Better Neural PDE Solvers Through Data-Free Mesh Movers

Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Ampere equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a two-branch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems.

Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. Due to the importance of Navier-Stokes equations, the development on efficient numerical schemes is important for both science and engineer. Recently, with the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations, which can accelerate the simulation or inferring process in a mesh-free and differentiable way. In this paper, we point out that the capability of existing deep Navier-Stokes informed methods is limited to handle non-smooth or fractional equations, which are two critical situations in reality. To this end, we propose the \emph{Deep Random Vortex Method} (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation. Specifically, the random vortex dynamics motivates a Monte Carlo based loss function for training the neural network, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVM not only can efficiently solve Navier-Stokes equations involving rough path, non-differentiable initial conditions and fractional operators, but also inherits the mesh-free and differentiable benefits of the deep-learning-based solver. We conduct experiments on the Cauchy problem, parametric solver learning, and the inverse problem of both 2-d and 3-d incompressible Navier-Stokes equations. The proposed method achieves accurate results for simulation and inference of Navier-Stokes equations. Especially for the cases that include singular initial conditions, DRVM significantly outperforms existing PINN method.

Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs

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.