Neural solver uses math to improve partial differential equation accuracy
Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning
Machine LearningComputational Engineering, Finance, and Science
Summary
Solving partial differential equations (PDEs) accurately is important for many science and engineering tasks. The authors created a new neural network called AFDONet that uses a special math theory, adaptive Fourier decomposition, to design its structure. This math-guided design helps the network solve complex PDEs more precisely and makes its inner workings easier to understand. It works well on curved spaces and problems with sharp changes. Overall, the authors show a better way to build neural solvers with clear mathematical foundations.
What this means in practice
- •For computational engineers: Improve simulations on complex geometries and surfaces by using an accurate neural solver grounded in established math theory.
- •For machine learning system builders: Design interpretable neural networks for solving PDEs that maintain high accuracy even on datasets with sharp gradients.
Authors
Zeyuan Song, Zheyu Jiang
Abstract
Accurate numerical solutions of partial differential equations (PDEs) are crucial in numerous science and engineering applications. In this work, we introduce a novel neural PDE solver named AFDONet, which incorporates neural operator learning and adaptive Fourier decomposition (AFD) theory for the first time into a specifically designed variational autoencoder (VAE) structure, to solve a general class of nonlinear PDEs on smooth manifolds. AFDONet is the first neural PDE solver whose architectural and component design is fully guided by an established mathematical framework (in this case, AFD theory), turning neural operator design from an art to a science. Thus, AFDONet also exhibits exceptional mathematical explainability and groundness, and enjoys several desired properties. Furthermore, AFDONet achieves outstanding solution accuracy and competitive computational efficiency in several benchmark problems. In particular, thanks to its deep connections with AFD theory, AFDONet shows superior performance in solving PDEs on i) arbitrary (Riemannian) manifolds, and ii) datasets with sharp gradients. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.