Cristopher Salvi: From Neural SDEs to Neural SPDEs, A rough paths perspective

Описание: Title: From Neural SDEs to Neural SPDEs, A rough paths perspective
Speaker: Cristopher Salvi
Abstract: Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling dynamical systems evolving under the influence of randomness. 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, SDEs, RDEs -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving at an arbitrary resolution, both in space and in time. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can be used to learn solution operators of SPDEs (a.k.a. It\^o maps) depending simultaneously on the initial condition and a realization of the driving noise. By transferring some of its operations to the spectral domain, we show how a Neural SPDE can be evaluated either calling an ODE solver or solving a fixed point problem, inheriting in both cases memory-efficient backpropagation capabilities for training provided by existing adjoint-based or implicit-differentiation-based methods. Experiments on various semilinear SPDEs (including stochastic Navier-Stokes) demonstrate how our model is capable of learning complex spatiotemporal dynamics with better accuracy and using only a modest amount of training data compared to all alternative models, and its evaluation is up to 3 orders of magnitude faster than traditional solvers.