Unveiling Multi-regime Patterns in SciML: Distinct Failure Modes and Regime-specific Optimization

Neural networks (NNs) trained under different hyperparameters can fall into distinct training ``regimes'', with models in the same regime showing homogeneous properties and models across regimes differing qualitatively. In this paper, we analyze multi-regime patterns in scientific machine learning (SciML) models by characterizing these regimes and the transitions between them. We show how different regimes affect trainability and generalization, and we demonstrate that loss-landscape analysis enables regime-based diagnostics to understand, evaluate, and improve SciML model training. Our analysis yields three key insights: (1) compared with computer vision (CV) tasks, SciML models exhibit significantly more pathological loss landscapes; (2) optimization methods are regime-specific -- different optimization strategies help in different regimes, but none is uniformly effective; and (3) SciML models exhibit fine-grained failure modes that challenge conventional interpretations of standard loss-landscape metrics. Using this study, we aim to unify our understanding of seemingly different failure modes across SciML tasks and obtain task-oblivious insights and methodologies for addressing these failures. We validate these findings across widely used SciML models, including physics-informed neural networks (PINNs), Fourier neural operators (FNOs), and Neural Ordinary Differential Equations (NeuralODEs), on benchmarks spanning representative ordinary and partial differential equations.