🤖 AI Summary
This work addresses the differentiability and portability bottlenecks in numerical solving of partial differential equations (PDEs) for incompressible computational fluid dynamics (CFD). Methodologically, it encodes classical PDE discretization schemes directly as analytical, weight-free convolutional layers and integrates Jacobi iteration with a U-Net architecture to realize a differentiable multigrid solver—entirely without training. The key contribution is the first formulation of a traditional CFD solver as an untrained, fully differentiable, and platform-agnostic neural network module. This enables seamless coupling with data-driven components to construct hybrid physics-AI models. Experiments demonstrate high accuracy and efficiency on the convection–diffusion equation, Burgers equation, and incompressible Navier–Stokes equations—matching the precision of established CFD solvers while incurring zero training cost.
📝 Abstract
Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to implement numerical discretisations as convolutional layers of a neural network, where the weights or filters are determined analytically rather than by training. Furthermore, we demonstrate that these systems can be solved entirely by functions in AI libraries, either by using Jacobi iteration or multigrid methods, the latter realised through a U-Net architecture. Some advantages of the Neural Physics approach are that (1) the methods are platform agnostic; (2) the resulting solvers are fully differentiable, ideal for optimisation tasks; and (3) writing CFD solvers as (untrained) neural networks means that they can be seamlessly integrated with trained neural networks to form hybrid models. We demonstrate the proposed approach on a number of test cases of increasing complexity from advection-diffusion problems, the non-linear Burgers equation to the Navier-Stokes equations. We validate the approach by comparing our results with solutions obtained from traditionally written code and common benchmarks from the literature. We show that the proposed methodology can solve all these problems using repurposed AI libraries in an efficient way, without training, and presents a new avenue to explore in the development of methods to solve PDEs with implicit methods.