提出新型优化器，让物理神经网络在强耦合系统中仍保持高精度。
Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization
- 用克罗内克预条件优化器解决多物理场耦合导致的精度下降问题。
- 在222组实验中，新方法使误差增长不超过2.3倍，远优于传统Adam方法。
- 适合研究复杂耦合系统建模的科研人员，尤其关注高精度数值模拟者。
物理信息神经网络（PINNs）为求解耦合多物理场系统提供无网格路径，但其精度随方程间耦合强度增强而系统性下降，仅靠逆梯度范数损失平衡无法可靠防止该失效。本研究揭示了耦合降低训练精度的原因，并识别出一种可消除依赖性的优化器结构，取代逐案调参。通过神经正切核分析，证明标准核的谱半径随耦合强度γ以Ω(γ²)增长，而块对角高斯-牛顿预条件可将其限制在网络数S以内，与γ无关；任何对角预条件均无法在任意耦合类型或损失权重下恢复此界。我们通过克罗内克预条件优化器SOAP结合逆梯度范数损失平衡（SOAP+GradNorm）实现该结构，在四个难度递增基准上完成222组实验。所有系统中，SOAP+GradNorm是唯一在各测试条件下误差衰减始终受限的方法：在弱耦合线性问题中保持原精度，在非线性Nernst-Planck-Poisson系统中误差增长不超过2.3倍，而基于Adam的方法使L₂误差持续高于0.1失败阈值。同一效果也出现在六残差、四网络、二维电渗流系统中，电双层分辨率达德拜长度ε=0.01的参考解。结果表明，耦合引起的精度损失本质是预条件器结构问题，而非损失权重，克罗内克预条件成为训练强耦合刚性多物理场系统中PINNs的关键结构杠杆。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) offer a mesh-free route to solving coupled multiphysics systems, but their accuracy degrades systematically as inter-equation coupling strengthens, and inverse-gradient-norm loss balancing alone does not reliably prevent this failure. This study explains why coupling degrades PINN training and identifies an optimizer structure that removes the dependence, replacing case-by-case tuning with a principled remedy. Through a neural tangent kernel analysis, we prove that the standard kernel's spectral radius grows as $Ω(γ^2)$ with coupling strength $γ$, whereas block-diagonal Gauss-Newton (GN) preconditioning bounds it by the number of networks $S$, independent of $γ$; no diagonal preconditioner recovers this bound for any coupling type or loss weighting. We realize block-diagonal GN preconditioning through the Kronecker-preconditioned optimizer SOAP combined with inverse-gradient-norm loss balancing (SOAP+GradNorm) and evaluate it across 222 experiments on four benchmarks of increasing difficulty. Across all systems, SOAP+GradNorm is the only configuration whose degradation remains bounded in every regime tested: it preserves weak-coupling accuracy in linear problems and limits degradation to $2.3\times$ in the nonlinear Nernst-Planck-Poisson system, whereas Adam-based training leaves the $L_2$ error above the 0.1 failure threshold. The same effect applies to a six-residual, four-network, 2D electro-osmotic flow where the electric double layer is resolved down to a Debye length of $\varepsilon = 0.01$ on an $x$-invariant reference solution. These results recast coupling-induced accuracy loss as a problem of the preconditioner's structure rather than loss weighting and identify Kronecker preconditioning as a structural lever for training PINNs on strongly coupled, stiff multiphysics systems.
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