $t >0$ and Arrhenius reaction term
\begin{equation} \label{eq:F_tubular}
	\mathcal{F}(\psi,\theta;\gamma) = \psi \exp  \left( \gamma - \dfrac{\gamma}{\theta} \right).
\end{equation}
The parameters are the Damk{\"o}hler number $\cD$, the P{\'e}clet number  $\ensuremath{\mathop{\mathrm{Pe}}}$, and the reaction rate $\gamma$. We set $\mathcal{D}=0.167$, $ \ensuremath{\mathop{\mathrm{Pe}}}=5$, and  $\gamma=25$. The reference temperature is $\theta_{\text{ref}}(x,t) \equiv 1$,  and the constants are $\mathcal{B}=0.5$ and  $\beta=2.5$. Robin boundary conditions are imposed on the left boundary of the domain as follows:
\begin{equation*}
\frac{\partial \psi}{\partial x}(0,t) = \ensuremath{\mathop{\mathrm{Pe}}} (\psi(0,t)-1), \qquad \frac{\partial \theta}{\partial x}(0,t) = \ensuremath{\mathop{\mathrm{Pe}}} (\theta(0,t)-1),
\end{equation*}
and the Neumann boundary conditions on the right boundary are
\begin{equation*}
\frac{\partial \psi}{\partial x}(1,t) = 0, \quad \frac{\partial \theta}{\partial x} (1,t) =0.
\end{equation*}
Moreover, the initial conditions are prescribed as
\begin{equation*}
	\psi(x,0) = \psi_0(x), \qquad \theta(x,0) = \theta_0(x).
\end{equation*}
The quantity of interest is the temperature oscillation at the reactor exit, i.e., the quantity
\begin{equation*}
	y(t) = \theta(x=1,t).
\end{equation*}
The governing PDE is discretized with a finite difference approximation (see \cite{zhou2012thesis} for details) leading to a discretized state $\dstate(t) \in \mathbb{R}^{198}$ which takes the form
\begin{align}\label{eq:fullDiscretedModel}
	\dot \dstate(t) & = \bA \dstate(t) + \boldf(\dstate(t)) +  \bB.
\end{align}
Observe that in this example the input $\bu(t)$ is constant, i.e., $\bu(t) \equiv 1$.
The Arrhenius nonlinear term~\eqref{eq:F_tubular} requires pointwise evaluations (local in space), which fits to our framework presented  in \Cref{sec:use_nonlin}. Thus, we can apply the proposed non-intrusive operator inference framework to obtain a ROM by including PDE-level information about the nonlinear terms into the learning framework. We build the snapshot matrix by collecting snapshots in the time interval $T  = \left(0,30\right]$s which are $\delta t = 10^{-3}$ apart.

\Cref{fig:TR_decaySV} shows the decay of the singular values of the snapshot matrix $\bS$ and the nonlinear snapshot matrix $\bF$.
\begin{figure}[!tb]
	{	\centering
		\setlength\fheight{3.0cm}
		\setlength\fwidth{.4\textwidth}
		\inputtikz{decay_singularValues}
		\caption{Tubular reactor example: Decay of the singular values of the snapshot matrix  and the nonlinear snapshot matrix.}
		\label{fig:TR_decaySV}
	}
\end{figure}
We use the leading $r=10$ POD modes in the projection matrix $\bV$. We construct ROMs both via explicitly computing the projected model terms, and via our non-intrusive approach. As in the previous example, we make use of DEIM to efficiently evaluate the reduced nonlinear terms in both ROMs. We select DEIM basis vectors for the nonlinear term corresponding to relative singular values up to $10^{-8}$.

Both ROMs are  simulated until $T = 60$s, which is 100\% longer than the training interval.   \Cref{fig:tubularresponse_a} shows the quantity of interest obtained from the FOM and the two ROMs. Note, that for this set of parameters, the model enters a self-excited limit-cycle oscillation. \Cref{fig:tubularresponse_b} shows the output error $| \by(t)- \hat\by(t) |$ for the two ROMs. Both models produce similar output errors. This competitiveness of the learned ROM with the projection-based POD model is remarkable, since after $t=30$s outputs are purely predictive as the ROMs exit the range of training data.

\Cref{fig:stateOpComp} shows the relative error in the approximated states for both ROMs with increasing ROM order. The learned ROM is more accurate than the POD-ROM in most cases. The errors for both models level-off after $r = 18$ at a relative state error of $10^{-7}$, which is again accurate enough for most applications, certainly for the temperature field considered in this example.

\begin{figure}[!tb]
	{\small
		\centering
		\begin{subfigure}[t]{0.5\textwidth}
			\centering
			\setlength\fheight{3.0cm}
			\setlength\fwidth{.7\textwidth}
			\inputtikz{outputs}
			\caption{Transient response.}
			\label{fig:tubularresponse_a}
		\end{subfigure}%
		\begin{subfigure}[t]{0.5\textwidth}
			\centering
			\setlength\fheight{3.0cm}
			\setlength\fwidth{.7\textwidth}
			\inputtikz{output_error}
			\caption{Absolute output error.}
			\label{fig:tubularresponse_b}
		\end{subfigure}
	}
	\caption{Tubular reactor example: A comparison of the transient behavior of the projection-based POD ROM and the learned ROM.}
	\label{fig:tubularresponse}
\end{figure}

\begin{figure}[!tb]
	{\footnotesize
		\centering
		\setlength\fheight{3.0cm}
		\setlength\fwidth{.4\textwidth}
		\inputtikz{state_error_convergence}
		\caption{Tubular reactor example: Relative state errors produced by both ROMs, i.e., $\|\Mstate -\bV \hat{\Mstate}^{\text{POD}}\|/ \| \Mstate \|$ and $\|\Mstate -\bV \hat{\Mstate}^{\text{NI}}\|/ \| \Mstate \|$.}
		\label{fig:stateOpComp}
	}
\end{figure}

\subsection{Batch Chromatography} \label{sec:numerics_batchChrom}

In \Cref{sec:BC_setup} we describe the PDE model and its discretization. \Cref{sec:BC_intrusive} presents the structure-preserving ROM generation in the projection-based setting, and \Cref{sec:BC_nonintrusive} describes how we enforce the coupling structure in the non-intrusive learning framework. \Cref{sec:BC_results} presents our numerical results.

