On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs (Communications in Computational Physics, 2020)
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computationalscience and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network isobtained by minimizing a loss function in which any prior knowledge of PDEs anddata are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes ofPDEs: linear second-order elliptic and parabolic. By adapting the Schauder approachand the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfiesthe initial/boundary conditions, the convergence mode becomes $H^1$. Computationalexamples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.
Keywords: Physics informed neural networks; convergence; Hölder regularization; elliptic and parabolic PDEs; Schauder approach.