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

2026-08-22

AI Summary

本文提出一种基于神经架构的方法，从时间序列数据中学习稀疏可解释的动力系统模型，通过多步预测损失优化提高模型稳定性和泛化能力。

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.

Problem

• time-series data
• dynamical systems
• interpretable models
• stability
• sparse models

Innovation

• Sparse models
• Interpretable dynamics
• Multi-step prediction loss
• Consistency under composition
• Chaos recovery