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 $O(h^{1/2})$ 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.