AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-20
Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations
The authors develop and analyze neural stochastic residual methods for fully nonlinear parabolic terminal problems under exact diffusion sampling. A quadratic stochastic residual enables global training and backward regression with scalar networks and losses involving only spatial derivatives, and the residual norm as well as value, gradient, and Hessian errors are bounded in terms of the losses and the time step. In ten-dimensional backward-regression tests, mean relative Hessian errors were 5–6% with the quadratic residual versus about 15% with the first-order residual, and tuned backward regression achieved Hessian accuracy comparable to physics-informed neural networks.