Physics informed cell representations for variational formulation of multiscale problems

TL;DR

This paper introduces physics-informed cell representations combined with multilevel grids and MLPs to enhance the speed and accuracy of solving multiscale PDEs using PINNs, addressing their limitations in multiscale problems.

Contribution

It proposes a novel cell-based MLP architecture with multilevel grids and variational loss for improved multiscale PDE solving, outperforming traditional PINNs.

Findings

Enhanced convergence speed in multiscale PDEs

Superior accuracy over conventional PINNs

Effective handling of nonlinear and high-frequency boundary conditions

Abstract

With the rapid advancement of graphical processing units, Physics-Informed Neural Networks (PINNs) are emerging as a promising tool for solving partial differential equations (PDEs). However, PINNs are not well suited for solving PDEs with multiscale features, particularly suffering from slow convergence and poor accuracy. To address this limitation of PINNs, this article proposes physics-informed cell representations for resolving multiscale Poisson problems using a model architecture consisting of multilevel multiresolution grids coupled with a multilayer perceptron (MLP). The grid parameters (i.e., the level-dependent feature vectors) and the MLP parameters (i.e., the weights and biases) are determined using gradient-descent based optimization. The variational (weak) form based loss function accelerates computation by allowing the linear interpolation of feature vectors within grid cells. This cell-based MLP model also facilitates the use of a decoupled training scheme for Dirichlet boundary conditions and a parameter-sharing scheme for periodic boundary conditions, delivering superior accuracy compared to conventional PINNs. Furthermore, the numerical examples highlight improved speed and accuracy in solving PDEs with nonlinear or high-frequency boundary conditions and provide insights into hyperparameter selection. In essence, by cell-based MLP model along with the parallel tiny-cuda-nn library, our implementation improves convergence speed and numerical accuracy.