Technique to solve partial differential equations
Physics-informed neural networks (PINNs), also referred to as Theory-Trained Neural Networks (TTNs), are a type of universal function approximators that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples.
Function approximation.
Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization.
Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. On the other hand, physics-informed neural networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training data that do not necessarily need to be large and complete. Potentially, an accurate solution of partial differential equations can be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINN may be used for finding an optimal solution with high fidelity.
PINNs allow for addressing a wide range of problems in computational science and represent a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a meshfree alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, the trained PINN network can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In addition, they allow for exploiting automatic differentiation (AD) to compute the required derivatives in the partial differential equations, a new class of differentiation techniques widely used to derive neural networks assessed to be superior to numerical or symbolic differentiation.
Modeling and computation.
A general nonlinear partial differential equation can be:
formula_1
where formula_2 denotes the solution, formula_3 is a nonlinear operator parameterized by formula_4, and formula_5 is a subset of formula_6. This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems:
Data-driven solution of partial differential equations.
The "data-driven solution of PDE" computes the hidden state formula_2 of the system given boundary data and/or measurements formula_8, and fixed model parameters formula_4. We solve:
formula_10.
By defining the residual formula_11 as
formula_12,
and approximating formula_2 by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of formula_2 and formula_11 can be then learned by minimizing the following loss function formula_16:
formula_17.
Where formula_18 is the error between the PINN formula_19 and the set of boundary conditions and measured data on the set of points formula_20 where the boundary conditions and data are defined, and formula_21 is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the partial differential equation during the training process.
This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE.
Data-driven discovery of partial differential equations.
Given noisy and incomplete measurements formula_8 of the state of the system, the "data-driven discovery of PDE" results in computing the unknown state formula_2 and learning model parameters formula_4 that best describe the observed data and it reads as follows:
formula_1.
By defining formula_11 as
formula_27,
and approximating formula_2 by a deep neural network, formula_11 results in a PINN. This network can be derived using automatic differentiation. The parameters of formula_2 and formula_11, together with the parameter formula_32 of the differential operator can be then learned by minimizing the following loss function formula_16:
formula_17.
Where formula_18, with formula_36 and formula_8 state solutions and measurements at sparse location formula_20, respectively and formula_21 residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process.
This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation.
Physics-informed neural networks for piece-wise function approximation.
PINN is unable to approximate PDEs that have strong non-linearity or sharp gradients that commonly occur in practical fluid flow problems. Piece-wise approximation has been an old practice in the field of numerical approximation. With the capability of approximating strong non-linearity extremely light weight PINNs are used to solve PDEs in much larger discrete subdomains that increases accuracy substantially and decreases computational load as well. DPINN (Distributed physics-informed neural networks) and DPIELM (Distributed physics-informed extreme learning machines) are generalizable space-time domain discretization for better approximation. DPIELM is an extremely fast and lightweight approximator with competitive accuracy. Domain scaling on the top has a special effect. Another school of thought is discretization for parallel computation to leverage usage of available computational resources. 
XPINNs is a generalized space-time domain decomposition approach for the physics-informed neural networks (PINNs) to solve nonlinear partial differential equations on arbitrary complex-geometry domains. The XPINNs further pushes the boundaries of both PINNs as well as Conservative PINNs (cPINNs), which is a spatial domain decomposition approach in the PINN framework tailored to conservation laws. Compared to PINN, the XPINN method has large representation and parallelization capacity due to the inherent property of deployment of multiple neural networks in the smaller subdomains. Unlike cPINN, XPINN can be extended to any type of PDEs. Moreover, the domain can be decomposed in any arbitrary way (in space and time), which is not possible in cPINN. Thus, XPINN offers both space and time parallelization, thereby reducing the training cost more effectively. The XPINN is particularly effective for the large-scale problems (involving large data set) as well as for the high-dimensional problems where single network based PINN is not adequate. The rigorous bounds on the errors resulting from the approximation of the nonlinear PDEs (incompressible Navier–Stokes equations) with PINNs and XPINNs are proved. However, DPINN debunks the use of residual (flux) matching at the domain interfaces as they hardly seem to improve the optimization. 
Physics-informed neural networks and theory of functional connections.
