Deep Neural Operator Learning for stochastic heat equations and applications
Abstract
In this paper, we establish a universal approximation theorem for solution operators of semilinear stochastic heat equations in spatial dimension d≥2, extending deep neural operator approximation results to SPDEs with distribution-valued Gaussian forcing. A smooth Fourier cutoff reduces the forcing to finitely many measurements, while Besov and Schauder estimates control the resulting solution error. The finite-dimensional solution map is then approximated by ReLU branch networks together with grid-based trunk functions, yielding explicit bounds on the numbers of forcing measurements and trunk functions and approximation in every finite stochastic moment. For linear equations, the reduction coincides with truncation of the first Wiener chaos. Lastly, as an application, we prove a high-probability local-in-time approximation theorem for stochastic Navier--Stokes equations, with finite measurements of both the initial condition and forcing and an explicit sensing rate.
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BibTeX
@misc{du2026deep,
title = {{Deep Neural Operator Learning for stochastic heat equations and applications}},
author = {Hengrong Du and Qi Feng and Weinan Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04955},
eprint = {2611.04955},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04955},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deep Neural Operator Learning for stochastic heat equations and applications
AU - Hengrong Du
AU - Qi Feng
AU - Weinan Wang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04955
UR - https://arxiv.org/abs/2611.04955
AB - In this paper, we establish a universal approximation theorem for solution operators of semilinear stochastic heat equations in spatial dimension d≥2, extending deep neural operator approximation results to SPDEs with distribution-valued Gaussian forcing. A smooth Fourier cutoff reduces the forcing to finitely many measurements, while Besov and Schauder estimates control the resulting solution error. The finite-dimensional solution map is then approximated by ReLU branch networks together with grid-based trunk functions, yielding explicit bounds on the numbers of forcing measurements and trunk functions and approximation in every finite stochastic moment. For linear equations, the reduction coincides with truncation of the first Wiener chaos. Lastly, as an application, we prove a high-probability local-in-time approximation theorem for stochastic Navier--Stokes equations, with finite measurements of both the initial condition and forcing and an explicit sensing rate.
ER -