A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs (Communications in Computational Physics, 2022)
In recent years, physical informed neural networks (PINNs) have beenshown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs forthe second order elliptic equations with Dirichlet boundary condition, by establishingthe upper bounds on the number of training samples, depth and width of the deepneural networks to achieve desired accuracy. The error of PINNs is decomposed intoapproximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) andthe statistical error is estimated by Rademacher complexity. We derive the bound onthe Rademacher complexity of the non-Lipschitz composition of gradient norm withReLU$^3$ network, which is of immense independent interest.