\subsubsection{Problem setup} \label{sec:BC_setup}
Chromatography is a separation and purification process used in the chemical and pharmaceutical industry, and we consider one of the simplest chromatography processes, the so-called batch chromatography.  For a detailed description of the process, we refer to \cite{guiochon2006fundamentals,morZhaFLetal14}. The primary principle of the batch chromatography process for binary separation is shown in~\Cref{fig:schematic_bc}. A mixture of products A and B is injected at the inlet of the column and then the feed mixture flows through the column. Since the to-be-separated solutes exhibit different adsorption affinities to the stationary phase, they move at different velocities, and thus separate from each other when exiting the column. At the column outlet, component A, which moves faster, is collected between $t_1$ and $t_2$, and later, component B is collected between $t_3$ and $t_4$. Here, time $t_1$ and $t_4$ are determined by a minimum concentration threshold that the detector can resolve, and time $t_2$ and $t_3$ are determined by the purity specifications imposed on the products.
\begin{figure}[!tb]
	\centering
	\includegraphics[width = 9cm]{Fig_BatchChrom.jpg}
	\caption{The figure shows a chromatography process, separating a mixture, c.f.~\cite{morwiki_bc}.}
	\label{fig:schematic_bc}
\end{figure}

The dynamics of the batch chromatographic column can be described by the following dimensionless coupled PDE system:
\begin{subequations}\label{eq:BC_pde}
	\begin{align}
		&\dfrac{\partial c_i}{\partial t} +  \dfrac{1-\epsilon}{\epsilon}\dfrac{\partial q_i}{\partial t} + \dfrac{\partial c_i}{\partial x} - \dfrac{1}{\ensuremath{\mathop{\mathrm{Pe}}}}\dfrac{\partial^2 c_i}{\partial x^2} = 0, \label{eq:BC_pde_c} \\
		&\dfrac{\partial q_i}{\partial t} =\dfrac{L}{Q/\epsilon A_c} \kappa_i \left(q_i^{Eq} - q_i\right),\label{eq:BC_pde_q}
	\end{align}
\end{subequations}
where $i \in \{1,2\}$, $x\in (0,L)$ is the spatial variable, and the liquid and solid phase concentrations of the compound $i$ are denoted by $c_i$ and $q_i$, respectively; the constants $\epsilon$, $L$ are the interstitial liquid velocity, the column porosity, and the column length, respectively; $\ensuremath{\mathop{\mathrm{Pe}}}$ denotes the P\'eclet number; $A_c$ denotes the cross-sectional area, defined as $\tfrac{\pi D^2}{4}$, where $D$ is the diameter of the tube. The constant $\kappa_{i}$ is the mass-transfer coefficient of component~$i$; moreover, $q^{\text{Eq}}_i$ is the adsorption equilibrium concentration calculated by the isotherm equation for the component~$i$, given as follows:
\begin{equation} \label{eq:BC_qi}
	q^{\text{Eq}}_i=\frac{H_{i,1}\,c_i}{1+\sum\limits_{j=1,2}K_{j,1}c_j^f\,c_j}+\frac{H_{i,2}\,c_i}{1+\sum\limits_{j=1,2}K_{j,2}c_j^f\,c_j},
\end{equation}
where $H_{i,1}$ and $H_{i,2}$ are the Henry constants, and $K_{j,1}$ and $K_{j,2}$ the thermodynamic coefficients.  Furthermore, the boundary conditions are specified by the Danckwerts relations:
\begin{equation}
	\left.\frac{\partial c_i}{\partial x}\right|_{x=0} = \ensuremath{\mathop{\mathrm{Pe}}}\,(\left.c_i\right|_{x=0}-c^{\text{in}}_i), \quad\quad \left.\frac{\partial c_i}{\partial x}\right|_{x=L}=0,
\end{equation}
where $c^{\text{in}}_i$ is the concentration of component $i$ at the inlet of the column.  A sigmoid-type injection profile is assumed, which is a differentiable approximation of a rectangular profile, given by
\begin{equation}
	c^{\text{in}}_i(t)= \dfrac{1}{1 + e^{-5(t-t_{\text{inj}  } )  }  },
\end{equation}
where $t_{\text{inj}}$ is the injection period. In addition, we have the following initial conditions:
\begin{equation}
	c_i(t=0,x)=q_i(t=0,x)=0,\quad x\in[0,\;L],\;i \in \{1,2\}.
\end{equation}

System \eqref{eq:BC_pde} is a coupled set of PDEs, and more importantly, \eqref{eq:BC_pde_c} involves the derivatives of $c_i$ and $q_i$. However, we can modify \eqref{eq:BC_pde_c} such that the resulting equation does not have a  $\partial q_i/\partial t$ term. For this, we set $\epsilon_c := \dfrac{\epsilon-1}{\epsilon}$ and insert \eqref{eq:BC_pde_q} by into \eqref{eq:BC_pde_c}, yielding
\begin{subequations}\label{eq:BC_pde1}
	\begin{align}
		\dfrac{\partial c_i}{\partial t} + \dfrac{\partial c_i}{\partial x} - \dfrac{1}{\ensuremath{\mathop{\mathrm{Pe}}}}\dfrac{\partial^2 c_i}{\partial x^2} & = \epsilon_c\kappa_i \left(q_i^{Eq} - q_i\right),\label{eq:BC_pde1c}\\
		\dfrac{\partial q_i}{\partial t} & = \kappa_i \left(q_i^{\text{Eq}} - q_i\right).\label{eq:BC_pde1q}
	\end{align}
\end{subequations}
A finite volume discretization of the governing equations yields a discretized model of the form:
\begin{equation}\label{eq:discretizedSys}
	\begin{bmatrix} \dot \bc_1 \\ \dot \bq_1 \\ \dot \bc_2 \\ \dot \bq_2 \end{bmatrix}  = \begin{bmatrix} \bA_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \bA_2  & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \bc_1 \\ \bq_1 \\ \bc_2 \\ \bq_2 \end{bmatrix} +  \begin{bmatrix} \bB \\ 0 \\ \bB \\ 0 \end{bmatrix} u(t) + \begin{bmatrix} \epsilon_c \\ 1 \end{bmatrix} \otimes \begin{bmatrix} \bff_1(\bc_1,\bq_1,\bc_2,\bq_2) \\ \bff_2(\bc_1,\bq_1,\bc_2,\bq_2) \end{bmatrix},
\end{equation}
where $\bc_1,\bq_1,\bc_2,\bq_2 \in \Rn$, $\bA_1,\bA_2\in \Rnn$, $\bB\in \Rn$, and $n$ is the number of degrees of freedom in the discretized field. The term $\bff_i(\bc_1,\bq_1,\bc_2,\bq_2)\in \Rn$ is the evaluation of the nonlinear term $\kappa_i \left(q_i^{\text{Eq}} - q_i\right)$ on the right-hand side of \eqref{eq:BC_pde1q} on the spatial grid. Note that this nonlinear term involves a rational function, see \eqref{eq:BC_qi}, which again translates to a nonlinear term that is local in space, and hence the nonlinear function fits the setting described in \Cref{sec:PDE}.

