Data-integrated neural networks for solving partial differential equations

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($10^{-5}$)).