Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs

Stochastic partial differential equations (SPDEs) are significant tools for modelling dynamics in many areas including atmospheric sciences and physics.Neural Operators, generations of neural networks with capability of learning maps between infinite-dimensional spaces, are strong tools for solving parametric PDEs. However, they lack of ability to modeling stochastic PDEs which usually have poor regularity \footnote{Roughly speaking, regularity describes the smoothness of a function.} due to the driving noise. As the theory of regularity structure has achieved great successes in the analysis of SPDEs and provides the concept \emph{model} that well-approximate SPDEs' solutions, we propose the Neural Operator with Regularity Structure (NORS) which incorporates the models for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic $\Phi^4_1$ model and the 2d stochastic Navier-Stokes equation, and the results demonstrate that the NORS is resolution-invariant, efficient, and can achieve one order of magnitude lower error with a modest amount of data.