用核包方法提升物理神经网络求解微分方程的稳定性和效率
KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks
- 改用再生核希尔伯特空间范数构造损失函数，替代传统L2损失
- 通过核包方法加速计算，在多种复杂方程上实现稳定求解
- 适合需要高稳定性与精度的科学计算场景
微分方程广泛应用于工程建模。尽管神经网络灵活，物理信息神经网络（PINNs）近年被提出用于求解复杂微分方程，并在多个应用中表现优异。然而，通常采用的L2损失函数在某些复杂方程上会导致数值解不准确且不稳定。本文提出一种新框架KP-PINNs，通过再生核希尔伯特空间（RKHS）范数重新表达损失函数，并利用核包（KP）方法加速计算。理论分析表明，KP-PINNs在各类微分方程下均保持稳定。数值实验验证其能高效、有效地求解微分方程。该框架为提升基于PINNs的科学计算求解器的稳定性和精度提供了新方向。
原文摘要 · Abstract (English)
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.
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