In the PINN framework, initial and boundary conditions are not analytically satisfied, thus they need to be included in the loss function of the network to be simultaneously learned with the differential equation (DE) unknown functions. Having competing objectives during the network's training can lead to unbalanced gradients while using gradient-based techniques, which causes PINNs to often struggle to accurately learn the underlying DE solution. This drawback is overcome by using functional interpolation techniques such as the Theory of functional connections (TFC)'s constrained expression, in the Deep-TFC framework, which reduces the solution search space of constrained problems to the subspace of neural network that analytically satisfies the constraints. A further improvement of PINN and functional interpolation approach is given by the Extreme Theory of Functional Connections (X-TFC) framework, where a single-layer Neural Network and the extreme learning machine training algorithm are employed. X-TFC allows to improve the accuracy and performance of regular PINNs, and its robustness and reliability are proved for stiff problems, optimal control, aerospace, and rarefied gas dynamics applications.
Physics-informed PointNet (PIPN) for multiple sets of irregular geometries.
Regular PINNs are only able to obtain the solution of a forward or inverse problem on a single geometry. It means that for any new geometry (computational domain), one must retrain a PINN. This limitation of regular PINNs imposes high computational costs, specifically for a comprehensive investigation of geometric parameters in industrial designs. Physics-informed PointNet (PIPN) is fundamentally the result of a combination of PINN's loss function with PointNet. In fact, instead of using a simple fully connected neural network, PIPN uses PointNet as the core of its neural network. PointNet has been primarily designed for deep learning of 3D object classification and segmentation by the research group of Leonidas J. Guibas. PointNet extracts geometric features of input computational domains in PIPN. Thus, PIPN is able to solve governing equations on multiple computational domains (rather than only a single domain) with irregular geometries, simultaneously. The effectiveness of PIPN has been shown for incompressible flow, heat transfer and linear elasticity.
Physics-informed neural networks (PINNs) for inverse computations.
Physics-informed neural networks (PINNs) have proven particularly effective in solving inverse problems within differential equations, demonstrating their applicability across science, engineering, and economics. They have shown useful for solving inverse problems in a variety of fields, including nano-optics, topology optimization/characterization, multiphase flow in porous media, and high-speed fluid flow. PINNs have demonstrated flexibility when dealing with noisy and uncertain observation datasets. They also demonstrated clear advantages in the inverse calculation of parameters for multi-fidelity datasets, meaning datasets with different quality, quantity, and types of observations. Uncertainties in calculations can be evaluated using ensemble-based or Bayesian-based calculations.
Physics-informed neural networks for elasticity problems.
Surrogate networks are intended for the unknown functions, namely, the components of the strain and the stress tensors as well as the unknown displacement field, respectively. The residual network provides the residuals of the partial differential equations (PDEs) and of the boundary conditions.The computational approach is based on principles of artificial intelligence.
Physics-informed neural networks (PINNs) with backward stochastic differential equation.
Deep backward stochastic differential equation method is a numerical method that combines deep learning with Backward stochastic differential equation (BSDE) to solve high-dimensional problems in financial mathematics. By leveraging the powerful function approximation capabilities of deep neural networks, deep BSDE addresses the computational challenges faced by traditional numerical methods like finite difference methods or Monte Carlo simulations, which struggle with the curse of dimensionality. Deep BSDE methods use neural networks to approximate solutions of high-dimensional partial differential equations (PDEs), effectively reducing the computational burden. Additionally, integrating Physics-informed neural networks (PINNs) into the deep BSDE framework enhances its capability by embedding the underlying physical laws into the neural network architecture, ensuring solutions adhere to governing stochastic differential equations, resulting in more accurate and reliable solutions.
Physics-informed neural networks for biology.
An extension or adaptation of PINNs are Biologically-informed neural networks (BINNs). BINNs introduce two key adaptations to the typical PINN framework: (i) the mechanistic terms of the governing PDE are replaced by neural networks, and (ii) the loss function formula_16 is modified to include formula_41, a term used to incorporate domain-specific knowledge that helps enforce biological applicability. For (i), this adaptation has the advantage of relaxing the need to specify the governing differential equation a priori, either explicitly or by using a library of candidate terms. Additionally, this approach circumvents the