发表机构
McGill University; California Institute of Technology; Aarhus University; University of Copenhagen（麦吉尔大学; 加州理工学院; 奥胡斯大学; 哥本哈根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对物理信息神经网络的病态目标问题，提出SS-ESOAP优化方法，在8个PDE基准中多数取得最低残差，在Boussinesq问题上优于Adam，为刚性高精度物理信息训练提供可扩展选项。
AI 中文摘要
物理信息神经网络（PINNs）常面临病态目标问题，限制了高精度训练。稠密拟牛顿方法可改善局部条件，但需要昂贵的优化器状态；而Kronecker分解方法如SOAP可扩展到更大的网络，但依赖周期性基更新。我们提出\textit{\textbackslash method}，它在SOAP风格预处理的基础上，增加了适配Kronecker几何的标量割线能量修正，以及自适应基更新后的方差状态下缩放。我们刻画了标量修正诱导的定向割线匹配，并给出了基变换过程中方差状态失配的界。在8个PDE基准测试中，\textit{\textbackslash method}在6个基准上取得了最低的最终残差，包括Burgers和Boussinesq问题，而SOAP系列基线在Gray-Scott和Ginzburg-Landau问题上表现更好。在Boussinesq问题上，\textit{\textbackslash method}在4.1小时内达到了$10^{-5}$的残差，峰值VRAM为9.2 GB，而Adam在14小时内未达到该目标。在4个代表性PDE上的三种子集$L^2$和$H^1$误差，支持了更低残差与更高解精度之间的关联。这些结果表明，\textit{\textbackslash method}是针对刚性、高精度物理信息训练的可扩展选项，而非现有优化器的统一替代品。
英文摘要
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.
Comments31 pages, 12 figures, 16 tables. Submitted to NeurIPS 2026
