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

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• TAPINN uses 5x fewer parameters than hypernetwork-based alternative while achieving better physics compliance

  80% confidence

• Linear probe on latent space achieves prognostics MSE of 3.5×10^-4 for regressing forcing parameter F0, confirming highly structured representation

  80% confidence

• HyperPINN suffers from memorization pathology, achieving lowest data MSE but high physics residual, overfitting trajectory points without satisfying governing ODE

  80% confidence

• TAPINN shows approximately 49% lower physics residual compared to baseline (0.082 vs. 0.160)

  80% confidence

• Standard MLPs struggle to approximate discontinuous or non-smooth parameter dependence due to spectral bias and singular Jacobians at bifurcation points

  80% confidence

• Joint training without alternating optimization yields significantly higher physics residual (~0.158), performing nearly identically to standard baseline, confirming alternating optimization is critical

  80% confidence

• Multi-Output baseline with Sobolev training exhibited gradient norms 2.14x higher on average with variance 2.18x larger, suggesting unstructured latent space exacerbates optimization pathologies

  80% confidence

• Standard PINNs often face challenges when modeling parameterized dynamical systems with sharp regime transitions, such as bifurcations

  80% confidence

• TAPINN achieves stable convergence with 2.18x lower gradient variance than a multi-output Sobolev Error baseline

  80% confidence

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