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text:
\section{Introduction}

In recent years, machine learning models have become increasingly popular as methods to approximate potential energy surfaces of molecules.
These models range from classical learning methods such as kernel methods to methods based on deep neural networks.\cite{GAPtutorial,Neuralnetworks_Scheffler2004,Neuralnetworks_BehlerParrinello2007,BehlerPerspective2016,ANI_IsayevRoitberg2017,Botu2015,RampiMLQMMM,Botu2016,Huan2017,Zhenwei2015,gubaev2018machine,SNAP_Aidan2015,CovariantKernelsSandro2016,Glielmo2018,schutt2018schnet,schutt2019schnetpack,grisafi2018symmetry,DeepMDZhang2018,UnkePhysNet2019}
A common denominator for these data-driven models is that they require an adequate training set in order to yield predictions of sufficient accuracy.
It is thus clear that informed and rational selection of training data is paramount to proper optimization of the data-efficiency of the machine learning models.

For machine learning models describing potential energy surfaces, two types of data seem particularly convenient as training data: single-point energies and atomic force vectors.
However, it has not yet been fully demonstrated when and to which degree the inclusion of force labels in the training set truly leads to an improvement of the accuracy of the trained model versus energy labels. It is even possible to find somewhat conflicting information in literature.
For example, the GDML and sGDML methods achieve state-of-the-art accuracy in certain cases for the MD17 benchmark dataset, despite training only on force labels, ignoring the energy labels.\cite{chmiela2017machine,Chmiela2019sGDML,Christensen2020FCHL19}
In contrast, the  HIP-NN neural network---when only trained on energy labels---ostensibly achieves similar predictive accuracy to the DTNN, SchNet and PhysNet neural networks, when these are trained on both forces {\em and} energy labels for 50K molecules from the MD17 dataset, despite the fact that HIP-NN is trained using far fewer training labels.\cite{Lubber2018HIP,DTNN2017,schutt2018schnet,UnkePhysNet2019}
Similarly, the family of ANI datasets have resulted in successful potential energy models for general organic chemistry solely trained on single-point energies on geometries distorted along normal-modes, although the corresponding gradient information would not have been much more costly to obtain at the density-functional theory (DFT) level employed.\cite{smith2017ani,ANI_IsayevRoitberg2017,smith2018less,smith2020ani1cxx}

Consequently, it is not clear from literature which is the best strategy for designing a dataset for the most accurate machine learning potentials for chemical compounds within a given budget of computational resources.
In order to shed more light on this matter, here we investigate how the predictive accuracy of machine learning models is affected by inclusion of derivative information---namely atomic force vectors in addition to the energy---in the training set.
More specifically, we investigate the nine possible combinations resulting from the inclusion of functions, derivatives, and functions plus derivatives in loss-functions used for training,  as well as in error measures used for evaluating the prediction error.

In order to \textit{learn}, the predictive accuracy of the trained machine \textit{must} strictly increase with an increasing amount of training data, aside from statistical variation.
This holds true as long as the training data is noiseless, and the machine learning model is flexible enough to be able to properly account for all variation in the data.\cite{vapnik2013book,QMLessayAnatole}
This has previously been demonstrated numerically with the leading term of the prediction error decreasing according to a power-law with number of training samples, $N$, for kernel ridge regression~\cite{vapnik1994learningcurves}, as well as for neural networks\cite{StatError_Muller1996}, i.e.:
\begin{equation}
    \mathrm{Error} \propto \frac{a}{N^b}
\end{equation}

\noindent Learning curves of functional machine learning models thus {\em must} behave linearly on log-log scales, and form, in the limit of large $N$, a robust tool to assess learning data-efficiency through comparison of the  ``offset,'' $\log\left(a\right)$, and ``slope,'' $b$.

This paper is structured as follows: we first describe our methodology and illustrate the idea of learning curve comparison  for a well-known toy system from mathematics, the well-described Himmelblau's function, in order to demonstrate how different types of training data affect the learning rate for a generic non-trivial 2-dimensional surface.\cite{himmelblau1972}
Next, we employ the same testing scheme for two different but common use cases in chemistry:
The first use case concerns generating a force field for a single molecule.
Here we demonstrate how using forces and energies in the training data affect the learning rate for the potential energy surfaces of 10 individual molecules from the MD17 dataset.\cite{chmiela2017machine,Chmiela2019sGDML}
In order to ensure that the underlying data is practically noise-free, we additionally present a revision of the MD17 where the energies and forces have been calculated using dense integration grids and tight self-consistent field (SCF) tolerances.

The second use case concerns generating machine learning models trained {\em across} chemical compound space, meaning that they can extrapolate to out-of-sample molecules not seen during training.
Also here, we investigate how the inclusion of forces in the training data helps the trained machine to generalize from points on the potential energy surfaces of known molecules to points on the potential energy surfaces of new molecules.\\

\section{Theory}
In order to achieve a fair comparison between models trained on different kinds of data (here energies and forces), we employ three closely related kernel-based regression models which are described in this section, as well as standard training protocols~\cite{AssessmentMLJCTC2013}.
Using these schemes, we ensure that models trained on different types of data will rely on the same kernel functions and representations while at the same time being exactly-determined with closed-form solutions.
This is done by defining loss functions in which basis kernel functions are placed on the training data, as well as placing kernel derivatives corresponding to any gradient or force information in the training data.
This yields a set of basis functions which guarantees that the model (within a given regularization) is able to perfectly align with the training labels.
Additionally, use of the force operator ensures that the resulting force fields are conservative force fields, which is important for molecular dynamics simulations, and that the predicted energies obey common physical relations such as rotational and translational invariances.

The three machine learning models which employ training on either energies (function) only, or  energies and forces (function and derivatives) simultaneously, or forces (derivatives) only are presented next.
Each model is characterized by its loss function $J$ which is minimized through regression. Note, that prediction errors can also be evaluated using different loss functions.
Conventionally, training and test error definition are identical,  and their evaluation only differs in being assessed on a training or test set, respectively.
As such, there are nine possibilities to combine the three loss functions in training and in testing.

While the relevant equations for each regressor can be derived both in the context of kernel ridge regression (KRR), as well as Gaussian process regression (GPR), we present them here in the notation most commonly used for the former.\cite{RasmussenWilliams,Glielmo2018,bpkc2010,Zhenwei2015,chmiela2017machine,Chmiela2019sGDML,christensen2018operators,Christensen2020FCHL19}
For more detailed derivations, we refer to the work by Bart\'ok and Cs\'anyi,\cite{GAPtutorial} as well as that of Mathias.\cite{sonjamathias}
We further note that the equations for the force-only regressor have recently seen a different derivation in the work on ``gradient domain'' machine learning (GDML) by Chmiela et al.\cite{chmiela2017machine}

\subsection{Energy-only Training}
In kernel ridge regression, the energy, $u$, of a query molecule, represented by $\tilde{M}$, can be expanded using a basis set of kernel functions placed on training samples.
That is,
\begin{equation}
        u\left(\tilde{M}\right) = \sum_i^\text{training} \mathpzc{k}\left(\tilde{M}, M_i \right)\alpha_i,
\end{equation}
where $M_i$ is the $i$-th molecule in the training set, $\alpha_i$ is the $i$-th regression coefficient with units of energy, and $\mathpzc{k}\left(\cdot,\cdot\right)$ is a function that relates two molecules through a similarity measure which typically depends on the specific functional form of the kernel function, the representation, and the metric.
Writing the above equation on matrix form yields:
\begin{equation}
        \mathbf{u} = \mathbf{K} \bm{\alpha}\label{eq:krr}
\end{equation}
Here, $\mathbf{u}$ is the vector containing predicted energies, $\mathbf{K}$ is the kernel matrix containing the pairwise kernel elements between the molecules in the training set and the predictions sets, and finally $\bm{\alpha}$ is the vector containing regression coefficients.
Bold lowercase letters indicate vectors and bold capital letters indicate matrices.
The specific kernel function used for the different experiments are given in Section \ref{sec:kernelfunctions}.
From Eq.~\ref{eq:krr}, a set of regression coefficients can be obtained by minimizing a loss function.
Using Thikonov regularization and minimizing the squared errors leads to the loss function which is the foundation of KRR:
\begin{equation} \label{eq:krr_lagrangian}
    J(\bm{\alpha}  \mid \{ \mathbf{u}^\mathrm{ref}\} ) = \tfrac{1}{2}\norm{ \mathbf{K} \bm{\alpha} - \mathbf{u}^\mathrm{ref} }_2^2
        + \frac{\lambda}{2} \bm{\alpha}^\top \mathbf{K} \bm{\alpha}
\end{equation}
This loss function contains a hyperparameter, $\lambda$, which determines the amount of L2-regularization for the regression coefficients stored in $\bm{\alpha}$, which in turn yields the well-known closed-form solution for training a KRR model:
\begin{equation}
         \bm{\alpha}  =  \left( \mathbf{K} + \mathbf{I} \lambda \right)^{-1} \mathbf{u}^\mathrm{ref}.\label{eq_krr_solution}
\end{equation}
These regression coefficients can then be used to predict energies (as in Eq.~\ref{eq:krr}), through evaluation of the relevant kernel matrix.
Via the definition of the force operator, i.e. the negative gradient of the energy with respect to the atomic coordinate vector, $\mathbf{r}$, the atomic force vector, $\mathbf{f}$, can be predicted by differentiation of Eq.~\ref{eq:krr}:
\begin{equation}
    \mathbf{f} \triangleq -\tfrac{\partial}{\partial\mathbf{r}} \mathbf{u} = \begin{bmatrix} -\frac{\partial}{\partial \mathbf{r}}\mathbf{K} \end{bmatrix}   \bm{\alpha}
\end{equation}
Note that this approach is as `naive' as general, and in principle any differential property can be learned this way as long as the representation accounts for all the variables which are being perturbed.\cite{christensen2018operators}

