DNN modeling of partial differential equations with incomplete data

We present a computational technique for modeling the evolution of partial differential equations (PDEs) with incomplete data. It is a significant extension of the recent work of data driven learning of PDEs, in the sense that we consider two forms of partial data: data are observed only on a subset of the domain, and data are observed only on a subset of the state variables. Both cases resemble more realistic data collection scenarios in real -world applications. Leveraging the recent work on modeling partially-observed dynamical systems, we present a deep neural network (DNN) structure that is suitable for PDE modeling with such kinds of incomplete data. In addition to the mathematical motivation for the DNN structure, we present an extensive set of numerical examples in both one -and two-dimensions to demonstrate the effectiveness of the proposed DNN modeling. In one example, the method can accurately predict the solution when data are only available in less than half (40%) of the domain.(c) 2023 Elsevier Inc. All rights reserved.