Deep Neural Network Approach to Forward-Inverse Problems
This provides a method for researchers in computational science to handle forward-inverse problems in differential equations, though it appears incremental as it builds on existing DNN approaches.
The paper tackles solving differential equations (DEs) and their inverse problems by proposing a unified deep neural network (DNN) architecture that approximates analytic solutions and model parameters simultaneously, with theoretical convergence proofs and numerical validation on various equations like 1D transport and 2D heat equations.
In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data, the inverse problem. That is, we provide a unified framework of DNN architecture that approximates an analytic solution and its model parameters simultaneously. The architecture consists of a feed forward DNN with non-linear activation functions depending on DEs, automatic differentiation, reduction of order, and gradient based optimization method. We also prove theoretically that the proposed DNN solution converges to an analytic solution in a suitable function space for fundamental DEs. Finally, we perform numerical experiments to validate the robustness of our simplistic DNN architecture for 1D transport equation, 2D heat equation, 2D wave equation, and the Lotka-Volterra system.