  Stochastic partial differential equations (SPDEs) are significant tools for
modeling 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 the ability to modeling SPDEs which usually
have poor regularity due to the driving noise. As the theory of regularity
structure has achieved great successes in analyzing SPDEs and provides the
concept model feature vectors that well-approximate SPDEs' solutions, we
propose the Neural Operator with Regularity Structure (NORS) which incorporates
the feature vectors for modeling dynamics driven by SPDEs. We conduct
experiments on various of SPDEs including the dynamic Phi41 model and the 2d
stochastic Navier-Stokes equation, and the results demonstrate that the NORS is
resolution-invariant, efficient, and achieves one order of magnitude lower
error with a modest amount of data.