\subsubsection{Maintaining the coupling structure: Projection-based ROM} \label{sec:BC_intrusive}
Classically, to obtain the basis for POD, one would stack all variables in one vector, i.e., $[\bc_1^\top, \bq_1^\top,\bc_2^\top,\bq_2^\top]^\top$ and determine the common subspace $\bV$ by taking the SVD of that data. However, by doing so, we observe that both methods produce unstable ROMs. This is potentially due to the fact that the governing PDEs are highly coupled and the resulting ROMs do not preserve the coupling topology structure. We present an approach that uses the knowledge about the topology structure of a coupled system in the ROM.

To preserve the coupled topology structure, we first compute a basis for each subsystem, i.e., $\bc_i \approx \bV_{\bc_i} \hat \bc_i$ and $\bq_i \approx \bV_{\bq_i} \hat \bq_i$, $i \in \{1,2\}$.  Next, we construct a projection matrix with a block structure, i.e., we approximate
\begin{equation} \label{eq:BC_V}
	\begin{bmatrix} \bc_1\\ \bq_1\\\bc_2\\\bq_2\end{bmatrix} \approx \begin{bmatrix} \bV_{\bc_1}  & & & \\ & \bV_{\bq_1} & & \\ & & \bV_{\bc_2} & \\ & & & \bV_{\bq_2} \end{bmatrix} \begin{bmatrix} \hat{\bc}_{1} \\ \hat{\bq}_{1}\\\hat{\bc}_{2}\\\hat{\bq}_{2}\end{bmatrix}.
\end{equation}
With the above projection matrix $\bV$, a projection-based ROM preserves this structure, i.e.,
\begin{equation}\label{eq:discretizedSysROM}
	\begin{bmatrix} \dot{\hat{\bc}}_1 \\ \dot{\hat{\bq}}_1 \\ \dot{\hat{\bc}}_2 \\ \dot{\hat{\bq}}_2 \end{bmatrix}  = \begin{bmatrix} \hat{\bA}_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \hat{\bA}_2  & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \hat{\bc}_1 \\ \hat{\bq}_1 \\ \hat{\bc}_2 \\ \hat{\bq}_2 \end{bmatrix} +  \begin{bmatrix} \hat{\bB}_1 \\ 0 \\ \hat{\bB}_2 \\ 0 \end{bmatrix} u(t) + \begin{bmatrix} \epsilon_c \\ 1 \end{bmatrix} \otimes \begin{bmatrix} \hat{\bff}_1(\hat{\bc}_1,\hat{\bq}_1,\hat{\bc}_2,\hat{\bq}_2) \\ \hat{\bff}_2(\hat{\bc}_1,\hat{\bq}_1,\hat{\bc}_2,\hat{\bq}_2) \end{bmatrix},
\end{equation}
where $\hat{\bc}_1,\hat{\bq}_1,\hat{\bc}_2,\hat{\bq}_2 \in \Rr$, $\hat{\bA}_1,\hat{\bA}_2\in \Rrr$, $\hat{\bB}_1,\hat{\bB}_2 \in \Rr$.
Note, that the dimension of the coupled ROM~\eqref{eq:discretizedSysROM} is $4r$, however, it has sparse structure, e.g., the linear system matrix has only $2r^2$ nonzero entries and the linear input matrix has only $2r$ nonzero entries. Moreover, it is also possible to use different ROM dimensions for each of the four physical states. Thus, enforcing the coupling structure does not necessarily lead to drastically more expensive ROMs.

\begin{remark} \label{rem:Stable}
	Maintaining the coupling structure in a ROM  of the  aggregate system is important for physical interpretability, and can have numerical implications as well~\cite{morBenF15,liao2007important,morReiS07,reis2008survey}.
	%
	For instance, structure-preserving ROMs can  lead to convergent feedback controllers for coupled systems~\cite{kramer16MORcontrolBurgers} and have implications on stability.
	%
	We observe in our numerical experiments on the batch chromatography example, that unstable ROMs arise when we do not preserve this structure. Conversely, when the coupling structure of the FOM is maintained in the ROMs, we obtain stable ROMs. A theoretical justification of this observation for nonlinear systems remains an open problem, though, for linear systems, stability analysis for coupled systems is performed in \cite{morReiS07}.
\end{remark}

\subsubsection{Maintaining the coupling structure: Learned ROM} \label{sec:BC_nonintrusive}
Having derived the form of the projection-based structure-preserving ROM leads to a strategy to enforce the coupling structure in the operator inference framework as follows. First, we compute the block-diagonal projection matrix $\bV$ from~\eqref{eq:BC_V}. Second, we collect projected state and time-derivative data:
\begin{equation}
	\hat{\Mstate}
	= \begin{bmatrix} \bV_{\bc_1}^\top \bC_1  \\ \bV_{\bq_1}^\top \bQ_1 \\\bV_{\bc_2}^\top \bC_2 \\ \bV_{\bq_2}^\top \bQ_2 \end{bmatrix}
	=: \begin{bmatrix} \hat{\bC}_1 \\ \hat{\bQ}_1 \\ \hat{\bC}_2 \\ \hat{\bQ}_2\end{bmatrix},
	\qquad
	\dot{\hat{\Mstate}}
	= \begin{bmatrix} \bV_{\bc_1}^\top \dot\bC_1  \\ \bV_{\bq_1}^\top \dot\bQ_1 \\ \bV_{\bc_2}^\top \dot\bC_2 \\ \bV_{\bq_2}^\top \dot\bQ_2 \end{bmatrix}
	=: \begin{bmatrix} \dot{\hat{\bC}}_1 \\ \dot{\hat{\bQ}}_1 \\ \dot{\hat{\bC}}_2 \\ \dot{\hat{\bQ}}_2\end{bmatrix}.
\end{equation}
The input matrix $\bU$ is assembled as given in~\eqref{eq:Xdot_X} and the nonlinear term matrix $\bF$ is built in a similar way as given in \eqref{eq:F}. We then solve \textit{separate} least-squares problems of the form
\begin{equation}\label{eq:separateLS}
	\min_{\hat \bA_{i},\hat{\bB_i}}  \ \| \dot{\hat\bC}_i -\epsilon_c\bV_{\bc_i}^\top \bF - \hat \bA_{i} \hat{\bC}_i - \hat \bB_{i} \bU \|_F,\quad i \in \{1,2\}.
\end{equation}
Recall the structure of the projection-based ROM in \eqref{eq:discretizedSysROM}. For $\dot{\hat\bq}_i$, $i \in \{1,2\}$, there are no linear terms on the right hand side, and so there is no need to solve a least-squares problem. The expressions for $\dot{\hat\bq}_i$ are
\begin{align*}
	\dot{\hat\bq}_i = \boldf_i(\hat\bc_1,\hat\bq_1,\hat\bc_2,\hat\bq_2) =  \bV^\top_{\bq_i}\boldf_i(\bV_{\bc_1}\bc_1,\bV_{\bq_1}\bq_1,\bV_{\bc_2}\bc_2,\bV_{\bq_2}\bq_2).
\end{align*}

