Symbolic Neural ODEs: Learning interpretable models from time-series data

Abstract:We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.

Submitted to the SIAM J. Dynamical Systems

Machine Learning (cs.LG); Systems and Control (eess.SY); Dynamical Systems (math.DS); Machine Learning (stat.ML)

arXiv:2608.22112 [cs.LG]

(or arXiv:2608.22112v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.22112 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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