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

arXiv: https://arxiv.org/abs/2507.17887 · HTML: https://mlsanity.com/paper/2507.17887/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 23 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Temporal Convolutional Networks: A Unified Approach to Action Segmentation](https://mlsanity.com/paper/1608.08242/) (2016)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs](https://mlsanity.com/paper/2204.06255/) (2022)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)
- [Functional SDE approximation inspired by a deep operator network architecture](https://mlsanity.com/paper/2402.03028/) (2024)

## Built on by (most cited first, 2 of 2)

- [Analysis of Fourier Neural Operators via Effective Field Theory](https://mlsanity.com/paper/2507.21833/) (2025)
- [Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations](https://mlsanity.com/paper/2610.18939/) (2026)

## Abstract (arXiv)

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.