KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks

2025-06-10

AI Summary

To address numerical instability and accuracy degradation in Physics-Informed Neural Networks (PINNs) when solving complex partial differential equations (PDEs)—stemming from the conventional L² loss—we propose a novel framework that reconstructs the loss function using the norm in a Reproducing Kernel Hilbert Space (RKHS). This work is the first to systematically embed RKHS theory into the PINN training objective. We theoretically establish that the resulting loss confers cross-equation stability. To ensure scalability, we integrate the Kernel Packet (KP) method for efficient low-rank approximation of the kernel matrix, substantially reducing computational overhead. Extensive experiments on strongly nonlinear and multiscale PDEs demonstrate that the proposed method achieves superior convergence accuracy and robustness compared to standard PINNs, while delivering significant computational speedup. This advances scientific computing by providing a highly stable, differentiable, and efficient PDE solver.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Differential equations are involved in modeling many engineering problems. Many efforts have been devoted to solving differential equations. Due to the flexibility of neural networks, Physics Informed Neural Networks (PINNs) have recently been proposed to solve complex differential equations and have demonstrated superior performance in many applications. While the L2 loss function is usually a default choice in PINNs, it has been shown that the corresponding numerical solution is incorrect and unstable for some complex equations. In this work, we propose a new PINNs framework named Kernel Packet accelerated PINNs (KP-PINNs), which gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet (KP) method to accelerate the computation. Theoretical results show that KP-PINNs can be stable across various differential equations. Numerical experiments illustrate that KP-PINNs can solve differential equations effectively and efficiently. This framework provides a promising direction for improving the stability and accuracy of PINNs-based solvers in scientific computing.

Problem

• Improving stability of PINNs for complex differential equations
• Accelerating loss computation using Kernel Packet method
• Enhancing accuracy of neural network-based equation solvers

Innovation

• KP-PINNs uses RKHS norm for loss function
• Kernel Packet method accelerates computation
• Improves stability and accuracy of PINNs