用多头+模约束提升PINN求解非线性微分方程效率
Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space
- 多头训练学习方程解的通用空间，而非单一解
- 引入模为1的隐空间正则化，加速收敛与迁移
- 适用于耦合、多尺度、刚性微分方程求解
非线性微分方程是描述自然现象的基本工具，但对刚性微分方程仍缺乏有效求解方法。本文提出一种基于物理信息神经网络（PINNs）的机器学习框架，用于求解非线性多尺度微分方程及反问题。该框架采用多头（Multi-Head, MH）训练策略，使网络学习给定方程组在一定变异性下的所有解的通用解空间，而非特定解。结合一种新提出的解空间模约束（Unimodular Regularization, UR）技术，显著提升了PINNs的求解效率。实验表明，该方法通过促进迁移学习，可高效求解非线性、耦合及多尺度微分方程。
原文摘要 · Abstract (English)
Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics-Informed Neural Networks (PINNs). This framework is based on what is called \textit{multi-head} (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multi-head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations.
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