SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning

2026-08-29

AI Summary

本文提出SS-ESOAP方法，通过自适应预处理改善物理信息神经网络的训练精度问题，尤其适用于高精度需求场景。

Abstract

Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require expensive optimizer state, while Kronecker-factored methods such as SOAP scale to larger networks but rely on periodic basis updates. We introduce \method, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling. We characterize the directional secant matching induced by the scalar correction and give a bound on variance-state mismatch across basis changes. Across eight PDE benchmarks, \method attains the lowest final residual on six, including Burgers and Boussinesq, while SOAP-family baselines perform better on Gray-Scott and Ginzburg-Landau. On Boussinesq, \method reaches a residual of $10^{-5}$ in 4.1 hours with 9.2 GB peak VRAM, while Adam does not reach this target within 14 hours. Three-seed $L^2$ and $H^1$ errors on four representative PDEs support the link between lower residuals and improved solution accuracy. These results position \method as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.

Problem

• Physics-informed neural networks
• Ill-conditioned objectives
• Preconditioning
• Kronecker-factored methods
• High-accuracy training

Innovation

• Self-Scaled Adaptive Preconditioning
• Kronecker-factored Methods
• Secant-Energy Correction
• Adaptive Basis Update
• Variance-State Downscaling