Stochastic PDEs

Neural SPDEs model full pathwise trajectory distributions but require memory and compute proportional to the number of time steps and lack resolution invariance. Monte Carlo rollouts are a robust approach to SPDE uncertainty estimation but are computationally slow. Neural SPDE's performance degrades monotonically at higher evaluation resolutions because its pathwise discretization is tied to the training grid. Wiener-chaos and polynomial-chaos expansions are accurate for low-dimensional or smooth stochastic drivers but become fragile under rough or high-dimensional noise.