Universal approximation property of neural stochastic differential equations

2025-03-20

AI Summary

This work investigates the universal approximation capability of neural stochastic differential equations (NSDEs) for general Itô-type diffusion SDEs. For continuous coefficient functions satisfying a global linear growth condition, we establish that NSDEs achieve local uniform approximation in path space. Our method integrates stochastic analysis (via Itô’s formula), functional approximation theory, and parameterized neural network design; leveraging compactness arguments in path space, we derive quantitative error bounds and explicit convergence rates for SDEs with regular coefficients. This work presents the first universal approximation theory linking multiple neural network architectures to NSDEs. By transcending conventional deterministic neural network approximation frameworks, our results provide a rigorous theoretical foundation and computationally tractable guarantees for modeling stochastic dynamical systems, pricing financial derivatives, and simulating complex physical processes.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Social Networks and Social Media: Computational social science

Abstract

We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential equations can approximate general stochastic differential equations, both of It^o diffusion type, arbitrarily well. Moreover, quantitative error estimates are derived for stochastic differential equations with sufficiently regular coefficients.

Problem

• Approximate continuous functions under growth constraints
• Model general stochastic differential equations accurately
• Provide error estimates for regular coefficient cases

Innovation

• Neural networks approximate continuous functions uniformly
• Neural SDEs approximate general SDEs effectively
• Quantitative error estimates for regular SDEs