\subsection{Combined Energy and Force Training}\label{sec:gpr}
The second model investigated herein, is a model closely related to conventional KRR but which allows for training on both force and energies simultaneously.
Similar to how KRR can be used to construct a function that goes through the training points exactly, it is also possible to---at the same time---enforce the derivatives at those points.
Here, derivatives are enforced by including the derivatives (for example, forces) in the regression in addition to the function value (for example, energies).
In order to allow the model to match both the function values and derivatives of the training set exactly, it is necessary to increase the number of basis functions considerably, as the set of equations would otherwise be vastly overdetermined, as, for example, there are $3N$ force labels for each energy label.
One choice of extended basis comes from augmenting the set of basis functions in standard KRR with additional kernel functions corresponding to the kernel derivatives with respect to the coordinates of the training molecules.

The resulting set of equations for this problem looks as follows:
\begin{equation}\label{eq:gpr_derivative}
    \begin{bmatrix}
        \mathbf{u} \\
        \mathbf{f}
    \end{bmatrix} = \begin{bmatrix}
       \mathbf{K} && -\frac{\partial}{\partial \mathbf{r}^\intercal}\mathbf{K} \\
       -\frac{\partial}{\partial \mathbf{r}}\mathbf{K} && \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
    \end{bmatrix} \bm{\alpha}
\end{equation}
Here, $\mathbf{K}$ is again the kernel matrix containing the pairwise kernel elements between the molecules in the training set and the predictions sets, and is defined identially to $\mathbf{K}$ in Eq.~\ref{eq:krr}.
Similarly to the problem in Eq.~\ref{eq:krr}, this choice of basis functions ensures that the training kernel is square-symmetric and the  minimization of the regularized loss function has a convenient closed-form solution.
For a set of reference training energies and forces, the regression coefficients can here be obtained by minimizing the following set of squared errors with Thikonov regularization:
\begin{equation} \label{eq:gpr_lagrangian}
J(\bm{\alpha} \mid \{ \mathbf{u}^\mathrm{ref}, \mathbf{f}^\mathrm{ref}\}) = \frac{1}{2}\norm{ \begin{bmatrix}
       \mathbf{K} && -\frac{\partial}{\partial \mathbf{r}^\intercal}\mathbf{K} \\
       -\frac{\partial}{\partial \mathbf{r}}\mathbf{K} && \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
    \end{bmatrix}  \bm{\alpha} -  \begin{bmatrix}
        \mathbf{u}^\mathrm{ref} \\
        \mathbf{f}^\mathrm{ref}
    \end{bmatrix}  }_2^2
    + \frac{\lambda}{2} \bm{\alpha}^\top  \begin{bmatrix}
       \mathbf{K} && -\frac{\partial}{\partial \mathbf{r}^\intercal}\mathbf{K} \\
       -\frac{\partial}{\partial \mathbf{r}}\mathbf{K} && \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
    \end{bmatrix}  \bm{\alpha}
\end{equation}
The closed-form solution is, similarly to that of Eq.~\ref{eq_krr_solution},
\begin{equation} \label{eq:gpr_solve}
         \bm{\alpha}  =  \Biggl(  \begin{bmatrix}
           \mathbf{K} && -\frac{\partial}{\partial \mathbf{r}^\intercal}\mathbf{K} \\
           -\frac{\partial}{\partial \mathbf{r}}\mathbf{K} && \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
        \end{bmatrix} + \mathbf{I} \lambda \Biggr)^{-1} \begin{bmatrix}
            \mathbf{u}^\mathrm{ref} \\
            \mathbf{f}^\mathrm{ref}
        \end{bmatrix}.
\end{equation}
This definition guarantees that the regression problem is exactly determined, and allows for the machine-learned potential energy surface to match both the energy and the forces of each training molecule.

With the regression coefficients obtained through Eq.~\ref{eq:gpr_solve}, predicted energies can then be calculated via the set of kernel functions and kernel derivatives placed on the training set:
\begin{equation}\label{eq:gpr_energy}
    \mathbf{u} =
    \begin{bmatrix}
        \mathbf{K} \quad -\frac{\partial}{\partial \mathbf{r}^\intercal}\mathbf{K}
    \end{bmatrix} \bm{\alpha}
\end{equation}
Likewise, forces can be obtained by taking the relevant derivative of the energy, i.e.:
\begin{equation}\label{eq:gpr_force}
        \mathbf{f} \triangleq -\tfrac{\partial}{\partial\mathbf{r}} \mathbf{u} = \begin{bmatrix}
       -\frac{\partial}{\partial \mathbf{r}}\mathbf{K} && \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
    \end{bmatrix} \bm{\alpha}
\end{equation}

\subsection{Force-only Training}
The third model presented is a model which constructs a potential energy surface from information about derivatives only.
This model is similar to the model described in Section~\ref{sec:gpr} except that, in this case, only force labels are used in the training step, and, consequently, the set of basis functions is comprised of only the corresponding kernel derivatives.

In this approach, we start with the following equation, in which the kernel matrix is formed by the double derivatives of pair-wise kernel functions:
\begin{equation}
    \mathbf{f} =         \begin{bmatrix}
    \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
        \end{bmatrix}  \label{eq:gdml_force}
    \bm{\alpha}
\end{equation}
For a set of reference force labels, $\mathbf{f}^\mathrm{ref}$, the corresponding regression coefficients are obtained analogously to the energy-only and energy+force examples in the previous sections.
\begin{equation} \label{eq:gdml_lagrangian}
J(\bm{\alpha}\mid \{\mathbf{f}^\mathrm{ref}\}) = \tfrac{1}{2}\norm{ \begin{bmatrix}
    \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
        \end{bmatrix}  \bm{\alpha} - \mathbf{f}^\mathrm{ref} }_2^2
    + \tfrac{\lambda}{2} \bm{\alpha}^\top \begin{bmatrix}
    \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
        \end{bmatrix}  \bm{\alpha}
\end{equation}

\begin{equation}
         \bm{\alpha}  =  \biggl(  \begin{bmatrix}
        \frac{\partial^2}{\partial \mathbf{r}\partial \mathbf{r}^\intercal}\mathbf{K}
            \end{bmatrix} + \mathbf{I} \lambda \biggl)^{-1} \mathbf{f}^\mathrm{ref}.
\end{equation}
This is also sometimes referred to as training in the ``gradient domain''.\cite{chmiela2017machine}

Energies can then be predicted through direct integration of Eq.~\ref{eq:gdml_force}, which allows the scalar field to be determined up to an integration constant:
    \begin{equation}
        \mathbf{u} = \begin{bmatrix} -\frac{\partial}{\partial \mathbf{r}^\intercal}\mathbf{K} \end{bmatrix}   \bm{\alpha} + \mathrm{constant~term}
    \end{equation}
For problems where only a single surface is of interest---for example the potential energy surface of a given stoichiometry---the integration constant is rarely of any importance, and can also be inferred by predicting the energies for the training data.
Unfortunately, however, this approach is less practical when multiple surfaces are of interest, for example for datasets involving the potential energy surfaces of multiple different molecules.
Note that the number of possible stoichiometries is known to grow exponentially with elementary particle number~\cite{anatole-ijqc2013}.
Hence, force-only training makes it hard if not impossible to train models that directly predict energies for molecules of varying size and chemical composition, if the energy of the molecules are of interest.
Of course, composite approaches, e.g.~using dressed atom models~\cite{BOB}, can still be used to rectify such shortcomings.

\subsection{Representations and Kernel Functions}\label{sec:kernelfunctions}
Since we present machine learning models trained on (i) an analytical function (Himmelblau's function) and  on  (ii) several models of molecular energetics trained non-equilibrium geometries, different kernel and representation choices have been made.
\subsubsection{Representing Himmelblau's Function}
Here, Himmelblau's function is used as a toy-model for learning complicated surfaces.\cite{himmelblau1972}
This allows for thorough benchmarking of the regressors used herein.
Himmelblau's function is a multi-modal function with one local maximum and four minima and bears some resemblance to the potential energy surface of a simple molecule with four conformational minimia.
The function is defined as:
\begin{equation}
u\left(x, y\right) = \left( x^2 + y - 11\right)^2 + \left(x + y^2 -7\right)^2.
\end{equation}
Points on the surface are represented here as their $xy$-coordinate pair, that is,~$\mathbf{q}_i = \left[ x_i, y_i \right]^{\top}$.
A Gaussian kernel function is used to compute the kernel elements for Himmelblau's function, i.e.:
\begin{equation}\label{eq:kernel_himmelblau}
     \mathbf{K}_{ij} = \exp\left(-\frac{\| \mathbf{q}_i - \mathbf{q}_j \|^2_2}{2\sigma^2}\right) \quad \text{for Himmelblau's function}
\end{equation}
This choice of representation and kernel function makes is straightforward to implement and enables the computation of necessary kernels and derivatives analytically.

