Robust Learning of Heterogeneous Dynamic Systems

2026-04-06

AI Summary

This work addresses the limitation of existing ordinary differential equation (ODE) approaches, which are typically confined to modeling a single dynamical system and struggle to learn shared patterns across multiple heterogeneous systems while ensuring generalization. The study introduces distributionally robust optimization into heterogeneous ODE modeling for the first time, constructing a unified robust model by maximizing worst-case rewards over a convex-combination uncertainty set of trajectory derivatives. A bilevel stabilization strategy is devised to enhance estimation stability. The resulting estimator admits an explicit weighted-average form, with theoretical guarantees on weight consistency, trajectory error bounds, and asymptotic validity of confidence intervals. Experiments demonstrate that the proposed method significantly outperforms existing approaches in both simulations and intracranial electroencephalography data analysis, achieving improved generalization and estimation stability.

Abstract

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.

Problem

• heterogeneous dynamic systems
• ordinary differential equations
• shared patterns
• robust learning
• multi-system modeling

Innovation

• distributionally robust learning
• heterogeneous ODE systems
• worst-case reward optimization
• bi-level stabilization
• trajectory estimation