通过Kronecker预条件优化实现多物理场物理信息神经网络的耦合鲁棒精度
Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization
AI总结：
针对多物理场耦合PINN在强耦合下精度退化问题，通过神经正切核分析证明标准NTK谱半径随耦合强度γ呈Ω(γ²)增长，而块对角Gauss-Newton预条件使谱半径有界于网络数S，结合Kronecker预条件优化器SOAP与逆梯度范数损失平衡实现耦合鲁棒精度。
摘要：
用于耦合多物理场系统的物理信息神经网络（PINN）在方程间耦合增强时会遭受系统性精度退化。我们通过神经正切核（NTK）分析为这一现象提供了理论解释：对于线性耦合系统，我们证明标准NTK的谱半径随耦合强度γ呈Ω(γ²)增长，缩小了稳定学习率，而块对角Gauss-Newton（GN）预条件产生预条件NTK $K_P = J H^{+} J^ op$（其中$H$是块对角GN Hessian矩阵），其谱半径以$S$（网络数量）为界，与γ无关。我们在对称、非对称和非线性耦合PDE系统上数值验证了Ω(γ²)增长，并确认在所有情况下$λ_{\max}(K_P) = S$。将Kronecker预条件优化器SOAP与逆梯度范数损失平衡（SOAP+GN）相结合，实现了耦合鲁棒精度：在跨越三个非线性递增的一维系统和一个二维电渗流基准的234个实验中，即使耦合参数变化一到两个数量级，SOAP+GN保持最终epoch的$L_2$退化≤1.1倍（强耦合与弱耦合误差之比），而Adam+GN则超过$10^2$倍。SOAP+GN进一步扩展到EDL分辨条件下的二维六PDE电渗流系统——这一所有先前PINN电动力学研究通过简化物理而避免的工况——而Adam+GN完全失败（$L_2 > 0.9$）。
英文摘要：
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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