NeSTR: Neural S-Transform Reconstruction for Gaussian SPDEs
Abstract
We introduce Neural S-Transform Reconstruction (NeSTR) for Gaussian stochastic partial differential equations (SPDEs). The learned object is not the random solution itself, but its S-Transform on a finite dimensional subspace of the white noise test space. For Wick type SPDEs, NeSTR yields a deterministic parametric PDE for the restricted S-Transform, and the stochastic solution is recovered from the Taylor coefficients of this transform at the origin. Thus the neural approximation is tied directly to the Wiener chaos expansion of the solution. The main novelty of NeSTR is an inverse S-Transform reconstruction principle combined with finite mode learning: we prove consistency of the recovered chaos coefficients, separate noise mode, chaos truncation, deterministic solver, and neural approximation errors, and give a deterministic learning formulation for the restricted transform. Numerical tests on an additive stochastic heat equation and a multiplicative Wick heat equation show that polynomial spectral representations in the transform variables provide stable access to higher order chaos coefficients. The resulting reconstruction separates the equation dependent deterministic coefficient fields from a reusable Gaussian--Hermite basis. Consequently, NeSTR learns an analytic generating function whose derivatives recover the Wiener chaos coefficients, rather than learning stochastic trajectories directly.
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BibTeX
@misc{agram2026nestr,
title = {{NeSTR: Neural S-Transform Reconstruction for Gaussian SPDEs}},
author = {Nacira Agram and Fred Espen Benth and Jan Rems},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23001},
eprint = {2609.23001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23001},
}
RIS
TY - UNPB
TI - NeSTR: Neural S-Transform Reconstruction for Gaussian SPDEs
AU - Nacira Agram
AU - Fred Espen Benth
AU - Jan Rems
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23001
UR - https://arxiv.org/abs/2609.23001
AB - We introduce Neural S-Transform Reconstruction (NeSTR) for Gaussian stochastic partial differential equations (SPDEs). The learned object is not the random solution itself, but its S-Transform on a finite dimensional subspace of the white noise test space. For Wick type SPDEs, NeSTR yields a deterministic parametric PDE for the restricted S-Transform, and the stochastic solution is recovered from the Taylor coefficients of this transform at the origin. Thus the neural approximation is tied directly to the Wiener chaos expansion of the solution. The main novelty of NeSTR is an inverse S-Transform reconstruction principle combined with finite mode learning: we prove consistency of the recovered chaos coefficients, separate noise mode, chaos truncation, deterministic solver, and neural approximation errors, and give a deterministic learning formulation for the restricted transform. Numerical tests on an additive stochastic heat equation and a multiplicative Wick heat equation show that polynomial spectral representations in the transform variables provide stable access to higher order chaos coefficients. The resulting reconstruction separates the equation dependent deterministic coefficient fields from a reusable Gaussian--Hermite basis. Consequently, NeSTR learns an analytic generating function whose derivatives recover the Wiener chaos coefficients, rather than learning stochastic trajectories directly.
ER -