Variational Physics-Informed Neural Networks For Solving Partial Differential Equations

By E. Kharazmi, Zhongqiang Zhang, G. Em Karniadakis. 2019

339 citations recorded by the source corpus.

Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a Petrov-Galerkin version of PINNs based on the nonlinear approximation of deep neural networks (DNNs) by selecting the {\em trial space} to be the space of neural networks and the {\em test space} to be the space of Legendre polynomials. We formulate the \textit{variational residual} of the PDE using the DNN approximation by incorporating the variational form of the problem into the loss function of the network and construct a \textit{variational physics-informed neural network} (VPINN). By integrating by parts the integrand in the variational form, we lower the order of the differential operators represented by the neural networks, hence effectively reducing the training cost in VPINNs while increasing their accuracy compared to PINNs that essentially employ delta test functions. For shallow networks with one hidden layer, we analytically obtain explicit forms of the \textit{variational residual}. We demonstrate the performance of the new formulation for several examples that show clear advantages of VPINNs over PINNs in terms of both accuracy and speed.

Subfields: Aerospace Engineering, Artificial Intelligence, Statistical and Nonlinear Physics

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Citations in the corpus

Understanding and mitigating gradient pathologies in physics-informed neural networks (2020, methodology): “A representative example of this approach is physics-informed neural networks [12, 13, 14, 15], in which the outputs of a neural network are constrained to approximately satisfy a system of partial differential equations (PDE) by using a regularization functionalR[fθ(x)] that typically corresponds t”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, background): “Some of the main contributions in this thread are based on: collocation [6–8], variational principle (deep Ritz) [9, 10], and Petrov-Galerkin domain-decomposition [11, 12].”
Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics (2020, methodology): “[18] E.”
Extended Physics-informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition based Deep Learning Framework for Nonlinear Partial Differential Equations (2020, background): “Thanks to their universal approximation properties, they can be exploited to construct alternative approaches for solving PDEs.”
hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition (2020, methodology): “It has been shown in [25] that in VPINNs the variational residuals can be obtained analytically for the case of shallow networks.”
Parallel Physics-Informed Neural Networks via Domain Decomposition (2021, background): “Due to their simplicity, PINNs have led to progress in many areas of computational science, see for examples [7, 8, 9, 10, 11, 12, 13, 14, 15].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “Loss function can be constructed using collocation points, weighted residuals derived by the Galerkin–Method [76], or energy based.”