\subsubsection{Results} \label{sec:BC_results}

For the numerical experiment, we choose the model parameters as follows: $\epsilon=0.4$, $L=10.5$, $\ensuremath{\mathop{\mathrm{Pe}}} =2000$, $D = 2.6$, $\kappa_{1}=\kappa_{2}=0.1$, $H_{1,1} = 3.728,  H_{1,2} = 0.3, H_{2,1}= 2.688,  H_{2,2} = 0.1$, $K_{1,1} = 46.6,  K_{1,2} = 33.6, K_{2,1}= 3000,  K_{2,2} =1000$, $Q=0.1018$, $c_1^f = c_2^f =  2.9\times 10^{-3}$, and $t_\text{inj}=1.3$s. For spatial discretization we choose $n= 400$ (thus the overall state is $4n=1600$).
The model is simulated until $T=10$s and snapshots are collected every $\delta t = 5\times 10^{-5}$s.
Such a fine time stepping allows us to obtain an accurate approximation of the time derivative; however, we take every $100$th snapshot of the state and time derivative while inferring the operators in  the least squares procedure.
\Cref{fig:BC_decaySV} shows the decay of the singular values of each component, showing a rather slow decay, which can be expected due to the transport nature of the problem. Next, we compute ROMs via intrusive POD and our non-intrusive approach with the same matrix $\bV$ as shown in~\eqref{eq:BC_V}, where each component is reduced to order $r = 22$. As in the previous two examples, we incorporate a DEIM approximation of the reduced nonlinear terms for both ROMs. We select DEIM basis vectors for the nonlinear term corresponding to relative singular values up to $10^{-10}$.
\Cref{fig:BC_response}, left, shows the quantity of interest obtained from the time-domain simulations of the FOM and the two ROMs.  \Cref{fig:BC_response}, right, shows the (mean) relative error of the two outputs. The results indicate that the non-intrusive and intrusive approaches yield ROMs of comparative accuracy.
\Cref{fig:BC_stateOpComp} shows the relative error in the approximated states for both ROMs with increasing ROM order. For ROM orders of $r<30$, both the intrusive POD ROM and the learned ROMs perform similarly. However, as $r$ increases further, the error for the non-intrusive models does not decrease as favorably as for POD intrusive models. This could be due to the condition number of the operator inference problem \eqref{eq:ROM_OPINF}. Adding regularization to the least-squares problem could improve this as seen in~\cite{SKHW2020_learning_ROMs_combustor} and also using data from several different simulations, similar to \cite[Sec 3.6]{Peherstorfer16DataDriven}.

\begin{figure}[!tb]
	{	\centering
		\setlength\fheight{3.0cm}
		\setlength\fwidth{.4\textwidth}
		\inputtikz{BC_DecaySV}
		\caption{Batch chromatography example: Decay of singular values of the collected snapshots for all four components separately and the nonlinear snapshot matrix.}
		\label{fig:BC_decaySV}
	}
\end{figure}

\begin{figure}[!tb]
	{\footnotesize
		\centering
		\setlength\fheight{3.0cm}
		\setlength\fwidth{.4\textwidth}
		\inputtikz{BC_Response}\hfill
		\inputtikz{BC_Error}
	}
	\caption{Batch chromatography example: A comparison of the POD intrusive model with the learned model of order $r = 4\times 22$.}
	\label{fig:BC_response}

\end{figure}

\begin{figure}[!tb]
	\centering
	\setlength\fheight{3.0cm}
	\setlength\fwidth{.4\textwidth}
	{\footnotesize
		\inputtikz{BC_DiffFullState_approxResidual}
	}
	\caption{Batch Chromatography example: Relative errors $\|\Mstate -\bV \hat{\Mstate}^{\text{POD}}\|/ \| \Mstate \| $ and $\|\Mstate - \bV\hat{\Mstate}^{\text{NI}}\| / \| \Mstate \|$.}
	\label{fig:BC_stateOpComp}
\end{figure}

\section{Conclusions} \label{sec:conclusion}
We have presented a data-driven model reduction method that non-intrusively learns reduced models from data collected of the full model states. The presented method applies to non-polynomial nonlinear models that are spatially local, and extends previous results for the polynomial case~\cite{Peherstorfer16DataDriven}. The method exploits the spatially local structure of the nonlinear terms to generate nonlinear snapshots from state data without requiring access to the discretization of the nonlinear term explicitly. Moreover, we presented a convergence result that shows that under mild assumptions on the time stepping scheme, and if the time step of the sampled data is sufficiently small, the non-intrusively learned reduced models converge to the same reduced models as obtained with intrusive model reduction methods. The numerical results show that the learned reduced models provide accurate predictions when compared to the full-model predictions. The three numerical experiments with reactive flows and chemical purification processes demonstrate that the method works well for strongly nonlinear systems, and further information in form of known structure can also be embedded into the learning framework.
Future research directions motivated by this work are: addressing the effect of measurement noise (from real data) on the learning procedure and ultimately the learned ROMs; developing a suitable regularization strategy (possibly problem dependent) that could improve the learned ROMs accuracy and stability; and including additional information, such as physical constraints in the least-squares problem as suggested by \cite{loiseau_brunton_2018}.

\section*{Acknowledgments}
This work was supported in part by the US Air Force Center of Excellence on Multi-Fidelity Modeling of Rocket Combustor Dynamics award FA9550-17-1-0195, the Air Force Office of Scientific Research MURI on managing multiple information sources of multi-physics systems awards FA9550-15-1-0038 and FA9550-18-1-0023, and the AEOLUS center under US Department of Energy Applied Mathematics MMICC award  DE-SC0019303. The fourth author was partially supported by the US Department of Energy, Office of Advanced Scientific Computing Research, Applied Mathematics Program, DOE Award DESC0019334 and the National Science Foundation under Grant No. 1901091.

