Theoretical Framework for Learning High-Dimensional Controlled Non-Linear Dynamical Systems Using Neural ODEs
Researchers have developed a theoretical framework for understanding how neural ordinary differential equations (neural ODEs) learn from data when trained with stochastic gradient descent. The work uses dynamical mean field theory to analyze training dynamics and derive learning curves in high-dimensional settings. This advances fundamental understanding of how neural networks generalize and train, with implications for multiple architectures including ResNets and generative models.
A new theoretical study provides mathematical foundations for neural ordinary differential equations, a framework that unifies continuous-time dynamical systems modeling with discrete deep learning. The researchers introduce a class of models specifically designed for theoretical analysis and solve their training dynamics using dynamical mean field theory, deriving learning curves in the high-dimensional limit. Neural ODEs are notable for their dual nature: inference dynamics that govern forward computation and training dynamics that control parameter optimization. This theoretical approach applies to multiple important architectures and paradigms, including residual networks (ResNets), autoregressive models for next-token generation, generative models, and recurrent neural networks used in theoretical neuroscience. The work bridges a gap between practical deep learning and rigorous mathematical analysis of how these systems learn and generalize.
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The study's own limitations, scope boundaries, and open questions are not detailed in the abstract provided. Specific experimental validation results, computational complexity analysis, and comparison with existing theoretical frameworks are not included in the available text.
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- arXiv stat.MLCenter
Theory of learning of high-dimensional controlled non-linear dynamical systems (I): models and methods
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