Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization

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 $\mathrm{\Omega }({\gamma }^2)$ with coupling strength $\gamma$, shrinking the stable learning rate, while block-diagonal Gauss–Newton (GN) preconditioning yields a preconditioned NTK $K_P=J{H}^{+}{J}^{\mathrm{\top }}$ 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 $\mathrm{\Theta }(\gamma )$, 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 $\mathrm{\Omega }(\gamma )$ iterations ($\mathrm{\Omega }({\gamma }^2)$ if fixed), whereas block-diagonal GN requires $O(1)$. We verify $\mathrm{\Omega }({\gamma }^2)$ 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 $\le 2.3\times$ 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 $\epsilon =0.01$ – a regime all prior PINN electrokinetics studies have avoided – where Adam+GradNorm fails entirely ($L_2>0.3$).