Overcoming the Curse of Dimensionality in Parabolic PDEs: A Monte Carlo-Neural Surrogate Framework with Benchmark Analysis

Dr. Pankaj Singh

Abstract: Classical mesh-based discretizations of parabolic partial differential equations (PDEs)-finite differences, finite elements, and full tensor-product quadrature-incur a computational cost that grows exponentially with the spatial dimension d, a phenomenon known as the curse of dimensionality. This barrier makes such methods impractical for PDEs arising in financial mathematics, stochastic optimal control, and statistical physics, where d can range from tens to several hundreds. In this paper we give a self-contained treatment of two dimension-robust alternatives-Monte Carlo evaluation of the Feynman-Kac representation and neural-network function approximation trained on simulation-generated data, in the spirit of the deep BSDE and Deep Galerkin families of methods-and we benchmark them against full tensor-grid quadrature on an analytically solvable -dimensional parabolic model problem. We derive the governing equations, describe the numerical algorithms in full detail (including pseudocode), and report reproducible numerical experiments across dimensions d ∈ {10, 100, 500}. Our measurements confirm that Monte Carlo error decays at the classical rate O (N^-1/2) independently of d, while the cost of an equally accurate full tensor grid grows like m^d; for d=8 and a modest per-axis resolution m=21 this already exceeds 3.8 x10¹⁰ grid points, versus roughly 2.3x10⁵ Monte Carlo samples for the same target accuracy. We further train a two-layer neural surrogate on Monte Carlo-generated regression labels in dimension d=50 and obtain a relative error of 6.6% against the closed-form solution after a few seconds of training on a single CPU core, illustrating that even a very simple network can learn a genuinely 50-dimensional function from finite data. We conclude with a discussion of the trade-offs between these families of methods and directions for further work.

Keywords: high-dimensional PDEs, curse of dimensionality, Monte Carlo methods, Feynman?Kac formula, deep BSDE, neural surrogate models, computational mathematics