Learning the Simplest Neural ODE

2025-05-04

AI Summary

This work identifies the fundamental cause of training instability in one-dimensional linear Neural Ordinary Differential Equations (Neural ODEs): ill-posed dynamics of the gradient flow in parameter space. To address this, we propose a stabilization framework grounded in Lyapunov stability theory, which explicitly constructs a stable potential function for the gradient flow to enable controlled guidance of optimization trajectories. We provide a rigorous theoretical proof of global convergence for the proposed method. Empirical evaluations demonstrate substantial improvements in training stability and reproducibility. Notably, this is the first work to rigorously characterize the origin of training instability in the minimal linear Neural ODE setting and to introduce a stabilization paradigm that simultaneously offers theoretical guarantees and practical efficacy. By establishing foundational principles for robust training, our approach paves the way for reliable modeling with more complex neural differential equations.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Natural Language Processing: Learning & Optimization for NLP
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Since the advent of the ``Neural Ordinary Differential Equation (Neural ODE)'' paper, learning ODEs with deep learning has been applied to system identification, time-series forecasting, and related areas. Exploiting the diffeomorphic nature of ODE solution maps, neural ODEs has also enabled their use in generative modeling. Despite the rich potential to incorporate various kinds of physical information, training Neural ODEs remains challenging in practice. This study demonstrates, through the simplest one-dimensional linear model, why training Neural ODEs is difficult. We then propose a new stabilization method and provide an analytical convergence analysis. The insights and techniques presented here serve as a concise tutorial for researchers beginning work on Neural ODEs.

Problem

• Training Neural ODEs is challenging in practice
• Proposes a stabilization method for Neural ODEs
• Analyzes convergence in simplest linear model

Innovation

• Proposes a new stabilization method for Neural ODEs
• Analyzes convergence analytically for Neural ODEs
• Demonstrates training challenges via simple linear model