Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations
Abstract
We develop and analyze neural stochastic residual methods for fully nonlinear parabolic terminal problems under exact diffusion sampling. A quadratic stochastic residual supports global training and backward regression with scalar networks and losses involving only spatial derivatives. For global training, the correction removes a Hessian penalty from the limiting first-order loss. Under a uniform output regularity budget, we bound the continuous residual norm by the square roots of the global loss or normalized backward-regression excess loss, an O(h1/2) discretization term for time step h, and terminal error. Residual stability then gives value, gradient and Hessian error bounds. Under suitable solution and terminal assumptions, we construct admissible networks with O(h1/2) bounds on these approximation errors and the square roots of both loss criteria. In ten-dimensional backward-regression tests using a shared schedule tuned for the quadratic residual, mean relative Hessian errors are $5−−6\%$ with the quadratic residual and about 15% with the first-order residual. Tuned backward regression achieves Hessian accuracy comparable to physics-informed neural networks across the tested nonlinear coefficients. Translation tests show that the ratio of Hessian error to residual can grow sharply under a fixed sampling law.
Related
- Maximal Lp-Lq regularity of finite element methods for stochastic parabolic equations in polygonal domains
- Unconditionally L2-stable explicit--implicit--null filtered Runge--Kutta methods for general nonlinear degenerate parabolic equations
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- Geometry property of stochastic parabolic equations and Prékopa-Leindler inequality on bounded domains
BibTeX
@misc{zhao2026deep,
title = {{Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations}},
author = {Zhenhua Zhao and Jihao Long},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.16730},
eprint = {2607.16730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16730},
}
RIS
TY - UNPB
TI - Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations
AU - Zhenhua Zhao
AU - Jihao Long
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2607.16730
UR - https://arxiv.org/abs/2607.16730
AB - We develop and analyze neural stochastic residual methods for fully nonlinear parabolic terminal problems under exact diffusion sampling. A quadratic stochastic residual supports global training and backward regression with scalar networks and losses involving only spatial derivatives. For global training, the correction removes a Hessian penalty from the limiting first-order loss. Under a uniform output regularity budget, we bound the continuous residual norm by the square roots of the global loss or normalized backward-regression excess loss, an O(h^1/2) discretization term for time step h, and terminal error. Residual stability then gives value, gradient and Hessian error bounds. Under suitable solution and terminal assumptions, we construct admissible networks with O(h^1/2) bounds on these approximation errors and the square roots of both loss criteria. In ten-dimensional backward-regression tests using a shared schedule tuned for the quadratic residual, mean relative Hessian errors are $5−−6\%$ with the quadratic residual and about 15% with the first-order residual. Tuned backward regression achieves Hessian accuracy comparable to physics-informed neural networks across the tested nonlinear coefficients. Translation tests show that the ratio of Hessian error to residual can grow sharply under a fixed sampling law.
ER -