Integral regularization PINNs for evolution equations

2025-03-31

AI Summary

To address the accuracy degradation in physics-informed neural networks (PINNs) caused by cumulative temporal discretization errors during long-term integration of evolutionary equations, this work proposes integral-regularized PINNs (IR-PINNs). Methodologically, we introduce— for the first time—subinterval integral residual constraints into the loss function and integrate a dynamic, solution-evolution-aware adaptive Monte Carlo sampling strategy. This dual design significantly enhances resolution of sharp gradients and rapid transient dynamics. Compared with standard PINNs and state-of-the-art variants, IR-PINNs reduce long-term simulation errors by 40–65% across multiple benchmark ODE and PDE problems, while simultaneously improving temporal stability and modeling fidelity. The framework establishes a new paradigm for high-fidelity, long-horizon prediction of evolutionary systems.

Technology Category

• Machine Learning: Evolutionary Learning
• Planning, Routing, and Scheduling: Temporal Planning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Addresses temporal error accumulation in PINNs for evolution equations
• Enhances long-time integration accuracy via integral regularization
• Improves dynamic system modeling with adaptive sampling techniques

Innovation

• Integral regularization enhances temporal accuracy
• Sub-interval constraints improve dynamic resolution
• Adaptive sampling refines collocation points dynamically