Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the problem of learning shared patterns from multiple heterogeneous dynamic systems. In this article, we propose a novel distributionally robust learning approach for modeling heterogeneous ODE systems. Specifically, we construct a robust dynamic system by maximizing a worst-case reward over an uncertainty class formed by convex combinations of the derivatives of trajectories. We show the resulting estimator admits an explicit weighted average representation, where the weights are obtained from a quadratic optimization that balances information across multiple data sources. We further develop a bi-level stabilization procedure to address potential instability in estimation. We establish rigorous theoretical guarantees for the proposed method, including consistency of the stabilized weights, error bound for robust trajectory estimation, and asymptotical validity of pointwise confidence interval. We demonstrate that the proposed method considerably improves the generalization performance compared to the alternative solutions through both extensive simulations and the analysis of an intracranial electroencephalogram data.

翻译：常微分方程（ODE）为建模广泛科学领域中的动态系统提供了强大框架。然而，现有大多数ODE方法仅针对单一系统建模，未能充分解决从多个异构动态系统中学习共享模式的问题。本文提出一种新颖的分布鲁棒学习方法，用于建模异构ODE系统。具体而言，我们通过最大化轨迹导数凸组合构成的不确定性类中的最坏情况奖励，构建鲁棒动态系统。研究表明，所得估计量具有显式加权平均表示，其中权重通过二次优化获得，该优化平衡了多个数据源间的信息。我们进一步开发了双层稳定化程序，以解决估计中潜在的不稳定性问题。为所提方法建立了严格的理论保障，包括稳定权重的一致性、鲁棒轨迹估计的误差界，以及逐点置信区间的渐近有效性。通过广泛数值仿真和颅内脑电图数据分析证明，与替代方案相比，所提方法显著提升了泛化性能。