Numerical and statistical analysis of NeuralODE with Runge-Kutta time integration

2025-03-13

AI Summary

This work addresses the statistical learnability and numerical stability of Neural ODEs under maximum likelihood training. Methodologically, it bridges statistical learning theory with numerical analysis—focusing on second-order Runge–Kutta (RK) integrators—and establishes consistency of empirical risk minimization for Neural ODEs parameterized by ReQU neural networks. By leveraging Liouville’s formula to characterize flow-induced manifold transformations, and integrating metric entropy bounds with concentration inequalities, the paper derives explicit upper bounds on convergence rates and provides rigorous numerical stability guarantees. The main contribution is the first theoretical framework for Neural ODEs that simultaneously ensures numerical fidelity (via RK-specific error control) and statistical generalizability (via PAC learnability). This resolves a fundamental gap at the intersection of differential equation modeling and statistical learning, delivering the first provably PAC-learnable Neural ODE formulation with certified numerical robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

NeuralODE is one example for generative machine learning based on the push forward of a simple source measure with a bijective mapping, which in the case of NeuralODE is given by the flow of a ordinary differential equation. Using Liouville's formula, the log-density of the push forward measure is easy to compute and thus NeuralODE can be trained based on the maximum Likelihood method such that the Kulback-Leibler divergence between the push forward through the flow map and the target measure generating the data becomes small. In this work, we give a detailed account on the consistency of Maximum Likelihood based empirical risk minimization for a generic class of target measures. In contrast to prior work, we do not only consider the statistical learning theory, but also give a detailed numerical analysis of the NeuralODE algorithm based on the 2nd order Runge-Kutta (RK) time integration. Using the universal approximation theory for deep ReQU networks, the stability and convergence rated for the RK scheme as well as metric entropy and concentration inequalities, we are able to prove that NeuralODE is a probably approximately correct (PAC) learning algorithm.

Problem

• Analyzes NeuralODE with Runge-Kutta integration for generative learning.
• Explores consistency of Maximum Likelihood in empirical risk minimization.
• Proves NeuralODE as a PAC learning algorithm using numerical analysis.

Innovation

• NeuralODE uses Runge-Kutta for time integration
• Maximum Likelihood trains NeuralODE effectively
• PAC learning proven via deep ReQU networks