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Integral regularization PINNs for evolution equations

Xiaodong Feng, Haojiong Shangguan, Tao Tang, Xiaoliang Wan

TL;DR

This paper introduces Integral Regularization PINNs (IR-PINNs) to tackle long-time integration of evolution equations by enforcing temporal correlations through an integral residual over temporal subintervals. By recasting the evolution equation into an integral form and coupling an integral residual with the standard residual, IR-PINNs improve temporal accuracy and mitigate error propagation across time; an adaptive sampling scheme further concentrates collocation points in regions with sharp dynamics. The authors validate IR-PINNs on multiple benchmarks, including chaotic Lorenz and Kuramoto-Sivashinsky systems, as well as Boussinesq-Burgers and time-dependent Fokker-Planck equations, showing substantial improvements over conventional PINNs and competitive or superior performance to other state-of-the-art approaches. The framework combines a time-marching strategy, Gaussian quadrature for the integral term, and a density-based adaptive sampling method, yielding robust long-time solutions with controlled computational cost. Overall, IR-PINNs offer a robust, accurate method for long-time evolution problems and open avenues for further investigation into the mechanisms behind integral-based regularization and more advanced adaptive sampling techniques.

Abstract

Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long-time integration for these equations remains a significant challenge. While physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDEs, they often suffer from temporal error accumulation, which limits their effectiveness in capturing long-time behaviors. To alleviate this issue, we propose integral regularization PINNs (IR-PINNs), a novel approach that enhances temporal accuracy by incorporating an integral-based residual term into the loss function. This method divides the entire time interval into smaller sub-intervals and enforces constraints over these sub-intervals, thereby improving the resolution and correlation of temporal dynamics. Furthermore, IR-PINNs leverage adaptive sampling to dynamically refine the distribution of collocation points based on the evolving solution, ensuring higher accuracy in regions with sharp gradients or rapid variations. Numerical experiments on benchmark problems demonstrate that IR-PINNs outperform original PINNs and other state-of-the-art methods in capturing long-time behaviors, offering a robust and accurate solution for evolution equations.

Integral regularization PINNs for evolution equations

TL;DR

This paper introduces Integral Regularization PINNs (IR-PINNs) to tackle long-time integration of evolution equations by enforcing temporal correlations through an integral residual over temporal subintervals. By recasting the evolution equation into an integral form and coupling an integral residual with the standard residual, IR-PINNs improve temporal accuracy and mitigate error propagation across time; an adaptive sampling scheme further concentrates collocation points in regions with sharp dynamics. The authors validate IR-PINNs on multiple benchmarks, including chaotic Lorenz and Kuramoto-Sivashinsky systems, as well as Boussinesq-Burgers and time-dependent Fokker-Planck equations, showing substantial improvements over conventional PINNs and competitive or superior performance to other state-of-the-art approaches. The framework combines a time-marching strategy, Gaussian quadrature for the integral term, and a density-based adaptive sampling method, yielding robust long-time solutions with controlled computational cost. Overall, IR-PINNs offer a robust, accurate method for long-time evolution problems and open avenues for further investigation into the mechanisms behind integral-based regularization and more advanced adaptive sampling techniques.

Abstract

Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long-time integration for these equations remains a significant challenge. While physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDEs, they often suffer from temporal error accumulation, which limits their effectiveness in capturing long-time behaviors. To alleviate this issue, we propose integral regularization PINNs (IR-PINNs), a novel approach that enhances temporal accuracy by incorporating an integral-based residual term into the loss function. This method divides the entire time interval into smaller sub-intervals and enforces constraints over these sub-intervals, thereby improving the resolution and correlation of temporal dynamics. Furthermore, IR-PINNs leverage adaptive sampling to dynamically refine the distribution of collocation points based on the evolving solution, ensuring higher accuracy in regions with sharp gradients or rapid variations. Numerical experiments on benchmark problems demonstrate that IR-PINNs outperform original PINNs and other state-of-the-art methods in capturing long-time behaviors, offering a robust and accurate solution for evolution equations.

Paper Structure

This paper contains 17 sections, 45 equations, 22 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Physics-informed neural networks
  3. Methodology
  4. Integral regularization PINNs (IR-PINNs)
  5. The choice of number of subintervals $N$:
  6. The choice of hyperparameter $\alpha$:
  7. Computational cost of regularization term:
  8. Adaptive sampling strategy for IR-PINNs
  9. Numerical experiments
  10. Lorentz system
  11. Kuramoto-Sivashinsky equation
  12. Boussinesq-Burgers equations
  13. Time-dependent Fokker-Planck equation
  14. Conclusion
  15. Time-marching strategy
  16. ...and 2 more sections

Figures (22)

  • Figure 1: Simple ODE: Left: Residual curve. Right: Reference solution versus numerical solution. Relative $L_2$ error: $7.2033e-01$.
  • Figure 2: Simple ODE: Left: Residual curve. Right: Reference solution versus numerical solution. Relative $L_2$ error: $1.3156e-03$.
  • Figure 3: Simple ODE: Left: Residual curve. Right: Reference solution versus numerical solution. Relative $L_2$ error: $4.9012e-03$.
  • Figure 4: Schematic diagram of integral regularization PINNs.
  • Figure 5: Schematic diagram of adaptive sampling.
  • ...and 17 more figures