Physics-informed neural networks
Physics-informed neural networks (PINNs) 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). They overcome the low data availability of some biological and engineering systems that makes most state-of-the-art machine learning techniques lack robustness, rendering them ineffective in these scenarios. 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 correctness 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 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 equations can be: formula_1 where formula_2 denotes the solution, formula_3 is a nonlinear operator parametrized 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: of partial differential equations. Data-driven solution of partial differential equations. The "data-driven solution of  PDE" results in computing the unknown state formula_2 of the system given noisy measurements formula_8 of the state and fixed model parameters formula_4 and it reads as follows: formula_10. By defining formula_11 as formula_12, and approximating formula_2 by a deep neural network, formula_11 results in a PINN. 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_17: formula_18. Where formula_19, with formula_20 and formula_8 state solutions and measurements at sparse location formula_22, respectively and formula_23 residual function. This second term requires the structured information represented  by the partial differential equations to be satisfied in the training process. This strategies allows assembling computationally efficient physics-informed surrogate models that may find application in data-driven forecasting of physical processes,model predictive control, multi-physics/multi-scale modeling and simulation. 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_29, 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_34 of the differential operator can be then learned by minimizing the following loss function formula_17: formula_18. Where formula_19, with formula_20 and formula_8 state solutions and measurements at sparse location formula_22, respectively and formula_23 residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategies 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. Extended physics-informed neural networks (XPINNs). 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.