KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks

TL;DR

KP-PINNs introduces a novel loss function based on RKHS norm and employs Kernel Packet acceleration, enhancing stability and efficiency in solving complex differential equations with physics-informed neural networks.

Contribution

The paper proposes KP-PINNs, a new framework that improves PINNs stability and efficiency using RKHS-based loss and Kernel Packet acceleration techniques.

Findings

KP-PINNs achieve stable solutions across various differential equations.

Numerical experiments demonstrate improved efficiency and accuracy.

The framework offers a promising approach for scientific computing 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.