Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs

Stochastic partial differential equations (SPDEs) are powerful mathematical models for modeling dynamics in many areas including atmospheric sciences and physics. Neural Operators are deep learning based approach which are proposed for solving parametric PDEs. As the theory of regularity structure has achieved great successes in analyzing SPDEs and provides the concept model feature vectors that wellapproximate SPDEs' solutions, we propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. Stochastic partial differential equations (SPDEs), which generalizes PDE via random force terms and coefficients, are significant tools for modeling dynamics in many areas including atmospheric sciences (Hasselmann, 1976), physics (Uhlenbeck & Ornstein, 1930), biology (Wilkinson, 2018), economics (Barone-Adesi & Whaley, 1987), etc. SPDEs are used to study statistical mechanics of the dynamics systems, e.g., stochastic Navier-Stokes equations models the statistics of turbulent flows (Buckmaster & Vicol, 2019) in atmospheric science and the Φ Since SPDEs relate to many scientific open problems, studying the solution of SPDEs from both mathematical proving and numerical methods is a hot research direction in both math and physics. Inspired by recent advances in using AI techniques to accelerate scientific computing, we study using deep learning methods for modeling the solution of SPDEs.