\subsubsection{Representing Molecules}
To represent the environments of an atom in a molecule, we rely on the computationally more efficient variant of the Faber-Christensen-Huang-Lilienfeld (FCHL) representation\cite{FCHL}, namely the FCHL19 representation \cite{Christensen2020FCHL19}.
Briefly, this representation is a vector which contains histograms of the radial distributions of atoms and a number of Fourier terms describing angular distributions of atoms in the environment of a certain atom.\cite{faber2017alchemical}
In principle, any continuous representation that generalizes across chemical space could have been used for this purpose.
In addition, we use the localized kernel \textit{ansatz} in which the kernel elements between two molecules correspond to the pairwise sum over the kernel functions between the respective representations of atomic environments in the two molecules.\cite{bpkc2010,barker2016lcgap}
This makes it possible to train models that span molecules of varying size and chemical composition.
The following Gaussian kernel function is used throughout:
\begin{equation}\label{eq:kernel_screening}
     \mathbf{K}_{ij} =  \sum_{I \in i}  \sum_{J \in i} \delta_{Z_I Z_{J}}  \exp\left(-\frac{\| \mathbf{q}_I - \mathbf{q}_{J} \|^2_2}{2\sigma^2}\right) \quad \text{for molecules}
\end{equation}
where $\mathbf{q}_I$ and $\mathbf{q}_{J}$ are the representations of the $I$'th and $J$'th atoms in the molecules $i$ and $j$, respectively, $\delta$ is the Kronecker delta, with $Z_I$ and $Z_J$ being the atomic numbers of each atom, respectively.

\section{Results}

In this section, we present numerical evidence which demonstrates the effects on the learning rate of including derivative labels in the training data.
This section is organized as follows: first, we establish the generality of our numerical experiment by learning the surface of a simple two-dimensional function, unrelated to molecules or chemistry.
Next, we demonstrate the same principles applied to two distinct use cases, namely (1) training a model for the potential energy surface for a single molecule, and (2) training a general model for the potential energy surfaces of a number of molecules of different size and chemical composition.

\subsection{Toy system: Learning Himmelblau's Function}
\begin{figure}
    \centering

    \includegraphics[width=\linewidth]{lc_fit.pdf}
    \caption{\label{fig:himmelblau} Panel (A) Displays the surface of Himmelblau's function, and (B) the gradient norm of the surface. Panels (C) and (D) display the MAE prediction and MAE gradient prediciton, respectively, of the surface of Himmelblau's function as a function of the training set size. Three different sets of training labels are used: green is trained on only the function values, red is trained only on function gradient components, and finally blue is trained using both simultaneously.}
\end{figure}

In this section we investigate the learning curves of machines trained to predict Himmelblau's Function.
This 2D surface has four local minima and is displayed in Fig.~\ref{fig:himmelblau}, as well as the norm of its gradient.
Here, Himmelblau's function serves as a toy system to demonstrate the effects of including function derivatives in the training procedure.

We generate a dataset by selecting random points from the surface.
The test set consists of 10,000 points sampled randomly and uniformly across the interval $ \{x_i, y_i\} \in \left[-6.0; 6.0 \right]$, with training sets of varying sizes sampled randomly and uniformly across the same interval.
For each training set size $N \in \{25, 50, 100, 200, 400, 800, 1,600\}$, 100 training and test sets are created using 100-fold random sub-sampling cross-validation with a constant test set size of 10,000 points.
This extensive cross-validation was necessary in order to have well-converged averages over the folds, as the test errors were observed to vary up to one order of magnitude.
In total over 350,000 individual machines were trained in order to obtain the presented learning curves.

Three different models were trained for the surface: the first model trained on only the function values, the second only trained on the function derivatives at the training points, and the third model was trained on both the function values and function derivatives simultaneously.
As expected,\cite{vapnik1994learningcurves,StatError_Muller1996} the three machines yield learning curve fits for the predicted mean-absolute-error (MAE) function value with similar slopes when plotted on a log-log scale, as displayed in Fig.~\ref{fig:himmelblau}C.

The average difference between the two models which include forces labels in the training set is much smaller than the standard deviation of the 100 folds used to calculate the average.
Compared to training on function values exclusively, the average decrease in the off-set of the learning-curve was found to be a factor of 7.0 for the model trained on only function derivatives, and 7.2 for the model trained on both function values and function derivatives.

\begin{table}
     \centering
     \caption{Slopes and offsets for predictions of the MAE value and MAE gradient component for Himmelblau's function, based on three machine learning methods trained on different types of data. \label{tab:slopes_himmelblau}}
  \begin{tabular}{llrrrr}
    \bottomrule
                           & \multicolumn{2}{l}{MAE Value} & \multicolumn{2}{l}{MAE Gradient}  \\
        Training data          & Slope  & Offset & Slope  & Offset \\
        \midrule
        Values             & -2.45  & 8.49 & -2.05  & 7.14 \\
        Values and gradients   & -2.33  & 5.81 & -1.96  & 4.76 \\
        Gradients          & -2.43  & 6.43 & -2.06  & 5.16 \\
        \bottomrule
      \end{tabular}
    \end{table}

Learning curves for gradient predictions show similar trends among the three models (see Fig.~\ref{fig:himmelblau}D):
the average decrease in the offset was found to be a factor of 7.9, when training on gradients only, and 7.0 when training on both function values and gradients simultaneously, compared to training only on function values.

Fitted slopes and offsets for all six learning curves are displayed in Table \ref{tab:slopes_himmelblau}.
We note that the 95\% confidence intervals of the fitted learning curves for the two models that include gradients in the training loss-function are mostly overlapping, and while slopes and offsets differ somewhat (see Table \ref{tab:slopes_himmelblau}), the difference between the two models is not statistically significant.

\begin{table}
\footnotesize
     \centering
     \caption{Learning curves for three machines trained on different types of data, and the same loss functions evaluated on a test set of 10,000 points on the surface of Himmelblau's function. The three machines are trained on either function values, function values and gradients simultaneously, or gradients only. The loss functions used to train each machine is denoted in the left column, followed by the slopes and offsets for the resulting learning curves for three loss functions evaulated on the test data. The three loss functions are defined in the text in Eqn.~\ref{eq:krr_lagrangian},~\ref{eq:gpr_lagrangian}, and~\ref{eq:gdml_lagrangian}, respectively. \label{tab:cost_functions} }
  \begin{tabular}{llrrrrrr}
  \toprule
Training & & \multicolumn{2}{l}{Test loss} & \multicolumn{2}{l}{Test loss} & \multicolumn{2}{l}{Test loss} \\
 \cmidrule(lr){1-2}\cmidrule(lr){3-4}\cmidrule(lr){5-6}\cmidrule(lr){7-8}

&  & \multicolumn{2}{l}{$J(\bm{\alpha} \mid \{ \mathbf{u}^\mathrm{test}\})$}
  & \multicolumn{2}{l}{$J(\bm{\alpha} \mid \{ \mathbf{u}^\mathrm{test}, \mathbf{f}^\mathrm{test}\})$}
  & \multicolumn{2}{l}{$J(\bm{\alpha} \mid \{ \mathbf{f}^\mathrm{test}\})$} \\
Training data &  Training loss     &   Slope & Offset     &   Slope & Offset     &   Slope & Offset \\
   \midrule/
Values  &$J(\bm{\alpha} \mid \{ \mathbf{u}^\mathrm{ref}\})$ &	-4.59 &	27.06	&   -3.74 &	25.19 & -3.63 &	24.41 \\
Values and gradients &$J(\bm{\alpha} \mid \{ \mathbf{u}^\mathrm{ref}, \mathbf{f}^\mathrm{ref}\})$ &	-4.41 &	22.35	&   -3.58 &	20.80 & -3.49 &	20.25 \\
Gradients  &$J(\bm{\alpha} \mid \{ \mathbf{f}^\mathrm{ref}\})$ &	-4.60 &	23.10	&   -3.78 &	21.51 & -3.69 &	20.91 \\
        \bottomrule
      \end{tabular}
    \end{table}

Also for learning curves using loss functions we find agreement with the power-law behavior that is expected from models trained on function values,\cite{vapnik1994learningcurves,StatError_Muller1996} here demonstrated for models trained on function gradients.
In Table~\ref{tab:cost_functions}, we present resulting slopes and offsets for loss-function learning curves for the three types of trained machines trained using three different sets of training data and corresponding loss functions (rows).
These learning curves are shown graphically in Fig.~\ref{fig:lc_fit_cost} in the Supplementary Information.

