Integral regularization PINNs for evolution equations
This work addresses a critical bottleneck in physics-informed neural networks for solving dynamic systems, offering a robust solution for researchers and practitioners in computational science and engineering.
The paper tackles the challenge of accurate long-time integration for evolution equations by proposing integral regularization PINNs (IR-PINNs), which incorporate an integral-based residual term and adaptive sampling to reduce temporal error accumulation, demonstrating superior performance over original PINNs and other state-of-the-art methods in numerical experiments.
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.