Data-integrated neural networks for solving partial differential equations

TL;DR

This paper introduces DataInNet, a neural network framework that integrates data and physical constraints to effectively solve complex PDEs with high accuracy, outperforming traditional methods.

Contribution

The work presents a novel neural network architecture that combines data integration with residual learning to enforce physical constraints in PDE solutions.

Findings

Achieves relative L2 error of about 10^{-6} on Helmholtz equations.

Successfully solves high-frequency PDEs with errors around 10^{-5}.

Outperforms existing PDE solving methods in numerical experiments.

Abstract

In this work, we propose data-integrated neural networks (DataInNet) for solving partial differential equations (PDEs), offering a novel approach to leveraging data (e.g., source terms, initial conditions, and boundary conditions). The core of this work lies in the integration of data into a unified network framework. DataInNet comprises two subnetworks: a data integration neural network responsible for accommodating and fusing various types of data, and a fully connected neural network dedicated to learning the residual physical information not captured by the data integration neural network. This network architecture inherently excludes function classes that violate known physical constraints, thereby substantially narrowing the solution search space. Numerical experiments demonstrate that the proposed DataInNet delivers superior performance on challenging problems, such as the Helmholtz equation (relative \(L^2\) error: O(\(10^{-6}\))) and PDEs with high frequency solutions (relative \(L^2\) error: O(\(10^{-5}\))).