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


That's a pretty good idea, I mean I think I like the nice thick wooden ones better but I could see me having & using both, once I get enough gear anyway, lol. I'm still very much a newbie Vaper, got a VAMO 3wks ago & have been really enjoying it, just ordered an Innokin iTaste MVP so it looks like I would need to have one of those custom made to fit it, I wonder how much it would cost?? Thanks for another great video Todd, keep em coming & VAPE ON!


text:
Travel Fantasy Slam in Ocean City, Maryland
Fantasy Slam in Ocean City, Maryland
Five billfish species (and then some) in one incredible day
By Jon Coudriet
The No Quarter team with a very rare fantasy slam of five billfish species, including a 345-pound swordfish. Courtesy Bo Thai
In the last few weeks, the fishing world has been buzzing with news from the White Marlin Open. Drama, polygraphs and over $2.8 million in winnings that will apparently be withdrawn from the top team have held everyone's attention. But the best way to get over bad news is with good news (and good fishing), which is exactly what happened when Capt. Kyle Peet headed out of Ocean City, Maryland, September 13, 2016.
Peet and his crew run the 54-foot Taylor Made No Quarter, which is based out of Sunset Marina. These boys are local and know the area as well as anyone; tuna fishing is their expertise, but as any Ocean City local will tell you, white marlin is a species you grow up catching there. With a scheduled charter for the team from Xtratuf boots on the books, there were rumors of a red-hot white marlin bite off Norfolk Canyon. Kyle and his mate Erik Mateer had to scramble all night before the 5 a.m. Tuesday morning departure to make sure they were well-stocked with bait.
"I said to Erik, I think we need to try and get a grand slam on this trip," Peet says. The table was set. The boat arrived at Norfolk Canyon at 9:30 a.m. and caught a triple out of four white marlin bites within the first five minutes of fishing.
No Quarter returns to Sunset Marina in Ocean City, Maryland, with some impressive rigger laundry. Courtesy Greg Steen
"Then at 3:30 p.m. we got a blind bite on the flat, and I immediately put the boat in a turn," Peet reports. "I saw a roughly 200-pound blue one greyhounding between two other boats. I don't think I ever backed the boat down that hard, as I knew it was going to be the most difficult part of our grand slam, getting the blue marlin. We got the release and then it seemed like a light switch went off — we were getting savage bites, left and right."
"At around 4 p.m. I was trolling away from the sun and something crushed the teaser. I had no idea what it was, then the left flat went off, and then the right long. When the flat line came tight, we saw him jump and I backed up right through the dredges — it was the sailfish we needed for the grand slam." Then around 6 p.m. they hooked what they thought was a small blue marlin, until they got it to the boat and realized it was a longbill spearfish. That gave them a royal slam: four billfish species in one day. They had planned an overnighter so they kept fishing until the sun went down. "From 6 p.m. until dark we burned them up, catching 11 white marlin out of 20 bites, including one I hooked in the complete dark," Peet reports.
The team now needed a swordfish to complete the fantasy slam, which is one of the rarest and most difficult challenges in sport fishing. Peet says: "I was marking so much bait you'd think you could walk on it, so I just shut the boat down right there and boy, was there bait: squid, balls of fish, I've never seen so many schools of blackfin tuna. I was sure that we would get a sword bite." They jigged up a few tinker mackerel for bait and sent one armed with a 9/0 Gamakatsu hook down to around 220 feet on a Shimano Tiagra 130.
"I got the grill out and cooked dinner, and then everyone went to bed," Peet says. "Erik and I were still up, just hanging out and talking about the day, when the rubber band snapped. At first the fish was just lying there, and we figured it was a shark, so we pushed the drag up and cranked about 50 feet. Then he took off. At that point we knew it was a swordfish."
The team's fifth and final billfish species, a 345-pound swordfish, comes aboard. Courtesy Bo Thai
"Finally, around 4:30 in the morning the fish started coming to the boat, and Erik says it's the biggest swordfish he's ever seen," Peet says. "I looked down, and it looked like a sea monster. We got a couple gaffs in him, and it was done — we had the fantasy slam.
The team witnessed incredible fishing during their trip offshore, where they found the Norfolk Canyon loaded with bait and billfish. Courtesy Fletcher Case
By now the sun was coming up, and the team talked about calling it a trip, but with such incredible fishing they decided to stay and catch a few more. So from 6 a.m. until 11 a.m. they caught eight more white marlin and another longbill spearfish, finally heading for home around 1 p.m. that afternoon. "It was just an epic trip, and I don't think it's even really set in yet," Peet says.
The final catch stats were: 23 white marlin, two longbill spearfish, one blue marlin, one sailfish and one 345-pound swordfish.
Another look at No Quarter, a 54-foot Taylor Made, as she heads home. Courtesy Greg Steen
, North America
Kona Winter
Thoughts on Tournament Pre-Fishing

redpajama_set_name: RedPajamaCommonCrawl


What was the defining technology that transformed Arpanet into the Internet?
Here's your next round of cool reads.
It's probably worth mentioning that when we introduced a new Q&A section last week, readers were emailing with questions and comments probably not realizing the material was from past columns, all the way back to the beginning (over a decade). Since the original Q&A author passed away a number of years ago, we obviously can't forward anything to him to weigh in on. So, we added Disqus commenting to the section so you can comment, rant, ask questions, criticize, correct, or whatever. So feel free to start or chime in on the discussion. After all, that's what makes it interesting.
And thanks once again to the great folks over at M.E. Labs!
Switching to the TCP/IP networking standard radically transformed the Internet from a network that connected military and university mainframes to the sprawling "information superhighway" that reaches into hundreds of millions of homes, businesses, and schools, and has radically transformed the lives of its users.
The digitally-programmable potentiometer (DPP) is a mixed signal, system-level control device performing a component-level function. The potentiometer adds variability to the analog circuit, while the digital controls add programmability. The DPP brings the high speed, programming, computation, and control of a processor to the variation of the potentiometer in a wide array of analog applications.
Our last episode provided a convenient way to hook up a serial connection and provide a power supply for use in electronics prototyping on a solderless breadboard. Now, let's use that for general-purpose I/O to read switches and light LEDs. Then, we'll use a seven-segment LED to make the "World's Smallest Moving Message Sign," then finally we'll create an Arduino compatible prototype on a breadboard.
Optoelectronic security circuits - which respond to either visible or invisible (usually infrared) light levels - come in three (3) basic types. Last time, we covered circuits of the visible light type. This time, we'll tackle IR 'light-beam' and PIR circuits.
It's been said that: Once you learn to use the Propeller in its parallel processing environment, you will be hard pressed to ever again use a conventional processor for the kind of tasks small microcontrollers are designed for.
It's that time again. The cutoff for the March issue of Nuts & Volts and SERVO is at hand. So if you want us to mail you your very own copy, all you need to do is get your subscription order in by 1/27/16. Sounds easy I know, but you'd be surprised at how many people wait too long and then have to wait for the next time around.


Big news! We have a new identity.
We're proud to announce that Compass Research has joined Bioclinica. We will continue providing the same great service under our new name, Bioclinica Research.
2017 BIOCLINICA RESEARCH. ALL RIGHTS RESERVED.


