Chaos into Order: Neural Framework for Expected Value Estimation of Stochastic Partial Differential Equations

Stochastic Partial Differential Equations (SPDEs) are fundamental to modeling complex systems in physics, finance, and engineering, yet their numerical estimation remains a formidable challenge. Traditional methods rely on discretization, introducing computational inefficiencies, and limiting applicability in high-dimensional settings. In this work, we introduce a novel neural framework for SPDE estimation that eliminates the need for discretization, enabling direct estimation of expected values across arbitrary spatio-temporal points. We develop and compare two distinct neural architectures: Loss Enforced Conditions (LEC), which integrates physical constraints into the loss function, and Model Enforced Conditions (MEC), which embeds these constraints directly into the network structure. Through extensive experiments on the stochastic heat equation, Burgers' equation, and Kardar-Parisi-Zhang (KPZ) equation, we reveal a trade-off: While LEC achieves superior residual minimization and generalization, MEC enforces initial conditions with absolute precision and exceptionally high accuracy in boundary condition enforcement. Our findings highlight the immense potential of neural-based SPDE solvers, particularly for high-dimensional problems where conventional techniques falter. By circumventing discretization and explicitly modeling uncertainty, our approach opens new avenues for solving SPDEs in fields ranging from quantitative finance to turbulence modeling. To the best of our knowledge, this is the first neural framework capable of directly estimating the expected values of SPDEs in an entirely non-discretized manner, offering a step forward in scientific computing.