Topological Out-of-Domain Generalization in Dynamical Systems Reconstruction

Predicting the behavior of dynamical systems (DS) in regimes not observed during training—such as across tipping points or bifurcations—is a fundamental challenge in scientific machine learning. While hierarchical and hyper-network-based reconstruction models have shown promise in learning families of systems, they typically fail to generalize to new dynamical regimes without additional fine-tuning. This paper investigates the structural roots of these failures by comparing the mathematical properties of standard machine learning architectures (like Neural ODEs and RNNs) with the known physical properties of dynamical systems.

The authors identify three primary structural discrepancies between standard reconstruction models and physical systems:

1. Dense Jacobian Dependence: Physical systems typically exhibit sparse control-parameter dependence, whereas unconstrained machine learning models often learn dense, spurious dependencies that lead to spectral divergence outside the training domain.
2. Entangled Scaling: In standard models, the shift in fixed-point locations (positional scaling) is mathematically entangled with the local growth rates (dynamical scaling), whereas these properties often vary independently in real physical systems.
3. Discretization Artifacts: The process of time discretization in discrete-time models can induce nonlinear parameter dependencies that are not present in the underlying physical system.

To address these, the authors propose a "feature splitting" strategy combined with low-rank and L1-regularization on the feature-coupling weights. This forces the model to align with the sparse, physically grounded pathways of the true system. They also derive a closed-form bound on the reliable extrapolation range, providing a theoretical limit for when these models can be trusted.

This work provides a principled path toward "zero-shot" out-of-domain generalization in dynamical systems reconstruction. By aligning the inductive biases of neural models with the structural properties of physical laws, the authors demonstrate that models can successfully forecast transitions into new dynamical regimes (e.g., the transition to chaos in the Lorenz-63 system) without needing retraining. This is a critical step toward creating scientific models that can reliably predict systemic shifts in fields like climate science, ecology, and neuroscience.

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