Supervised Metric Regularization Through Alternating Optimization for Multi-Regime Physics-Informed Neural Networks

Standard Physics-Informed Neural Networks (PINNs) often face challenges when modeling parameterized dynamical systems with sharp regime transitions, such as bifurcations. In these scenarios, the continuous mapping from parameters to solutions can result in spectral bias or "mode collapse", where the network averages distinct physical behaviors. We propose a Topology-Aware PINN (TAPINN) that aims to mitigate this challenge by structuring the latent space via Supervised Metric Regularization. Unlike standard parametric PINNs that map physical parameters directly to solutions, our method conditions the solver on a latent state optimized to reflect the metric-based separation between regimes, showing ~49% lower physics residual (0.082 vs. 0.160). We train this architecture using a phase-based Alternating Optimization (AO) schedule to manage gradient conflicts between the metric and physics objectives. Preliminary experiments on the Duffing Oscillator demonstrate that while standard baselines suffer from spectral bias and high-capacity Hypernetworks overfit (memorizing data while violating physics), our approach achieves stable convergence with 2.18x lower gradient variance than a multi-output Sobolev Error baseline, and 5x fewer parameters than a hypernetwork-based alternative.