  Stochastic partial differential equations (SPDEs) are the mathematical tool
of choice for modelling spatiotemporal PDE-dynamics under the influence of
randomness. Based on the notion of mild solution of an SPDE, we introduce a
novel neural architecture to learn solution operators of PDEs with (possibly
stochastic) forcing from partially observed data. The proposed Neural SPDE
model provides an extension to two popular classes of physics-inspired
architectures. On the one hand, it extends Neural CDEs and variants --
continuous-time analogues of RNNs -- in that it is capable of processing
incoming sequential information arriving at arbitrary spatial resolutions. On
the other hand, it extends Neural Operators -- generalizations of neural
networks to model mappings between spaces of functions -- in that it can
parameterize solution operators of SPDEs depending simultaneously on the
initial condition and a realization of the driving noise. By performing
operations in the spectral domain, we show how a Neural SPDE can be evaluated
in two ways, either by calling an ODE solver (emulating a spectral Galerkin
scheme), or by solving a fixed point problem. Experiments on various semilinear
SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the
Neural SPDE model is capable of learning complex spatiotemporal dynamics in a
resolution-invariant way, with better accuracy and lighter training data
requirements compared to alternative models, and up to 3 orders of magnitude
faster than traditional solvers.
