Neural Stochastic PDEs
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice to model dynamical systems evolving under the influence of randomness. By formulating the search for mild solutions of an SPDE as a fixed-point problem, we introduce the Neural SPDE model to learn solution operators of SPDEs from partially observed data. Neural SPDEs provide an extension to two classes of physics-inspired neural architectures. On the one hand, our model extends the class of neural controlled, stochastic, rough differential equation (Neural CDE, SDE, RDE) models – continuous-time analogues of RNNs – in that it is capable of processing incoming sequential information even when the latter evolves in an infinite dimensional state space of functions. On the other hand, Neural SPDEs extend Neural Operators (NOs) – generalizations of neural networks to model mappings between spaces of functions – in that they can be used to learn solution operators of SPDEs (aka Ito maps) depending simultaneously on the initial condition and a realization of the driving noise (while there is no natural mechanism allowing to do so with NOs). The Neural SPDE model is resolution-invariant (in that it can be trained efficiently on coarser grids and then deployed on finer grids without sacrificing performance), it may utilize memory-efficient implicit-differentiation-based backpropagation and, once trained, its evaluation is up to 3 orders of magnitude faster than traditional SPDE solvers. Through extensive experiments on various semilinear SPDEs, including the 2D stochastic Navier-Stokes equations, we demonstrate how Neural SPDEs are capable of learning complex spatiotemporal dynamics with better accuracy and using only a modest amount of training data compared to all alternative models.
Cris Salvi is organiser of the Rough Path Interest Group. In 2021 he was awarded a DPhil in Mathematics at the University of Oxford. He recently moved to Imperial College London where he is Chapman Fellow in Mathematics.
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