Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization
About
Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation through neural tangent kernel (NTK) analysis: for linearly coupled systems, the standard NTK's spectral radius grows as $\Omega(\gamma^2)$ with coupling strength $\gamma$, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK $K_P = JH^{+}J^\top$ whose spectral radius is bounded by $S$ (number of networks), independent of $\gamma$. Adam's diagonal preconditioning destroys this projector structure -- inflating $\lambda_{\max}$ far above $S$ for any coupling type -- and its residual-dynamics kernel grows as $\Theta(\gamma)$, placing its stable learning rate strictly between gradient descent and GN. For one-way coupling the limitation is class-wide: no diagonal preconditioner, fixed or adaptive, halves the driving residual in fewer than $\Omega(\gamma)$ iterations ($\Omega(\gamma^2)$ if fixed), whereas block-diagonal GN requires $O(1)$. We verify $\Omega(\gamma^2)$ growth across linearly coupled benchmarks and confirm $\lambda_{\max}(K_P) = S$ in all three 1D systems, including nonlinearly coupled NP+P. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GradNorm) yields coupling-robust accuracy: across 222 experiments spanning three 1D systems and a 2D electroosmotic flow benchmark, SOAP+GradNorm maintains final-epoch $L_2$ accuracy across coupling strengths, with $\leq 2.3\times$ degradation in nonlinear NP+P while Adam+GradNorm fails ($L_2 > 0.1$). SOAP+GradNorm further scales to a 2D, 6-PDE electroosmotic flow at EDL-resolved conditions down to $\varepsilon = 0.01$ -- a regime all prior PINN electrokinetics studies have avoided -- where Adam+GradNorm fails entirely ($L_2 > 0.3$).