A hybrid FEM-PINN method for time-dependent partial differential equations

TL;DR

This paper introduces a hybrid FEM-PINN approach for solving time-dependent PDEs, combining finite element methods in time with neural networks for spatial coefficients, improving accuracy and efficiency.

Contribution

The paper proposes a novel hybrid FEM-PINN method that integrates finite element basis functions with neural networks for solving evolution PDEs, including an adaptive sampling strategy.

Findings

Reduces statistical errors in time integration.

Neural network outputs serve as reduced spatial basis functions.

Demonstrates improved accuracy and efficiency in numerical experiments.

Abstract

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network's output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments.