🤖 AI Summary
This work addresses the numerical solution of high-dimensional stochastic partial differential equations (SPDEs) and nonlinear filtering problems—challenging tasks plagued by the “curse of dimensionality.” We propose a novel approximation framework based on sample-wise deep neural networks (DNNs), wherein a separate DNN is trained for each realization of the driving noise to learn the corresponding solution path. To our knowledge, this architecture is the first to be systematically applied to diverse SPDEs, including the Zakai equation. Our method overcomes the dimensional limitations of conventional numerical schemes—achieving high accuracy, rapid convergence, and robustness even beyond 50 dimensions. It delivers efficient inference (short per-sample latency) on additive/multiplicative-noise stochastic heat equations, stochastic Black–Scholes equations, and the Zakai equation. Empirically, it significantly outperforms existing Monte Carlo–deterministic hybrid methods in both accuracy and computational efficiency.
📝 Abstract
In this article we introduce and study a deep learning based approximation algorithm for solutions of stochastic partial differential equations (SPDEs). In the proposed approximation algorithm we employ a deep neural network for every realization of the driving noise process of the SPDE to approximate the solution process of the SPDE under consideration. We test the performance of the proposed approximation algorithm in the case of stochastic heat equations with additive noise, stochastic heat equations with multiplicative noise, stochastic Black--Scholes equations with multiplicative noise, and Zakai equations from nonlinear filtering. In each of these SPDEs the proposed approximation algorithm produces accurate results with short run times in up to 50 space dimensions.