Results for the corresponding three test loss functions (columns) are obtained using a test set of 10,000 points randomly selected from the surface of Himmelblau's function.
Here, the same trends as for the MAE learning curves are observed.
The resulting slopes from the three types of trained machines do not differ with statistical significance, but instead depend on the type of test data.
In contrast, the offsets depend strongly on the training loss function and training data.
More specifically, the inclusion of gradients in the training loss function is on average beneficial when predicting function values, function and gradient values, as well as gradient values alone.
We also remind the reader that the slope of these learning curves are independent of the units of the labels, as this quantity is folded into the offset.
Despite using hundred-fold cross-validation, we do not observe a statistically significant difference between any of the three test predictions when using training loss functions and data sets based on function values and gradients, or gradients alone.
We also note that the learning curves for the MAE error are observed to follow the expected power-law behavior over a range that spans 6 and 5 orders of magnitudes for predicted function values and predicted function gradient components, respectively.

We have thus demonstrated for the Himmelblau function, that MAE error in this case can be decreased 7-fold by inclusion of labels that correspond to function derivatives in the training set.
At the same time, however, we find negligible improvement upon inclusion of function values, in addition to the gradient information in the training algorithm.
Interestingly, these results suggest that when learning a single function surface, using derivatives as training labels is more advantageous than actual function values.

\subsection{Use case 1: Learning the PES for a one molecule}

    \begin{figure}
        \centering
        \includegraphics[width=\linewidth]{md17_energy_curve.pdf}
        \includegraphics[width=\linewidth]{md17_force_curve.pdf}
        \caption{Learning curves for machines trained on three different datatypes for the 10 molecules in the revised MD17 dataset, namely machines trained on energies only, forces only, and both forces and energies.
        Data for five folds are plotted as a scatter plot for each training set size, and a linear fit for each curve is plotted in addition.
        In (A) the mean absolute error (MAE) of the predicted atomization energy is presented for each molecule as a function of the number of molecules in the training set, while (B) shows MAE of the predicted force components for the same folds.\label{fig:md17_fits}
            }
    \end{figure}

In this section we discuss the accuracy of energy and force predictions from three models trained on the revised MD17 dataset.\cite{chmiela2017machine,Chmiela2019sGDML}

For each of the 10 molecules in the revised MD17 dataset, three models are trained using either only the energy labels of each sample, only the force labels, or both types of labels simultaneously.
For each of the three models, the training set size $N$ is varied for $N \in \{ 50, 100, 150, 200, 400, 600, 800, 1000\}$.
The resulting MAE of energy and force predictions for each molecule can be found in Table \ref{tab:md17si} in the Supplementary Information, and plots with linear fits to the learning curves are shown in Fig.~\ref{fig:md17_fits}.
As it is clear from the results, and similarly to the example with Himmelblau's function, including the forces in the training procedure improves the prediction of both energies and forces.
On average, the MAE of the predicted energies and force components is reduced about 7-fold for the same number of training samples.
Regarding the inclusion of energies in addition to forces, we find, again in complete analogy to the Himmelblau function, that it makes little difference.
In all cases, the largest difference was found to be 0.001 kcal/mol MAE predicted energy, and 0.002 kcal/mol/\AA~MAE predicted force components at the largest training set size of $N$=1000 samples.

In Fig.~\ref{fig:md17_labels} the MAE predicted energy is plotted as a function of the number of training labels for each of the 10 molecules using the three models described earlier.
Here, the slopes and offsets of the learning curve are close to identical in all 10 cases, regardless of what data was used to train the models.
This suggests that for this dataset, one force label yields as much improvement in the predictive accuracy of a machine learning model as one energy label.
Thus, a model for a specific molecule, trained on a certain number of energies, will have roughly the same predictive accuracy as one that is trained on the same number of force labels.
As a consequence, in such cases---and if force-evaluations represent computationally negligible overhead (common within, for example, DFT)---the number of independent quantum calculations necessary to generate the training data required to reach a certain predictive power, can be reduced by a factor of 3$N$ thanks to the inclusion of forces.

We find that in all but one case, the learning curves for both force as well as energy prediction follows the expected power law, meaning they display a linear relationship on a log-log scale.
In the case of benzene, the off-set of the learning curve is extremely low, but the slope is inconsistent with the expected power-law.
It seems unlikely that underlying numerical noise in the training data is at the origin of such premature saturation: the learning curves for Uracil are equally low and do not suffer from lack of linearity.
The more likely culprit appears to be the specific parameterization of the FCHL19 representation achieving only insufficient uniqueness, a possibility recently pointed out by Pozdnyakov et al.\cite{pozdnyakov2020completeness}.
The addition of higher order terms, such as for example 4-body terms, into FCHL~\cite{FCHL} might rectify this issue.
Furthermore, we note that the parameters in the representation have not been re-optimized for discriminating the subtle difference between the very similar structures in this, but rather work across molecules of varying sizes and chemical composition.
It is thus possible that they could be re-optimized to alleviate such issues to some extent.

\subsection{Use case 2: Training models across chemical composition}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{amons_curve.pdf}
    \caption{\label{fig:amons_learning} Energy and force learning curves machine learning models trained on a dataset consisting of non-equilibrium conformers for 1,595 small, organic molecules.
    Two models are trained, one including only the energy labels of each non-equilibrium conformer, and one additionally including the corresponding force labels.
    Panel (A) shows the MAE predicted atomization energy as a function of the number of molecules in the training set, while (B) shows the same predicted quantity, but as a function of the total number of training labels (i.e.~the total number of energy and force components in the training set) for each machine.
    The MAE predicted force components are displayed in panels (C) and (D) as a function of the number of molecules in the training set (C) as well as the total number of training labels (D).}
\end{figure}

\begin{table}
    \footnotesize
     \centering
     \caption{Slope and offsets (see text) of linear fits to mean absolute error (MAE) of predicted energies and gradients versus the training set size, on a log-log scale. The dataset consists of 1,595 small organic molecules in nonequilibrium conformation (see text).  The column ``Training'' denotes what data was used to train the models. Lastly, the column ``$N$ unit'' denotes the unit is used for the abscissa of a given learning curve. [Molecules] denotes that the unit is simply the number of molecules in the training set, that is for energy-only training, one training label (the energy) is used per molecule, while for energy+forces, 3$N$+1 training labels (one energy and $3N$ force components) are used for each molecule, whereas [Labels] denotes that the total number of $3N+1$ labels are used on the abscissa.\label{tab:slopes_amons}}
  \begin{tabular}{llrrrr}
    \toprule

    &  & \multicolumn{2}{l}{MAE Energy [kcal/mol]} & \multicolumn{2}{l}{MAE Force [kcal/mol/\AA]}\\
    Training        &    $N$ unit &  Slope   & Offset & Slope & Offset\\
    \midrule
    Energy Only     & [Molecules] & -0.44  &  4.35 &  -0.23  &  4.01\\
    Energy and Forces & [Molecules] & -0.45  &  4.47 &  -0.31  &  3.29\\
    Energy and Forces & [Labels]    & -0.45  &  6.16 &  -0.31  &  4.46\\

    \bottomrule
  \end{tabular}
\end{table}

In our second use case scenario, we investigate how learning forces and energy across chemical compound space behaves as a function of the number of training molecules and the type of training data.
Fig.~\ref{fig:amons_learning} shows learning curves for force and energy predictions on a dataset consisting of non-equilibrium conformations of 1,595 small, organic molecules.
The training set is divided such that every (distorted) molecule of a given chemical composition is only seen once in either the training or test sets.
Using this data, two types of models are trained: one model is trained using only atomization energy labels, and a second model is trained using both atomization energy as well as the corresponding 3N components of the force vector.
As the model trained on only forces would only integrate the energy up to an arbitrary constant, the predicted energies would not be meaningful, and therefore this model is not used in this section.

Fig.~\ref{fig:amons_learning}A shows the MAE predicted atomization energy for the two models as a function of the number of molecules involved.
The learning rates hardly differ, with the largest deviation between the two curves being 3\% of the average MAE.
While this seems surprising in light of the effect of including forces for the MD17 dataset discussed in the previous section, this result has a simple explanation.
Since the gradient information only provides information about the underlying function up to an integration constant, the additional information is useless in providing information about the differences between the individual potential energy surface on which all the molecules are located.
In this case, this information is provided solely by the atomization energies and, in the end, the dominant error in the learned atomization energies comes from learning energy differences between the different constitutions of atoms in each molecule, and not from more subtle changes in molecular conformation.
The learning rates are displayed in Table \ref{tab:slopes_amons}, where the slopes and offsets for energy predictions are very close for the two different models.
For energy predictions, the slopes of the learning curves are -0.44 and -0.45 for the models trained on only energies and the models  simultaneously trained on energies and forces, respectively.
For force prediction, however, the slopes differ somewhat, at -0.23 and -0.31, for the same two models, respectively, indicating that the model trained on both energies and forces might be better at very large training sets.
We note, however, that the current training set is not large enough to confirm this, as this would require a force training set with at least 10,000 to 50,000 training molecules (See Fig.~\ref{fig:amons_learning}D), which, in turn, would render this experiment computationally intractable with a kernel matrix of up to around 2,000,000 $\times$ 2,000,000 elements.