This patient has an injury to his right ankle. He also has abrasions to his right lower leg and a laceration to his right thigh. Based on the MOI this patient is an unlikely candidate for spine injury.
A usability test was performed and the patient is able to bear weight. The minor bleeding was controlled with direct pressure. The wounds will be further assessed in camp where there are more materials for wound care.
Once in camp we will irrigate and clean the abrasion and dress the abrasion in a clean dry gauze dressing. The laceration is not gaping and appears clean. It will be irrigated with water, then closed with "steri strips" and dressed with a "micro thin" dressing.
Additionally, we will RICE the ankle injury and, if needed, support with tape.
Inability to bear weight on the ankle develops.
A thorough assessment is the standard for any patient. This patient presented with a couple of injuries that could be treated differently based on what exactly the assessment revealed.
A usability test can help to determine the extent of the musculoskeletal injury. We don't try to diagnose the injury or grade a sprain or the strain. We decide if an injury is usable or not. If usable, we use RICE therapy and may tape for support. If unusable we immobilize and evacuate.
For this patient and single rescuer the usability test may help determine whether this patient will self-evacuate to camp or if the rescuer will need to go for help to evacuate. A good thorough assessment can help determine whether this is a trip-ending injury or one that can be taken care of with some diligence in camp. The patient may not be climbing, but they could certainly sit and enjoy a sunset while the dishes are being done!
A review of our wilderness protocols regarding laceration closure shows that this wound can indeed be effectively closed. A laceration wider than a half inch would need to be "packed open" and eventually assessed by a physician. Paying particular attention to the prevention of infection will go a long way to ensure this wound heals without incident in the field.
You and your partner make the decision to continue car camping. The soft tissue injuries seem to be healing without complication. The ankle injury, moving into shades of burgundy and taupe, is still painful but with decreased swelling. RICE treatment in the evening and a supportive tape wrap during the day works to keep you both active. Although his mobility keeps him from climbing, he is able to belay you on a 5.10 off-width from which lesser individuals have been spit like so many watermelon seeds at the county fair.


Since graduating from the University of Southern California with degrees in French horn and voice, Jonathan Mack's career as a lyric tenor has taken him throughout the United States, Europe, and Australia as a recital, concert, and opera singer. For four years, Mr. Mack lived in Germany with his family where he was the leading lyric tenor for the opera houses of Kiel and Dortmund. His guest engagements took him throughout West Germany and France.
Jonathan has performed over fifty roles during his eighteen seasons with the Los Angeles Opera, including Ferrando in Mozart's Cosi fan tutte, Kudrjas in Janacek's Katya Kabanova, Quint in Britten's Turn of the Screw, and Orpheus in Offenbach's Orpheus in the Underworld. Appearances with other companies include Belmonte in Mozart's Abduction for Netherlands Opera, Lysander in Britten's A Midsummer Night's Dream for Kentucky Opera, Bob Boles in Britten's Peter Grimes for Vancouver Opera, and the Steersman in Wagner's Flying Dutchman for Opera Columbus. Most recently he made his sixth appearance with the San Luis Obispo Mozart Festival singing Tamino in The Magic Flute.
His concert work includes engagements with the London Symphony Orchestra, Chautauqua Festivals, the Carmel Bach Festival, the Los Angeles Philharmonic, and the Minnesota Orchestra with conductors including Giulini, Mehta, Previn, Boulez, Rattle, Hogwood, and Tilson-Thomas. Recent performances include a tour of the province of Quebec in Beethoven's ninth symphony with the Montreal Metropolitan and Frank Ticheli's Pulitzer prize nominated first Symphony for Festival Miami. He returns to Florida this summer for two weeks of concerts including Mozart's Requiem, and to San Luis Obispo for Stravinski's Les Noces.
In demand as a studio singer, Jonathan has sung on over 100 productions for film, radio, and television, including the 2004 Grammy Awards. Jonathan is on the voice faculty at the University of Southern California and Chapman University.
Mr. Mack is featured as a soloist on six recordings: Gagliano's La Daphne for ABC Records; choral lieder by Brahms and Schumann with the William Hall Chorale for Klavier Records; three recordings on the Nonesuch label: two with the Los Angeles Vocal Arts Ensemble of Brahms' Liebeslieder Waltzes, Rossini's Sins of my Old Age, and one with the Los Angeles Philharmonic under Andre Previn of William Kraft's Contextures II. Most recently on a Crystal CD, Mr. Mack recorded John Biggs Songs of Laughter, Love, and Tears.
"He is instantly sympathetic, authoritative, and secure. He inflects every line with point and unexpected force, and he makes every word count. A sensitive actor, he is a musician of remarkable refinement and the owner of an extraordinarily pliant, sweet and ever growing tenor." - Martin Bernheimer, Los Angeles Times.


text:
Hit Mania Champions 2010 è una compilation di artisti vari facente parte della serie Hit Mania.

La compilation è mixata dal dj Mauro Miclini.

Tracce

Voci correlate
Hit Mania
Musica dance
Musica elettronica
Musica house

Collegamenti esterni

redpajama_set_name: RedPajamaWikipedia


For those who are at lost to the meaning of the word 'media buying', allow me to define what media buying is.
Media buying is an advertising process which is commonly used in the attainment of media spaces set at obtaining the ideal placement for your money's worth. It aims to help companies provide consumers with enough insight of their product to generate mass response thereby, increasing the volume of sales. It is a vital function of the marketing process, since most company owners generally have ideas on how they want to market their product, but do not have the knowledge, time, and clout to implement them.
Truth is–we can DIY most of the things we want to do or dream of doing. You can try Google searching ways to put your idea into fruition, and maybe-just maybe-come across materials, which could REALLY help you; some would even go as far as to say you do not need to spend lots of money in doing so. We all dream of that Eureka moment when we can shout to the world that we found the perfect solution, however, reality bites, because when it comes to advertising you could probably come up with enough media tricks to showcase your products, but as far as trial and error go – it would mean trying out several ways to do it RIGHT, which would entail not just more money, but wasted time and effort which you could have allocated for other things.
And, the worst thing you could probably do is come up with an advertising failure, one that could put your company and its products at a disadvantage. If you want your business to be taken seriously, then let a professional media buyer do it for you by engaging the service of a media buying agency, which could render your company the right leverage in the media advertising arena. There are a lot of media buyers out there, but take time to ensure that you are getting the best media buyer, who could equip you with an ideal media and advertising advantage by taking that extra mile with you.
Do you want to be the company whose advertisement falls into the advertising disaster category or do you want to provide your audience-the consumer-an enticing awareness of what your company can offer? Regrets do always happen at the end of the day so make sure we are always aware with our decision.
All the services provided by Ideal Media Management are performed in-house, meaning you don't have to source any of your creative, marketing, or advertising needs elsewhere. From Strategic planning, branding to copywriting and basic proof reading, there is no requirement too small or too large.
At Ideal Media Management, we are a full service agency, meaning we are a virtual one stop shop for all your media, marketing, and design needs.


text:
Hugh Latimer (Thurcaston, (Leicester), kb. 1490 – 1555. október 16.) angol reformátor, protestáns mártír, akit máglyán égettek meg.

