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

2025-07-23

AI Summary

This work addresses the challenge of learning dynamics for non-periodic inputs and non-Markovian stochastic systems—such as path-dependent stochastic differential equations (SDEs) and Lipschitz transformations of fractional Brownian motion. We propose the Mirror-Filled Fourier Neural Operator (MFNO), the first Fourier neural operator framework incorporating mirror padding to suppress boundary artifacts and enhance modeling capability for non-periodic signals. Theoretically, leveraging a Wong–Zakai-type theorem and multi-level approximation analysis, we rigorously prove that MFNO can approximate the solution operators of both classes of non-Markovian processes to arbitrary accuracy. Empirically, MFNO significantly outperforms LSTM, TCN, and DeepONet in resolution generalization and generates sample paths orders of magnitude faster than conventional numerical solvers, while maintaining high accuracy, strong generalization, and computational efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Approximating solutions for path-dependent stochastic differential equations
• Handling non-periodic inputs in neural operators
• Improving resolution generalization in stochastic system dynamics

Innovation

• Mirror-padded Fourier neural operator (MFNO)
• Handles non-periodic inputs via mirror padding
• Approximates path-dependent stochastic equations accurately