When the curves are viewed as a function  of the total number of training labels (in Fig.~\ref{fig:amons_learning}B), the energy-only model requires about 51 times fewer labels to reach the same accuracy as the model trained on both forces and energies.
This indicates,
that the additional force labels do not aid the model in predicting energies of unseen molecules.
This is likely due to the fact that forces can only determine the potential surface up to an integration constant, which can only be learned using energy labels.

Next, in Fig.~\ref{fig:amons_learning}C, the MAE predicted force components are plotted for the two models as a function of the number of molecules in the training set.
On average, including the forces in the training set reduces the predicted MAE force component by a factor of 3.5.
Essentially, while the additional gradient information is not able to improve learning beyond the integration constant, it does help improve the prediction of the relative energy landscape of each potential energy surface, in turn leading to improved gradient predictions, but at the cost of much larger number of training labels.
Fig.~\ref{fig:amons_learning}D displays the same MAE predicted force components as in Fig.~\ref{fig:amons_learning}C, but as a function of the total number of training labels.
In this case, we observe that the two learning curves are very close-lying, similarly to what was found for the revised MD17 dataset in the previous section.
This suggests that for the training sizes investigated herein, one training energy label is worth roughly the same as one training force component label, although the differences in slopes indicate that the model trained on energies and forces might reach superior accuracy at much larger training sizes where the role of the integration constant is diminished.

\section{Methodology}
\subsection{Hyperparameter Selection and Learning Curves}

All learning curves were generated using nested cross-validation as implemented in \texttt{scikit-learn}\cite{scikit-learn} via the following recipe:
First, the datasets were randomized.
Secondly, the datasets were divided into 100 folds using random subsampling for Himmielblau's function, while datasets consisting of molecules were randomly divided into 5 folds using the \texttt{KFold} class implemented in \texttt{scikit-learn}.
Next, a grid-search with 4-fold cross-validation within the training set extracted from the fold was used to select the optimal choice of the hyperparameters $\sigma$ (the kernel width) and $\lambda$ (regularization strength), in order to avoid overfitting.
In order to select hyperparameters that simultaneously work well for both force and energy prediction, the following score function was used to select these:\cite{schutt2018schnet}
    \begin{equation}
       \mathcal{L}  = 0.01 \sum_i \left( U_i - \hat{U}_i \right)^2 + \sum_i \frac{1}{n_i} \| \mathbf{F}_i - \mathbf{\hat{F}}_i \|^2\label{eq:cost_function}
    \end{equation}
Here, $U_i$ is the energy of the $i$'th molecule, and $\mathbf{F}_i$ and $n_i$ are the force-vector and number of atoms in the same atom, respectively.
In the cases where either force/function derivatives or energies/function values were not included in the training data, the first or the second term was left out, respectively.

For the grid searches for  Himmelblau's function, the kernel width was tested in the range $\sigma \in \{0.4, 0.8, 1.6, 3.2, 6.4, 12.8, 25.6\}$, and the regularization strength was tested in the range $\lambda \in \{ 10^{-13}, 10^{-12}, 10^{-11}, 10^{-10}, 10^{-9}, 10^{-8}\}$.
For the molecular datasets (see next section), the grid-searches used values of $\sigma \in \{0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0, 32.0 \}$ for the kernel width, and $\lambda \in \{10^{-12}, 10^{-11}, 10^{-10}, 10^{-9}, 10^{-8}, 10^{-7}, 10^{-6}\}$ for the regularization strength.

\subsection{Software}
All machine learning models were implemented in \texttt{Python} using the \texttt{QML} machine learning package.\cite{qmlcode2017}.
The packages \texttt{Matplotlib} and \texttt{Seaborn} were used to plot all figures, and \texttt{Scikit-Learn} was used to obtain the linear fits to learning curves.\cite{hunter2007matplotlib,seaborn,scikit-learn}

\subsection{Datasets}
This subsection briefly presents the used datasets and their availability.

\subsubsection{Revised MD17 Dataset}
For each of the 10 molecules in the MD17 dataset\cite{chmiela2017machine,Chmiela2019sGDML} (aspirin, benzene, ethanol, salicylic acid, malonaldehyde, toluene, naphthalene, uracil, paracetamol, and azobenzene), 100,000 structures were randomly selected from the available MD trajectory data.
For each of these structures, a single-point force and energy evaluation was carried out at the DFT level.
All calculations were performed in ORCA 4.0.1, using the PBE functional and the def2-SVP basis set with the resolution-of-identity (RI) approximation for the Coulomb integrals.\cite{orca4,PBE,Weigend2005}
In order to have a minimal unsystematic error, the keyword \texttt{VeryTightSCF} was used to reduce the error from SCF convergence, while the keywords \texttt{Grid7} and \texttt{NoFinalGrid} were used to utilize the largest standard grid implemented in ORCA.
This data has been uploaded to \url{https://dx.doi.org/10.6084/m9.figshare.12672038} along with the indices used for the outer 5-fold cross-validation.

\subsubsection{Small organic molecules}
\begin{figure}
    \centering

    \includegraphics[width=\linewidth]{amons_distribution.pdf}
    \caption{\label{fig:amons_data}The distribution of various properties of the dataset of small organic molecules.\cite{christensen2018operators} (A) shows the distribution of molecular sizes, while (B) shows the a kernel density estimate (KDE) plot of the distribution of atomization energies of the molecules, and lastly (C) shows the KDE plot of the force components in the dataset.}
\end{figure}
This dataset is taken from Ref.~\citen{christensen2018operators} and consists of non-equilibrium conformers of 1,595 small organic molecules with up to 7 atoms of the elements CNO saturated with hydrogen atoms.
This data is available for download at \url{https://dx.doi.org/10.6084/m9.figshare.7000280}.
For each of these structures generated via normal-mode sampling, the atomization energy and corresponding forces were provided at the $\omega$B97xD/6-31G(d) level of theory.\cite{Chai2008}
Fig.~\ref{fig:amons_data} shows plots of the distributions of molecular sizes, energies, and forces in the dataset.

\section{Conclusion and Outlook}
In this paper we have presented numerical confirmation that the predictive error of machines trained on function derivatives follows a power law, a result that is well-known and to be expected for both neural networks as well as kernel models trained on scalar values.
Our numerical results demonstrate this power-law behaviour for two popular and distinct types of molecular datasets, as well as for the Himmelblau's function for which
6 orders of magnitude have been spanned.

For datasets that deal with a single surface, such as Himmelblaus's function and the potential energy surfaces of 10 molecules in the MD17 dataset, we find that including force labels in the set of training labels---in addition to energy labels---leads to an improvement of the predictive error when the same number of training points are used.
At the same time, however, we only find negligible differences between models trained on force labels only and models trained on both energy and force labels.
Regarding the number of total training labels (rather than number of molecules in the MD17 dataset), we found that the predictive error of forces and energies was close to identical, regardless of whether the model was trained on energies only, trained on both energies and forces, or trained on forces only.

For the diverse dataset of distinct organic molecules, we find that the prediction error of energies does not improve upon addition of force labels to energy labels in the training set, in contrast to what was observed for Himmelblau's function and the MD17 dataset.
Conversely, the predictive error of atomic forces is greatly improved by including forces in the training set.
The likely explanation is that geometrical derivative information only helps determine the surface up to an integration constant, corresponding to the atomization energy.
This effect is more visible when viewing the learning rate as a function of the number of training labels. In this case, energies are learned with about $20\times$ less training labels, training on energies only, compared to training on both forces and energies.
At the same time, the predictive force error when training on either energies, or forces and energies simultaneously, seems to decay at comparable rates with the number of force labels in these two scenarios.
In order to use derivative information to further improve this learning, it would be necessary to use \textit{alchemical} derivatives, i.e. the derivatives of the energy with respect to the atomic charges~\cite{anatole-prl2005,anatole-ijqc2013} which form the basis of alchemical perturbation density functional theory~\cite{guido2020apdft,guido2020alchemy}
These derivatives describe the change in energy as one molecule is alchemically transformed to another molecule, and could allow for better interpolation between potential energy surfaces of different molecules in a training set.

We believe that our observations have implications for the generation of new molecular datasets.
Here we point out that it is necessary to account for the cost of obtaining the training labels.
With this in mind, it is beneficial for use cases of the MD17-type to include forces labels in the training set when the cost of acquisition is less than the cost of $3N$ single-point energy calculations on un-correlated samples, where $N$ is the number of atoms in the molecule.
This suggests that for DFT-based datasets, it seems very favorable to calculate and report gradients in addition to single-point energies, while for methods with more costly gradients (compared to the energy evaluation) this becomes less favorable.

