提出自适应预条件方法，提升物理信息神经网络训练精度与效率。
SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning
- 引入基于Kronecker结构的标量修正与自适应基更新机制
- 在8个偏微分方程测试中6个达到最低残差，如Boussinesq残差达10⁻⁵
- 适合高精度、高刚性物理模拟训练，不替代通用优化器
物理信息神经网络（PINNs）常因目标函数病态而难以实现高精度训练。密集拟牛顿方法虽改善局部条件，但需高昂的优化器状态开销；而类似SOAP的张量分解方法虽可扩展至大规模网络，却依赖周期性基更新。本文提出 extmethod，通过在SOAP式预条件中引入适配张量几何的标量割线能量修正，并结合自适应基更新与方差状态降维，提升训练稳定性。理论分析了标量修正引发的方向割线匹配特性，并给出了基变化时方差状态偏差的上界。在八个偏微分方程基准测试中， extmethod在六个任务（包括Burgers和Boussinesq）上达到最低残差；在Boussinesq问题中，4.1小时内实现10⁻⁵残差，峰值显存9.2 GB，而Adam在14小时内仍未收敛。四个代表性问题的三组种子L²与H¹误差验证了更低残差对应更高解精度。结果表明， extmethod是高刚性、高精度物理信息训练的可扩展方案，而非现有优化器的统一替代品。
原文摘要 · Abstract (English)
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.
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