Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments

TL;DR

This paper presents the mirror-padded Fourier neural operator (MFNO), a neural network that effectively learns stochastic system dynamics, with proven approximation capabilities and superior empirical performance over traditional models.

Contribution

Introduction of MFNO, an operator neural network that handles non-periodic inputs and approximates solutions of complex stochastic differential equations with theoretical guarantees.

Findings

MFNO can approximate solutions of path-dependent stochastic differential equations.

MFNO demonstrates strong resolution generalization compared to LSTMs, TCNs, and DeepONet.

MFNO achieves faster sample path generation with comparable or better accuracy.

Abstract

This paper introduces an operator-based neural network, the mirror-padded Fourier neural operator (MFNO), designed to learn the dynamics of stochastic systems. MFNO extends the standard Fourier neural operator (FNO) by incorporating mirror padding, enabling it to handle non-periodic inputs. We rigorously prove that MFNOs can approximate solutions of path-dependent stochastic differential equations and Lipschitz transformations of fractional Brownian motions to an arbitrary degree of accuracy. Our theoretical analysis builds on Wong--Zakai type theorems and various approximation techniques. Empirically, the MFNO exhibits strong resolution generalization--a property rarely seen in standard architectures such as LSTMs, TCNs, and DeepONet. Furthermore, our model achieves performance that is comparable or superior to these baselines while offering significantly faster sample path generation than classical numerical schemes.