发表机构
KTH Royal Institute of Technology（瑞典皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出FALM-PINN框架，解耦PINN的表示学习与系数拟合，可高效求解高频、非线性PDE，数值实验显示其相对L²误差较基准低两个数量级。
AI 中文摘要
物理信息神经网络（PINNs）常常无法准确求解具有高频或多尺度解的偏微分方程（PDEs）以及强非线性问题，这一难点源于两个因素：一是频谱偏差，即神经网络对高频特征欠拟合的倾向；二是表示-系数耦合，即在单个非凸优化目标中表示学习与系数拟合的纠缠。本研究提出傅里叶增强交替列文伯格-马夸尔特PINN（FALM-PINN），这是一种将表示学习与系数拟合解耦的优化框架：上层问题学习傅里叶增强基，以高频分量丰富潜在空间；下层问题通过拟合该基上的投影系数来解决耦合，使用列文伯格-马夸尔特算法求解非线性最小二乘问题。该框架适用于一般非线性和耦合PDE系统，对线性PDEs则简化为单步凸优化问题，且我们证明了两种情况下交替训练方案的全局收敛性。对多个具有挑战性的高频和非线性PDEs的数值实例表明，FALM-PINN的相对L²误差比最先进的基准低两个数量级。
英文摘要
Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative $L^2$ errors up to two orders of magnitude lower than state-of-the-art baselines.
Comments53 pages, 18 figures, 6 tables