Életpályája
VIII. Henrik angol királynak Aragóniai Katalinnal való vitájában tűnt fel először, amennyiben mint a cambridge-i egyetem részéről kiküldött vizsgáló bizottság tagja, a király mellett nyilatkozott. A király hálából udvari káplánjává, majd 1535-ben worcesteri püspökké tette. Amikor 1543-ban VIII. Henrik nőül vette utolsó hitvesét, a kétszeres özvegy, 32 esztendős és gyermektelen Parr Katalin-t, az új királyné, aki titokban a protestáns hit szabályait követte, egy igen merész és váratlan lépésként pont Hugh Latimer-t nevezte ki udvari lelkészévé.

Gardiner püspök, az elkötelezett katolikus egyházi ember, a leendő I. Mária angol királynő lelkes támogatója, kezdettől fogva azon volt, hogy akár hamis vádakkal is, de befeketítse Parr Katalin-t VIII. Henrik előtt.

Amikor Henrik uralkodásának a vége felé a reakció pártja került uralomra, a betegsége miatt Londonba utazott Latimert elfogatták, és őt titkos törvényszék elé állítva, a Towerbe záratták. Amikor VI. Eduárd lépett Anglia trónjára, Latimer kiszabadult, de bár a parlament közbelépésére visszanyerte püspöki székét, nem foglalta azt el többé. Eduárd halálával a többi reformátorral együtt őt is újra börtönbe vetették 1554-ben, és a következő évben reformátor-társával, Nicholas Ridley-vel együtt megégették.

Művei
Egyházi beszédeit ismételten kiadták Londonban 1825-ben két, majd 1845-ben négy kötetben.

Irodalom
Életrajza megjelent Tulloch Leaders of the Reformation című, 1859-ben kiadott művében és önállóan Latimer címen, Demans által 1869-ben kiadott életrajzában.

Források

Vértanúk
Protestáns egyházi személyek
Reformátorok
1555-ben elhunyt személyek


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First Metric Houses
Last Button A/B Telephone Box
Caird Shipyard
Argyll Factory
Dechmont Range
Blair's Ferry
Blair's Ferry shoreline and cottage
� Leslie Barrie
Jetty remains at Blair's Ferry
Blair's Ferry is the name of a small settlement to the south of Tighnabruaich and Kames on the western shore of the Kyles of Bute, and was probably named after an 18th century ferry which sailed across the Kyles of Bute to Kilmichael on the Isle of Bute, to the east, where the remains of a ferryman's house and inn are reported, together with those of a stone breast built in 1769.
The remains of a stone jetty can still be found on the shore at Blair's Ferry, near a cottage and outbuilding which are still in use.
Concrete ramp
Ramp from the north, insignia is to right
Approximately 300 metres to the north of the remains of the old stone jetty at Blair's Ferry is a modern concrete ramp, which has been identified as a World War II artefact. The ramp and surrounding area would have been used for training in landing craft operations in advance of operations such as the D-Day landings, and is one of a number of similar locations which were created and used by CTC Inveraray for such exercises.
Local lore tells of Winston Churchill landing on the ramp during a wartime visit to the area, and although this event remains uncorroborated, he did make a documented visit to HMS Quebec in 1941, a naval base which was located a short distance to the north.
The site has one heavily built concrete ramp with side reinforcements of over 30 centimetres thickness, and leads east from the road into the water. The southern reinforcement of the ramp is banked in shingle, while that to the north is exposed, and carries a military insignia, further described below.
A significant amount of brick and concrete debris lie on the beach near the ramp, and may have come from buildings associated with the installation and its training activities. However, no evidence has been found of such buildings.
The verge opposite the ramp has been reinforced with concrete which extends for some distance to the north and south of the ramp area, and may have been hard standing which served as a muster point for vehicles waiting to access the ramp and landing craft which would have been operating from it. This would also have avoided destroying the soft verge as heavy military vehicles manoeuvred on the area.
Seating for tourists, and a waste water treatment plant serving Kames and Tighnabruaich now occupy part of the area across the road from the ramp, which is now used a slipway.
Ramp insignia
I Corps insignia
The northern part of the ramp has a large military insignia impressed into its surface, measuring almost 2 feet (60 cm) per side. This has been identified as the insignia of I Corps (One Corps), which would have been expected to have been training in the area as it was involved in the early part of the D-Day landings, and is referred to as a spearhead unit that was landed early, with the specific remit of advancing on, and capturing the city of Caen, in Operation Charnwood.[1][2]
The road passing the ramp
Reinforced cable buried on beach
Northern reinforcement with insignia
Looking east along the ramp
Viewed from the south
1 ⇑ Caen_spreads - ww2_caen.pdf Retrieved April 01, 2013.
2 ⇑ Second World War Commemorative Booklets - Publications - Inside Government - GOV.UK Retrieved April 01, 2013.
Blair's Ferry SP
Concrete slipway SP
Blairs Ferry, stone jetty
Kilmichael area
Categories: World War II | Training Area | Concrete Beach | CTC Inveraray | Ferry
Page last modified on Monday, 01 April 2013, at 10:28