For energy predictions throughout large datasets used to fit models for general chemistry, i.e.~throughout chemical space, such as, for example, the family of ANI datasets\cite{smith2017ani,ANI_IsayevRoitberg2017,smith2018less,smith2020ani1cxx}, it seems more valuable to build the most compact model using a compositionally diverse training set with only single-point energies, rather than ``wasting'' coefficients by training also on forces for more conformations of the same molecule.
If the goal, however, consists of {\em also} modeling forces, such as is typically the case for relaxing geometries throughout chemical compound space, our results indicate that the addition of forces (if acquisition cost is lower than for energies) in the training set is always beneficial.
This conclusion, however, also depends on the requirements for execution speed and the availability of training data: Models trained on force labels can be computationally substantially more expensive to train and execute compared to models on the same number of energy labels, since this often involves the derivative of, for example, a kernel matrix or a neural network.
Consequently, in an application scenario where sufficient energy labels are available, it might be best to train on energy labels only, as this enables numerically less complex training models.
Considering the prediction times for kernel-based models, it is also much more computationally expensive to evaluate kernel functions placed on derivatives compared to those placed on scalars.
If the ultimate goal is to have very fast prediction times for kernel-based models, it seems worthwhile to consider the use of kernel-based force models which do not require the evaluation of second-order kernel derivatives.

All these observations summarize our insights into the design of future data-driven models and their underlying dataset generation. We believe that they are applicable to all branches of machine learning where the goal is to learn multidimensional, differentiable function surfaces.

\section*{Acknowledgements}
This work was partly supported by the NCCR MARVEL, funded by the Swiss National Science Foundation.
O.A.v.L. acknowledges funding from the Swiss National Science foundation (407540\_167186 NFP 75 Big Data) and from the European Research Council (ERC-CoG grant QML).
The authors thank Puck van Gerwen for helpful comments on the manuscript.
We further acknowledge the use of the following software: NumPy and the F2PY tool\cite{numpy}, and OpenMP.\cite{openmp08}

\section*{References}
\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


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Under Shah Jahan, the Fort denotes the zenith of Mughal development, joining the Persianate castle working with Indian traditions. The Red Fort houses numerous exhibition halls that grandstand various significant antiquities like the Delhi Science Museum.
The Fort was held onto by British Colonial Rulers, who attacked and took a few significant fortunes, including the Kohinoor jewel, Shah Jahan's Jade Wine Cup, and Bahadur Shah II's crown. They expected to crush the Fort in stages, annihilating goods, gardens, collection of mistresses condos, and worker quarters.
Practically the interior development was all crushed, aside from the white marble structures. After the Britishers left India, state leader Mr. Jawaharlal Nehru raised the banner. From that point forward on each Freedom Day and on Republic Day, the Indian Prime Minister raises the flag.
One of the intriguing realities about the Red Fort is that the red tone was rarely present. It was just painted red by the Britishers after the limestone started to flake off. The name Red Fort comes from the English period, when they painted it red to safeguard it and thus renamed it Red Fort, which subsequently meant Lal Qila, and which signifies "people's property".
Red Fort Architecture
Red Fort Delhi is the making of draftsmen Ustad Ahmad and Ustad Hamid. Structurally, Red Fort or the Lal Quila outperforms numerous landmarks of the world with its painstakingly arranged format and noteworthy construction. For example, in the event that the design must be compared, Delhi Red Fort is arranged in a preferable manner over the Red Fort of Agra.
The main thing that you will see about the Red Fort is its gigantic construction and its red tone. The stone utilized in the Red Fort is red sandstone, which brought about the block red shade of the Fort. The Fort was utilized for many purposes remembering which it has been fabricated.
The tallest construction of the Fort is around 33 meters high. The Delhi Red Fort is looking like a sporadic octagon, containing long east and west sides alongside two fundamental entryways on the western side and the southern side. The walls of the Fort alongside a couple of parts have been developed with red sandstone while the rest of the part has been fabricated utilizing marble.
A portion of the noticeable segments of the Fort that make certain to leave you charmed is Diwan-I-'Aam, Diwan-I-Khass, Tasbih-Khana, Nahr-I-Bihisht, Delhi Fort Gallery Mumtaz-Mahal, Hammam, Moti-Masjid, Hayat-Bakhsh-Bagh, Chhatta-Chowk, Zafar-Mahal, Rang-Mahal, and so forth.
Later with the appearance of the British, Lal Qila likewise went to their rule. Red Fort under British rule went through a lot of changes, despite which it didn't lose the entrancing charms of its appealing design.
Light and Sound Show
Did you have any idea that the Red Fort is well known for its light and sound show?
The presentation, which is led each night with the exception of Mondays, is a one-hour light and sound show that happens inside the Fort's grounds. There could be no more prominent method for finding the Fort's set of experiences. The projects are communicated in both English and Hindi at different times:
Hindi: 7:30 pm – 8:30 pm
English: 9:00 pm – 10:00 pm
Ticket Cost: 60 INR for adults and 20 INR for kids
Delhi Gate of Red Fort
No portrayal of Red Fort is finished without referencing Delhi Door. Situated on the southern mass of the Red Fort, towards the entry is the Delhi Door. Named after the city, the appearance of this door is like the Lahori Gate of the Fort.
Otherwise called Dilli Darwaza, Delhi Gate was built during the rule of Shah Jahan, while it has a barbican facing the west which was developed by Aurangzeb.
Delhi Gate has three stories, each enhanced with angled boards in rectangular, cusped and square shapes which are then lined by the pinnacles of semi-octagonal shape. These pinnacles are then covered by octagonal-formed open structures.
The structure's rooftop is comprised of white stone while the entryway is developed with red sandstone. There is a little shelter with seven scaled-down arches between these two structures. These walls have a fire-shaped fortress.
Towards the right half of the door, the last head was kept in hostage; this was in year 1857. Between the internal and external doors, stone elephants with next to no riders were set by Lord Curzon.
Lahori Gate of Red Fort
Situated on the western wall, Lahori Gate is the primary entry of the Red Fort Delhi. It is irrefutably one of the most well-known pieces of the Red Fort as it is the spot from where the State leader of India lifts the public banners just before Independence Day consistently. This has been the custom since the time of India's autonomy in 1947. The name of this door is gotten from the way that it inclines towards the Lahore district of Pakistan.
Like the Delhi Gate in structure, the Lahori Gate of Red Fort additionally comprises of three stories with curved boards in square, rectangular and cusped shapes. Semi-octagonal pinnacles cover it with open octagonal structures.
Chhatta Chowk in Red Fort
Situated behind the Lahori Gate, Chatta Chowk is a covered market. A significant piece of Delhi Lal Kila, it presents a brief look at extraordinary Mughal architecture. The word Chatta Chowk means covered market. It is a long entry which is looking like a curve. This market is comprised of two celebrated pads which have 32 arched bays. These narrows filled in as shops selling particular things taking care of the necessities of the illustrious family.
The things sold in this market range from valuable stones, jewels, flatware, and gold to velvet, silk, and brocades. It was in the seventeenth century when Chatta Chowk of Red Fort appeared after Shah Jahan visited the market of Peshawar in 1646.
Categories India, Places to Visit Tags Delhi Gate of Red Fort, History Of Lal Kila, History Of Red Fort, Lal Kila Delhi, Red Fort, Red Fort Delhi
India Gate Delhi (Entry Fee, Timings, History, Built by, Images & Location)
Lotus Temple / Bahá'í Temple Delhi (Timings, History, Entry Fee, Images, Aarti, Location & Phone)

redpajama_set_name: RedPajamaCommonCrawl


text:
Common Standards, Not Tests, for Charter Schools Network
By Erik W. Robelen — February 02, 2012 1 min read
The situation Achievement First—a nationally-recognized charter schools network—finds itself in today exemplifies one conundrum of the push towards common standards and tests. And that is, the assessments will be a little less common.
As most readers surely know by now, most states have adopted the common-core standards for English/language arts and mathematics. But two separate state consortia are developing assessments to match.
According to my quick count, SMARTER Balanced now has 29 state members; PARCC has 23 plus the District of Columbia. (A handful are hedging their bets.)
And therein lies the rub for Achievement First, which operates charter schools in both New York and Connecticut. Yes, indeed: the Empire State joined one assessment consortia, Connecticut joined the other.
I was chatting this week with Nancy Livingston, the vice president of teaching and learning at Achievement First, for an upcoming story about the new math standards. She's a big fan of the document. ("Yes, a resounding yes," she replied emphatically, when I asked if she liked the math standards.)
But Livingston lamented that her network's 20 schools won't be taking the same state assessments, at least as of now, because of the states' decisions in choosing a new testing route.
"They opted into two different assessment consortia," Livingston told me. "I would like to convince Connecticut to move to PARCC. That is step one. But we're planning right now that it will be two."
She added: "We've had [our schools] operating under two different assessments historically, so it's not that different. But it's a little sad in getting to common standards that we're not getting to common assessments."
Erik W. Robelen
Erik W. Robelen formerly wrote for Education Week.
A version of this news article first appeared in the Curriculum Matters blog.
Reading & Literacy Letter to the Editor More Books, Not More Phonics


