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

Emily C. Ehrhardt, Hanno Gottschalk, Tobias J. Riedlinger

TL;DR

The paper develops a rigorous framework for NeuralODEs with second-order Runge-Kutta time integration by linking ODE flow regularity (via Beckmann's problem and Liouville's formula) to statistical learning theory. It introduces ReQU neural networks to approximate vector fields, derives model-error bounds for log-determinant differences, and provides comprehensive generalization bounds using metric entropy and concentration inequalities, culminating in PAC and PAC-like learnability results for the target distribution class $\mathcal{T}_{d,k}$ and its Hölder-bounded extension $\mathcal{T}_{d,k,H}$. The work highlights the interplay between numerical integration accuracy, network capacity, and statistical learnability, showing that NeuralODEs can be PAC-learnable under realistic regularity assumptions, albeit with sample complexity that grows exponentially in certain regimes. Overall, it offers a rigorous bridge between numerical analysis of ODE solvers and probabilistic learning guarantees for flow-based generative models with neural vector fields.

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.

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

TL;DR

The paper develops a rigorous framework for NeuralODEs with second-order Runge-Kutta time integration by linking ODE flow regularity (via Beckmann's problem and Liouville's formula) to statistical learning theory. It introduces ReQU neural networks to approximate vector fields, derives model-error bounds for log-determinant differences, and provides comprehensive generalization bounds using metric entropy and concentration inequalities, culminating in PAC and PAC-like learnability results for the target distribution class and its Hölder-bounded extension . The work highlights the interplay between numerical integration accuracy, network capacity, and statistical learnability, showing that NeuralODEs can be PAC-learnable under realistic regularity assumptions, albeit with sample complexity that grows exponentially in certain regimes. Overall, it offers a rigorous bridge between numerical analysis of ODE solvers and probabilistic learning guarantees for flow-based generative models with neural vector fields.

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.

Paper Structure

This paper contains 19 sections, 20 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Existence of generative flows
  3. Flow Maps and Liouville's Formula
  4. Beckmann's problem
  5. Unconstrained formulation of the Beckmann problem
  6. Potential theory and Schauder estimates
  7. Regular solutions to Beckmann's problem
  8. Generative Learning with Neural ODEs
  9. Neural ODEs and Runge-Kutta Integration
  10. Neural Networks with ReQU Activation
  11. Learning Objective
  12. Statistical Learning
  13. Empirical Risk Minimization
  14. Main Results
  15. Model Error Bounds
  16. ...and 4 more sections

Key Result

Lemma 2.1

(Representation of Log-Determinant) \newlabellem:logdet-formula0 For $s,t \in \mathbb{R}, k \in \mathbb{N}, k\geq 2$ let $\Phi_{s,t}$ be the flow generated by $\xi \in C^{1}(\mathbb{R}^d \times \mathbb{R}, \mathbb{R}^d)$ with $\xi(\cdot , \tau)$ globally Lipschitz for all $\tau \in \mathbb{R}$. The

Theorems & Definitions (43)

  • Lemma 2.1
  • Proof 1
  • Theorem 2.3: Existence and $C^{2,\alpha}$ Schauder estimate nardi2015schauder
  • Theorem 2.4: $C^{k,\alpha}$ Schauder estimate agmon1959estimates
  • Proof 2
  • Theorem 2.5
  • Proof 3
  • Theorem 3.1
  • Lemma 3.2
  • Proof 4
  • ...and 33 more