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Deep neural network approach to forward-inverse problems. (English) Zbl 1442.35474
Summary: 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 [A. G. Baydin et al., J. Mach. Learn. Res. 18, Paper No. 153, 43 p. (2018; Zbl 06982909)], 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.
MSC:
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Software:
Adam; DiffSharp; PyTorch
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