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

Enzo Nicolás Spotorno, JOSAFAT RIBEIRO LEAL FILHO, Antonio Augusto Medeiros Frohlich

Keywords: Physics-Informed Neural Networks (PINNs), Topology-Aware PINN (TAPINN), Supervised Metric Regularization, Alternating Optimization, Multi-Regime Dynamics, Bifurcations, Spectral Bias, Mode Collapse, Triplet Loss, Latent Space Structuring, Duffing Oscillator, LSTM Encoder.

TL;DR: TAPINN structures latent space via metric learning and alternating optimization to handle bifurcations. It prevents mode collapse, cutting physics error by 49% with 5x fewer parameters than Hypernetwork baselines.

Abstract: 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 $\approx 49\%$ 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 $\mathbf{2.18\times}$ lower gradient variance than a multi-output Sobolev Error baseline, and significantly fewer parameters than a hypernetwork-based alternative.

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Journal Notes: Due to the awesome suggestions from the reviewers, several extensions are in progress already that will significantly improve the paper's rigor and quality. Since, the reviewers seem to have agreed on the relevancy of the work, we believe that with these expansions the paper will be significant for the community and publishing on the IOP focus collection would enable the dissemination of this idea, given it is a highly qualified venue.

Submission Number: 77