用新方法让物理神经网络求解特征值快500倍，精度还高。
Fast PINN Eigensolvers via Biconvex Reformulation
- 把特征对求解转为双凸优化，用交替凸搜索加速收敛
- 实验显示比传统梯度训练快500倍，且保证收敛性
- 适合需要快速模拟系统稳定性与模态的工程场景
特征值问题具有独特的前向-逆向结构，是表征系统热响应、稳定性与自然模态的基础。物理信息神经网络（PINNs）提供了一种无网格求解方案，但通常比经典数值方法慢数个数量级。本文提出一种重构的PINN方法，将特征对搜索转化为双凸优化问题，通过解析最优更新实现对特征值与特征函数的快速、可证明收敛的交替凸搜索（ACS）。数值实验表明，PINN-ACS在保持高精度的同时，收敛速度相比梯度驱动的PINN训练最快提升500倍。代码已开源于 https://github.com/NeurIPS-ML4PS-2025/PINN_ACS_CODES。
原文摘要 · Abstract (English)
Eigenvalue problems have a distinctive forward-inverse structure and are fundamental to characterizing a system's thermal response, stability, and natural modes. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative for solving such problems but are often orders of magnitude slower than classical numerical schemes. In this paper, we introduce a reformulated PINN approach that casts the search for eigenpairs as a biconvex optimization problem, enabling fast and provably convergent alternating convex search (ACS) over eigenvalues and eigenfunctions using analytically optimal updates. Numerical experiments show that PINN-ACS attains high accuracy with convergence speeds up to 500$\times$ faster than gradient-based PINN training. We release our codes at https://github.com/NeurIPS-ML4PS-2025/PINN_ACS_CODES.
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