Theoretical Framework for Learning High-Dimensional Controlled Non-Linear Dynamical Systems Using Neural ODEs
A new preprint introduces a mathematically rigorous class of models to analyze how neural ordinary differential equations (neural ODEs) are trained via online stochastic gradient descent, solving their training dynamics using dynamical mean field theory. Neural ODEs have emerged as a unifying framework connecting continuous-time dynamical systems with modern deep learning, but a solid theoretical foundation for their learning behavior has been lacking. The work derives explicit learning curves in the high-dimensional limit, potentially advancing understanding of generalization and optimization across a broad range of neural network architectures.
Posted to arXiv in June 2026, this theoretical machine learning paper addresses a gap in the formal understanding of neural ordinary differential equations (neural ODEs), a framework that bridges continuous-time dynamical systems and data-driven deep learning. The authors highlight a dual dynamical structure inherent to neural ODEs: inference dynamics governing forward computation and training dynamics governing parameter optimization. By introducing a tractable class of high-dimensional controlled nonlinear dynamical systems, the researchers apply dynamical mean field theory — a technique borrowed from statistical physics — to exactly solve the training dynamics under online stochastic gradient descent. The result is closed-form learning curves in the high-dimensional limit, offering quantitative predictions about how these models generalize. The framework is designed to be broadly applicable, covering architectures such as ResNets, autoregressive language models, generative models, and recurrent neural networks used in theoretical neuroscience. The paper is 28 pages with 2 figures and is labeled as part one of a series, suggesting further theoretical developments are forthcoming.
What's missing
As a preprint, this work has not yet undergone formal peer review. It is unclear how well the high-dimensional asymptotic results translate to practically sized models, and empirical validation against real-world neural ODE training runs is not described in the abstract.
What different sources said
- arXiv stat.MLCenter
Theory of learning of high-dimensional controlled non-linear dynamical systems (I): models and methods
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