发表机构
University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出符号神经常微分方程框架，通过多步训练结合稀疏正则化，从时间序列数据中学习稳定、可泛化的稀疏可解释动力系统模型，可准确恢复多类系统的动力学特性。
AI 中文摘要
我们提出一种机器学习框架，可直接从时间序列数据中识别稀疏、可解释的动力系统模型。该方法用神经网络架构对底层向量场进行参数化，并通过最小化有限时间范围内的多步预测损失进行训练。为确保数值可处理性，我们优化了预测步骤的平均绝对误差目标，并在训练过程中逐步增加时间范围。该公式的一个关键特性是，它在学习到的动力学的重复组合下强制执行一致性，因此与基于向量场单步回归的方法相比，识别出的模型表现出显著更高的稳定性。当结合促进稀疏性的正则化时，这会产生简约模型，能够泛化到训练数据之外。我们证明可准确恢复表现出广泛行为的系统，包括稳定和不稳定不动点、周期轨道以及混沌吸引子。对于混沌系统，尽管长期轨迹预测受初始条件敏感性的固有限制，但我们表明多步训练产生的模型具有准确的短期动力学，并在长期统计特性（包括均值、方差和李雅普诺夫指数）上表现出强一致性。此外，我们建立了将轨迹误差与统计精度关联起来的理论边界，为这种行为的原则性解释迈出了一步。
英文摘要
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.
CommentsSubmitted to the SIAM J. Dynamical Systems