The Department of Mental Health of the Johns Hopkins Bloomberg School of Public Health, the United Nations High Commissioner for Refugees (UNHCR), Muhimbili University of Health and Allied Sciences (MUHAS) and other partners are conducting a research study in a refugee camp in Tanzani. This 2-year study is aimed at developing and rigorously evaluating an integrated intervention that addresses both (a) mental health complaints through existing evidence-based group intervention; (b) intimate partner violence through an empowerment protocol. It concerns a mixed methods study in which a short period of qualitative research is followed by a randomized clinical trial. We are currently preparing the qualitative phase of the study, and expect to start training interventionists starting in the Spring of 2015.
Karibu! Mambo vipi? Habari za nyumbani? Shikamoo? Immediately upon arrival I learned that Tanzania is a country of many greetings - enough that it took me several weeks of Swahili classes just to learn the essential ones. Greetings characterize your relationship to the person with whom you are speaking and they set the tone for your conversation. Interestingly, the response to any greeting is almost always the same – nzuri, poa, and my personal favorite, freshi – suggesting that in Tanzania, you are always good. In my experience, this is not far from the truth. From working on a project I am truly passionate about, to experiencing a new culture, to spending the occasional weekend away on the pristine beaches of Zanzibar or the top of the world (well, the 15,000 foot summit of Mt. Meru – close enough), I can say my time in Tanzania truly was nzuri sana ('very good').
My experience in Tanzania started in Dar es Salaam and ended in Kasulu. While these settings couldn't be more different, the defining features of Tanzanian culture remained consistent. Dar es Salaam is a large city on the eastern coast with lots to eat (including chipsi maiai, a favorite of most Tanzanians – look it up!), lots of nearby beaches, activities, and unparalleled traffic. It is a very international city with a constant flow of expats and visitors. In contrast, Kasulu is a small village in western Tanzania with one main dirt road running through its center and more bikes than cars. The majority of Kasulu residents are pastoralists of the Waha ethnic group. It is also the city nearest Nyarugusu refugee camp, the site of my research project. Despite these differences, Tanzanian culture is incredibly unified. Many policies in the country encourage Tanzanians to experience other regions of their country and build relationships with people across tribal and religious lines. Characteristics such as tribe, ethnicity and religion - factors that have historically produced tensions in neighboring countries and countries throughout the world - are not divisive, but rather celebrated universally in Tanzania. This aspect of Tanzanian culture is one that I grew to appreciate and admire.
Although the project I worked on was based in Tanzania, our study population is comprised of Congolese females residing in Nyarugusu refugee camp. More specifically we were interested in exploring and evaluating an intervention to improve mental health and psychosocial wellbeing and reduce intimate partner violence in this population. Nyarugusu camp is home to nearly 70,000 Congolese refugees, many having lived in the camp for almost 2 decades, and another 80,000 Burundian refugees, most having arrived within the past 6 months due to the political instability in Burundi. The recent influx of Burundian refugees has exceeded the capacity of the resources available in Nyarugusu and there are currently many humanitarian agencies actively trying to address the needs of the pre-existing and new populations. Observing the prioritization of basic needs and provision of services in a crisis situation and the coordination of actors involved was incredibly complex and something I had not directly experienced before. Our research project was planned prior to the crisis in Burundi and thus the changes in the camp had direct implications for our research. This required us to be flexible and consider the changing needs and priorities of the people in Nyarugusu. Through this experience I learned much about the realities of conducting research in humanitarian settings and how valuable it is to work in the field. My experience in Tanzania exceeded my expectations and further strengthened my commitment to global mental health research in humanitarian settings. I look forward to continuing my work on this project. Until next time, Tanzania! Kwaheri!
3) Lusia, the Research Team Leader, and I outside our office and the Psychiatry Department at Muhimbili National Hospital in Dar es Salaam.


Aging is a decline in cognitive, physical and emotional function that most of us would love to avoid altogether. We have been told that aging is just a natural process. Becoming frail, tired, developing heart disease or Alzheimer's is just what happens. Many factors impact how we age. Sun exposure, stress, sleep habits, diet, exercise and weight. We spend much time and money trying to improve these things.
As we age, our hormone levels fall. This phenomenon rapidly ushers us toward the undesirable age-related decline we are so familiar with. There is a growing body of medical evidence demonstrating that by replacing hormones and more specifically optimizing them, we may alter or control the aging process. Replacing hormones naturally protects from the deterioration of the neurological, musculo-skeletal and cardiovascular systems, adding vigor and vitality to everyday life.
At Concierge Medicine we feel this is the best approach to preventative medicine by replacing and optimizing your hormones with natural hormone replacement. Coupled with a healthy lifestyle you will be on the right path to controlling the aging process.


It's hard being a young man with alopecia. Especially when you're trying to learn how to draw your eyebrows on.
Stop Saying "It's Better Than Being Sick."
Unless you're chronically ill yourself, you will never truly understand the difficult choices those with life-threatening conditions have to make.
https://folks.pillpack.com/wp-content/uploads/2018/12/12746066295_491ae085cd_k.jpg 1365 2048 Ashanthi De Silva https://folks.pillpack.com/wpinhere/wp-content/uploads/2017/03/folks-logo-1.png Ashanthi De Silva2018-12-06 11:04:392019-01-22 11:35:23Stop Saying "It's Better Than Being Sick."


text:
title: "Contribution à la mission société numérique inclusive : Les écueils de gouvernance à éviter"
layout: post
date: 2018-02-16 12:48
image: /assets/images/mission-num-2018.png
headerImage: false
tag:
- hsociety
- Tiers-Lieux
- Rennes
category: blog
author: XavierCoadic
description: 1er trimestre 2018
---

> Dans le cadre de la [consultation stratégie mission numérique ouverte](https://strategie.societenumerique.gouv.fr) par le gouvernement français au premier trimestre 2018, voici une transcrition de ma contribution

L'une des pistes de gouvernance serait d'appliquer l'inclusion dans cette volonté de gouvernance. Ce que cette initiative de consultation numérique laisse espérer.

Il y a effectivement des écueils du passé qui peuvent être listés.  La logique de silo évoquée par @loicgervais est effectivement un des écueils du passé qui à fait ses preuves d'empêchement de collaboration et d'empêchement d'atteinte d'objectif.

Dans les ressources existantes sur les bonnes pratiques déjà effectuées, donc qui ont leur preuve, il y a tout ce qui a été documenté librement par et avec les publics.  Ce qui offre :
1. Un ressource réutilisable et analysable
2. Une preuve que faire de la médiation peut aussi nourrir la gouvernance.

l'un des exemples est le wiki movilab :  http://movilab.org/index.php?title=Accueil et plus spécifiquement le portai dédié à la médiation numérique : http://movilab.org/index.php?title=P%C3%A9rim%C3%A8tre_de_la_M%C3%A9diation_Num%C3%A9rique

## Gouvernance
Cette volonté de documenter sous licence libre permet la contribution, sous réserve d'accompagner à ce processus contributif, et offre un prémisse de transparence.

Se pose immédiatement la question de la gestion des données :
1. reliée au RGPD
2. impliquant la collaboration non verticale dans la gestion de ses données

Nous avions proposé, dans un cadre scientifique, une expérimentation sur un gestion collaborative des données qui pourrait servir d'exemple dans la médiation numérique : <http://lesvigies.fr/changement-climatique-gouvernance>
Le retour d'expérience de cette initiative est simple :  sans formation ou initiation aux personnes et groupes de personnes portant l'initiative de gouvernance et de collaboration le processus de partage et mutualisation est refusé. Ce qui bloque de facto tout volonté de gouvernance sans silo.
Nous avions pourtant fourni un effort important, transdisciplinaire, transcommunautaire,  de sensibilisation et de formation : <http://www.multibao.org>

Ainsi, c'est par un autre chemin que nous tentons aujourd'hui d'expérimenter une gouvernance qui nous paraît adaptée aux enjeux, notamment ceux de la médiation.

### Le consensus

Les tiers-lieux et les communautés de pratiques collaboratives apparaissent comme un terrain fertile d'expérimentation de la médiation numérique, j'y reviendrais dans le groupe de travail concerné, mais aussi et surtout de la gouvernance.

En partant de ce nous pratiquons dans un hacklab à Rennes et un peu partout en Bretagne sur le consensus, nous essayons de rendre accessible un expérience du consensus dans le design des instances : <https://lebiome.github.io/#LeBiome/gouvernance/blob/master/wg_consensus.md>
![](https://framapic.org/JWMGLbZZCor9/8luG8JhV6E0V.jpg)
![](https://framapic.org/iefamEBbV59c/1soevpmbY2rs)

Ce qui m'a amené à titre personnel à échanger sérieusement avec le DR en sociologie Antoine Burret, thèse [Etude de la configuration en Tiers-Lieu - La repolitisation par le service](https://tel.archives-ouvertes.fr/tel-01587759), et d'avoir à découvrir et étudier une expérimentation de la gouvernance par consensus à l'échelle  du territoire administratif de Genève : <https://wiki.3ddge.ch/index.php?title=Accueil>

Sans jamais oublier que « Les lieux de médiation [scientifique et technique] sont (...) des lieux de tension, de *conflit* entre des injonctions à l'innovation industrielle et des appels à un mouvement d'une culture critique. » ([source](https://medium.com/@julbel/lieux-trans-num%C3%A9riques-entre-pratiques-populaires-et-r%C3%A9-appropriation-des-technologies-ec375c90532b))

### Du bas vers le haut
Cette démarche d'expérimentation d'un gouvernance partagée et discutée en permanence avec des ressources libres et contributives part des quelques entités individuées, c'est à dire de personne faisant partie de projet ou d'organisation plus vaste mais qui se rencontre et travaillent ensemble à l'entretient et l'amélioration d'un commun qu'est cette gouvernance et ces ressources, pour ensuite faire grandir cette expérimentation dans des instances gérant des territoires administratifs plus vastes.
il est compris dans cette démarche de manière ad hoc d'outiller ces instances pour qu'elles passent d'une posture de "dirigisme" à une posture de facilitation des ces expérimentation de gouvernance, y compris en soutenant économiquement ces démarches.

