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

Xiaodong Feng, Haojiong Shangguan, Tao Tang, Xiaoliang Wan, Tao Zhou

TL;DR

This work uses 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, and has developed an adaptive sampling strategy that refines the training set.

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.

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

TL;DR

This work uses 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, and has developed an adaptive sampling strategy that refines the training set.

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.
Paper Structure (24 sections, 70 equations, 15 figures, 9 tables, 2 algorithms)

This paper contains 24 sections, 70 equations, 15 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contribution
  4. Preliminaries
  5. Physics-informed neural networks (PINNs)
  6. Continuous time finite element method
  7. Galerkin projection
  8. Collocation projection
  9. Methodology
  10. A hybrid FEM-PINN method
  11. Some remarks on the hybrid form
  12. Deep adaptive sampling method
  13. Model $p_S(s)$
  14. Model $p_{X|S}(\textbf{x}|s)$
  15. Adaptive sampling approach
  16. ...and 9 more sections

Figures (15)

  • Figure 1: Schematic of the proposed approach.
  • Figure 2: Exact solutions. From left to right: $\beta=10,30$ and $50$.
  • Figure 3: Convection equation. Numerical results with linear finite element discretization.
  • Figure 4: Convection equation ($\beta=50$). Linear finite element basis functions versus quadratic finite element basis functions.
  • Figure 5: Allen Cahn equation. From left to right: the prediction and absolute error. Relative $L_2$ error is 5.08e-3 (Quadratic basis function, $N_{\mathrm{segment}}$=4, $N$=30).
  • ...and 10 more figures

Theorems & Definitions (1)

  • Remark 3.1