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Home D: Arts & Entertainment Art Watch: Take a break from the heat & enjoy exhibits across Chesco
Art Watch: Take a break from the heat & enjoy exhibits across Chesco
Jul 8th, 2015 · 0 Comment
By Lele Galer, Columnist, The Times
The Church Street Gallery in West Chester is exhibiting the wonderful oil paintings of local West Chester artist, Terry DeAngelo, for their July show entitled "Quiet Truths." The central image for the show is a four foot square painting called "Shot", with hundreds of black birds flying from the branches of a large tree against a canvas of cerulean blue.
I asked DeAngelo why he called it "Shot", and the answer is so obvious, I felt like an idiot. He said, "Have you ever been in the woods and seen one of those trees filled with black birds? This is what happens when they hear a gunshot…" That is wonderful! He said that he looks at painting as "a kind of adventure, a journey…" which Carol Giblin, co-owner of Church Street Gallery, has interpreted as "Quiet Truths."
The "Shot" heard round West Chester this week is by Terry DeAngelo.
I own one of DeAngelo's paintings and I love it. He has the light sensitivity and careful touch of a realist painter, but his composition allows for large spaces of color and makes his work very contemporary. He is a delightful person; very humble about his talent, and wonderful to talk with, so I think you should come to the reception and say hello. The show opening reception is Friday, July 11, at 5pm (please note that they never seem to have their receptions on First Friday! They go for second Friday instead.)
Several terrific locals shows are closing soon. In downtown West Chester, The Art Trust Gallery at Meridian Bank's show "Patterns" is in its last weeks. Please stop by and catch this delightful, small, eclectic collection of themed art works before it closes July 17th. Also, remember to visit the wonderful Horace Pippin show "The Way I See It" at the Brandywine Conservancy and Museum of Art . The last day of the landmark show is July 19! Horace Pippin lived and painted in West Chester and is famous for his direct portrayal of African American life in the 1940s and 50s, as well as his images of war. As a wounded war veteran of World War One, he learned to guide his painting arm with his left hand, painting or burning in his figures with deliberation and care. Come visit the show before it closes (and make sure to see the permanent collection too!)
In Kennett Square this week, Galer Estate Vineyard and Winery hosts its Thursday 'Pop Up' Art Show and will feature the photography of Sean Davis. Sean has recently begun a massive photographic documentation project that spans 30 miles along the Pennsylvania, Route 1 corridor from Avondale to the Main Line. In each township and borough along the way, he's capturing the beauty that makes each area unique. The images of specific buildings and landscapes brings a brief, written history of the image to tie in what the viewer is looking at. I am looking forward to seeing his images of our beautiful area! Several wineries in the area feature artist shows, so if there are other artists out there that would like to show at a winery for a day or more, just call one up and see if they have a spot on the calendar.
In Phoenixville, The Art Gallery at Franklin Commons is continuing their exhibition of "Flight" through July 16th. Some of the artists participating in the "Flight" Exhibit include Katee Boyle, Daisy Cohen, John Pompeo, Lynnette Shelley, Bob Hakun, and David Rankin. The Art Gallery at Franklin Commons is starting an educational and dialogue enriching Seeing Art by Numbers. This program is to encourage the understanding of and conversation around art, which becomes easier with some vocabulary. Through this new way of seeing and understanding artwork in the gallery, conversation is encouraged, and can be helpful to any level of artist or non-artists alike. The August Exhibit is "Roots", so if you are an artist that is inspired by the title, contact Katie Naber, curator, to see if your work may be included in the next show!
In Cochranville, art is literally rocks at The Street Road Artist's Space, located in an industrial building on the intersection of routes 41 and 926. This summer they are exhibiting "Sailing Stones" in their brand new 4500 foot barn. Sailing Stones shows work by 350+ contributors to the Street Road Rocks project, and by artists José Avila (Delaware), Egidija Čiricaitė (UK/Lithuania), Julia Dooley (Pennsylvania), Kaori Homma (UK/Japan), and their new participation in L.A.N.D., an ongoing project by Danish collective N55.
Each stone art piece takes a single rock as a starting point to an art work, and invites a conversation about permanence and transience, nature and art as well as the inclusivity of art in terms of medium and participation. The exhibit is opem Fridays and Saturdays 10am-4pm and by appointment. Along with their "Sailing Stones," summer exhibits include "Suburban Landscapes" by Brian Richmond, and "Crisis Farm: Seed to Table" by artists Maryann Worrell and Doug Mott. If you have a cool rock, you might consider bringing it to the show and see how yours might be included in the collection!
"Shack" by Sean Davis .
Many of you may have seen the news announcement about the sale of 4 masterpieces from The Delaware Art Museum's permanent collection that were sold in order to pay for the museum's renovation debt. Sales from collections are always met with negative press, but in a way, it should remind us how very precious each picture is in a museum and give us a greater appreciation of the artworks when we go to a museum. I am thrilled that the Delaware Art Museum is solvent and I am looking forward to a great year of exhibitions ahead.
Lele Galer is a local artist who has chaired numerous art shows, taught art history and studio art, public art and has chaired, written and taught the Art in Action Art Appreciation series for the UCFD schools for the past 12 years. She worked at the Metropolitan Museum of Art, and wrote for the Associated Press in Rome. She has been dedicated to Art History and art education for most of her adult life. Lele and her husband Brad own Galer Estate Winery in Kennett Square and she is on the Board of the Chester County Art Association and The Delaware Valley Art League.
Tags:Art Gallery at Franklin Commons · Art Trust Gallery at Meridian Bank · Art Watch · Brandywine Conservancy and Museum of Art · Church Street GalleryChester County Art Shows · Galer Estate Vineyard
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Transcriptomics/RNA analysis can be defined as the study of the transcriptome or the complete set of RNA transcripts which are produced by the genome, under specific circumstances, environment or in a specific cell - using high-throughput methods, such as microarray analysis. Globally increasing R&D activities in RNA sequencing and trascriptomics is expected to remain key growth driver for the RNA analysis market during the period of study.
This report identifies the RNA analysis/transcriptomics market size in for the year 2015-2017, and forecast of the same for year 2022. It also highlights the potential growth opportunities in the coming years, while also reviewing the market drivers, restraints, growth indicators, challenges, market dynamics, competitive landscape, and other key aspects with respect to RNA analysis/transcriptomics market.
Geographically North America dominated the RNA analysis/transcriptomics market owing to presence of global bio tech companies and higher investment into R& activities in the region. North America was followed by Europe and Asia-Pacific as the second and third largest markets for RNA analysis/transcriptomics. However Asia-Pacific is expected to remain the fastest growing market during the period of study driven by eonomic developments, increasing presence of bio tech companies and growing government funding to support new product developments are the key factors driving the growth of market in developing regions.
11.7. F. Hoffmann-La Roche Ltd.


Welcome to our newest Ruffians FC Fan!
Ruff Gaming » Ruffians FC » Ruffians FC - The Dressing Room » Welcome to our newest Ruffians FC Fan!
I just want to say a massive congrats to Rich (X Bulldog) and his fiancee Michelle on the birth of there son today - Tyler!
So from me (Uncle Ruff) and the rest of the Ruffians FC lads a massive congrats to you Rich!
Won't be long until hes scoring more goals than his oldman for the Mighty Ruff FC!!
Congrats to you both! Give him a controller now! He is probs better than me!
Change your life m8, greatest feeling in the world.
Great News! Congratulations to you both!
8 Re: Welcome to our newest Ruffians FC Fan!


6th International Scientific Conference on Bioaerosols, Fungi, Bacteria, Mycotoxins in Indoor & Outdoor Environments & Human Health – Call for Abstracts is Now Open!
Johns Hopkins Says Crazy Po' Black Mothers Worsen Child Asthma (As Opposed To Their Substandard Housing ????) Psychologists & "Non-profit" Medical Assoc Strike Again!


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