Error Analysis of the Nonconforming P1 Finite Element Method to the Sequential Regularization Formulation for Unsteady Navier-Stokes Equations

In this paper we investigate the nonconforming P_(1) finite element ap-proximation to the sequential regularization method for unsteady Navier-Stokes equations.We provide error estimates for a full discretization scheme.Typi-cally,conforming P_(1) finite element methods lead to error bounds that depend inversely on the penalty parameter ∈.We obtain an ∈-uniform error bound by utilizing the nonconforming P_(1) finite element method in this paper.Numerical examples are given to verify theoretical results.

A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs

In recent years,physical informed neural networks(PINNs)have been shown 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 for the second order elliptic equations with Dirichlet boundary condition,by establishing the upper bounds on the number of training samples,depth and width of the deep neural networks to achieve desired accuracy.The error of PINNs is decomposed into approximation error and statistical error,where the approximation error is given in C2 norm with ReLU^(3)networks(deep network with activation function max{0,x^(3)})and the statistical error is estimated by Rademacher complexity.We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU^(3)network,which is of immense independent interest.

Convergence Rate Analysis for Deep Ritz Method

Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.