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This research advances the application of Fourier neural operators to stochastic Volterra integral equations, establishing rigorous convergence guarantees for systems with memory. By validating mirror-padded architectures against piecewise linear Brownian motion approximations, the study offers a robust framework for predicting complex random trajectories with high generalization capabilities. The work bridges theoretical probability analysis with practical deep learning methods, setting a new standard for operator learning in stochastic dynamics.
Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations
arXiv:2610.18939v1 Announce Type: cross Abstract: This paper studies Wong--Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong--Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.
Abstract:This paper studies Wong--Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong--Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.
Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR)
arXiv:2610.18939 [stat.ML]
(or arXiv:2610.18939v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.18939 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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