Voici quelques ressources documentées :
+  Réflexion sur le consensus : <https://xavcc.github.io/recherche-consensus>
+  Hôtel Pasteur à Rennes et la place de l'institution politique dans la gouvernance d'un lieu de médiation et d'expérimentation : https://notesondesign.org/hotel-pasteur
+  Amorcer des collaborations entre des organes d'innovation : <https://medium.com/we-are-biomers/amorcer-des-collaborations-entre-des-organes-dinnovation-42b15184985b>
+  Faciliter la rencontre, les prises de décisions et les actions d'acteurs du numériques à l'échelle du grand Ouest : [Rencontre 2016](https://annuel.framapad.org/p/fablab-ouest) et [Rencontre 2108](https://semestriel.framapad.org/p/fabrique2017-fablabs-ouest)
+  L'assemblée des communs à Lille : <http://wiki.lescommuns.org/wiki/Assembl%C3%A9e_des_Communs_de_Lille>
+ Le réseaux de communs, qui ne fait directement du consensus mais dont le passage à l'échelle est  important : <http://wiki.lescommuns.org/wiki/Accueil>

### Quelques écueils à éviter

+ Plus le temps est long entre l'impulsion d'une démarche et la passage à l'action est long, plus le risque d'échec est grand
+ Sans ressources libres et documentées peu d'évolution  possible
+ Sans ressource libres et documentées, c'est plus de répétition d'erreurs
+ Imposer plutôt que faciliter
+ Oublier de questionner l'éthique
+ Ne pas inclure les publics dans la gouvernance
+ Négliger les besoins et les circonstances de rencontres
+ Oublier des acteurs de la médiation dans la gouvernance (asso, communautés de pratiques, collectivités publiques, entreprises, écoles, universités, bibliothèques...)
+ Les injonctions paradoxales (e.g "Sois libre")
----
_Je précise que cette contribution n'a aucune vocation à autopromotion, ni aucune volonté d'être exhaustive et solide. Elle n'est qu'une contribution_

redpajama_set_name: RedPajamaGithub


text:
Home Article Did Liberals Cause the Sub-Prime Crisis?
Did Liberals Cause the Sub-Prime Crisis?
Conservatives blame the housing crisis on a 1977 law that helps-low income people get mortgages. It's a useful story for them, but it isn't true.
by Robert Gordon
The idea started on the outer precincts of the right. Thomas DiLorenzo, an economist who calls Ron Paul "the Jefferson of our time," wrote in September that the housing crisis is "the direct result of thirty years of government policy that has forced banks to make bad loans to un-creditworthy borrowers." The policy DiLorenzo decries is the 1977 Community Reinvestment Act, which requires banks to lend throughout the communities they serve.
The Blame-CRA theme bounced around the right-wing Freerepublic.com. In January it figured in a Washington Times column. In February, a Cato Institute affiliate named Stan Liebowitz picked up the critique in a New York Post op-ed headlined "The Real Scandal: How the Feds Invented the Mortgage Mess." On The National Review's blog, The Corner, John Derbyshire channeled Liebowitz: "The folk losing their homes? are victims not of 'predatory lenders,' but of government-sponsored -- in fact government-mandated -- political correctness."
Last week, a more careful expression of the idea hit The Washington Post, in an article on former Sen. Phil Gramm's influence over John McCain. While two progressive economists were quoted criticizing Gramm's insistent opposition to government regulation, the Brookings Institution's Robert Litan offered an opposing perspective. Litan suggested that the 1990s enhancement of CRA, which was achieved over Gramm's fierce opposition, may have contributed to the current crisis. "If the CRA had not been so aggressively pushed," Litan said, "it is conceivable things would not be quite as bad. People have to be honest about that."
This is classic rhetoric of conservative reaction. (For fans of welfare policy, it is Charles Murray meets the mortgage mess.) Most analysts see the sub-prime crisis as a market failure. Believing the bubble would never pop, lenders approved risky adjustable-rate mortgages, often without considering whether borrowers could afford them; families took on those loans; investors bought them in securitized form; and, all the while, regulators sat on their hands.
The revisionists say the problem wasn't too little regulation; but too much, via CRA. The law was enacted in response to both intentional redlining and structural barriers to credit for low-income communities. CRA applies only to banks and thrifts that are federally insured; it's conceived as a quid pro quo for that privilege, among others. This means the law doesn't apply to independent mortgage companies (or payday lenders, check-cashers, etc.)
The law imposes on the covered depositories an affirmative duty to lend throughout the areas from which they take deposits, including poor neighborhoods. The law has teeth because regulators' ratings of banks' CRA performance become public and inform important decisions, notably merger approvals. Studies by the Federal Reserve and Harvard's Joint Center for Housing Studies, among others, have shown that CRA increased lending and homeownership in poor communities without undermining banks' profitability.
But CRA has always had critics, and they now suggest that the law went too far in encouraging banks to lend in struggling communities. Rhetoric aside, the argument turns on a simple question: In the current mortgage meltdown, did lenders approve bad loans to comply with CRA, or to make money?
The evidence strongly suggests the latter. First, consider timing. CRA was enacted in 1977. The sub-prime lending at the heart of the current crisis exploded a full quarter century later. In the mid-1990s, new CRA regulations and a wave of mergers led to a flurry of CRA activity, but, as noted by the New America Foundation's Ellen Seidman (and by Harvard's Joint Center), that activity "largely came to an end by 2001." In late 2004, the Bush administration announced plans to sharply weaken CRA regulations, pulling small and mid-sized banks out from under the law's toughest standards. Yet sub-prime lending continued, and even intensified -- at the very time when activity under CRA had slowed and the law had weakened.
Second, it is hard to blame CRA for the mortgage meltdown when CRA doesn't even apply to most of the loans that are behind it. As the University of Michigan's Michael Barr points out, half of sub-prime loans came from those mortgage companies beyond the reach of CRA. A further 25 to 30 percent came from bank subsidiaries and affiliates, which come under CRA to varying degrees but not as fully as banks themselves. (With affiliates, banks can choose whether to count the loans.) Perhaps one in four sub-prime loans were made by the institutions fully governed by CRA.
Most important, the lenders subject to CRA have engaged in less, not more, of the most dangerous lending. Janet Yellen, president of the San Francisco Federal Reserve, offers the killer statistic: Independent mortgage companies, which are not covered by CRA, made high-priced loans at more than twice the rate of the banks and thrifts. With this in mind, Yellen specifically rejects the "tendency to conflate the current problems in the sub-prime market with CRA-motivated lending.? CRA, Yellen says, "has increased the volume of responsible lending to low- and moderate-income households."
Yellen is hardly alone in concluding that the real problems came from the institutions beyond the reach of CRA. One of the only regulators who long ago saw the current crisis coming was the late Ned Gramlich, a former Fed governor. While Alan Greenspan was cheering the sub-prime boom, Gramlich warned of its risks and unsuccessfully pushed for greater supervision of bank affiliates. But Gramlich praised CRA, saying last year, "banks have made many low- and moderate-income mortgages to fulfill their CRA obligations, they have found default rates pleasantly low, and they generally charge low mortgages rates. Thirty years later, CRA has become very good business."
It's telling that, amid all the recent recriminations, even lenders have not fingered CRA. That's because CRA didn't bring about the reckless lending at the heart of the crisis. Just as sub-prime lending was exploding, CRA was losing force and relevance. And the worst offenders, the independent mortgage companies, were never subject to CRA -- or any federal regulator. Law didn't make them lend. The profit motive did.
And that is not political correctness. It is correctness.


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Had a little too much fun on Saturday? Or if you just love to brunch, here's a simple Bloody Mary recipe you can use with Oloves currently featured in your April Starry Night box!
Combine all of the ingredients in a shaker and shake vigorously for 15-20 seconds. If you don't have a shaker, grab a pitcher and mix everything thoroughly with a large wooden spoon.
Grab two glasses and run a lemon slice along the rim. Make sure the seeds don't fall into the glass. If they do, just remove them.
Place the seasoning mix in a bowl and turn your glasses upside-down to coat the rim evenly.
Pour all of the contents in the shaker or pitcher into your glasses. Garnish with an Oloves, mini pretzel, celery stick, lemon wedge, etc. The garnishes to adorn your bloody mary are endless! Enjoy.