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

Enzo Nicolas Spotorno, Josafat Ribeiro Leal, Antonio Augusto Frohlich

TL;DR

This paper tackles the difficulty of modeling parameterized dynamical systems with sharp regime transitions in PINNs due to spectral bias and ill-conditioned Jacobians near bifurcations. It introduces TAPINN, a topology-aware PINN that uses an LSTM encoder to map short observed trajectories to a latent $z$, with a Generator $G(t,z)$ that reconstructs the trajectory; training uses a composite loss $L_{total}=L_{data}+\alpha L_{physics}+\beta L_{metric}$ and a triplet-based metric loss with margin $m=0.2$. A phase-based Alternating Optimization schedule first aligns the latent metric structure (Phase I) and then enforces physics (Phase II), with interleaved updates to stabilize training. On the Duffing oscillator, TAPINN achieves a physics residual of $0.082$ using $8{,}003$ parameters—approximately $5\times$ fewer than HyperPINN—and yields a Prognostics MSE of $3.5\times 10^{-4}$ for predicting $F_0$ from $z$, indicating a structured, generalizable latent representation and improved robustness over standard 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 ~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.

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

TL;DR

This paper tackles the difficulty of modeling parameterized dynamical systems with sharp regime transitions in PINNs due to spectral bias and ill-conditioned Jacobians near bifurcations. It introduces TAPINN, a topology-aware PINN that uses an LSTM encoder to map short observed trajectories to a latent , with a Generator that reconstructs the trajectory; training uses a composite loss and a triplet-based metric loss with margin . A phase-based Alternating Optimization schedule first aligns the latent metric structure (Phase I) and then enforces physics (Phase II), with interleaved updates to stabilize training. On the Duffing oscillator, TAPINN achieves a physics residual of using parameters—approximately fewer than HyperPINN—and yields a Prognostics MSE of for predicting from , indicating a structured, generalizable latent representation and improved robustness over standard 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 ~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.
Paper Structure (4 sections, 1 figure, 1 table)

This paper contains 4 sections, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Experiments and Results
  4. Conclusion

Figures (1)

  • Figure 1: Qualitative analysis of the Topology-Aware training process: (a) Gradient norms during training. The Multi-Output baseline (orange) shows significant spikes near regime transitions compared to ours (blue). (b) t-SNE visualization of the learned embeddings $z$, colored by regime ($F_0$). Distinct clusters emerge without discrete supervision.