Dnn Modeling of Partial Differential Equations with Incomplete Data

23 Pages Posted: 14 Mar 2023

See all articles by Victor Churchill

Victor Churchill

The Ohio State University

Yuan Chen

The Ohio State University

Zhongshu Xu

The Ohio State University

Dongbin Xiu

The Ohio State University

Abstract

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.

Keywords: Data driven modeling, deep neural networks, reduced PDE systems, incomplete data

Suggested Citation

Churchill, Victor and Chen, Yuan and Xu, Zhongshu and Xiu, Dongbin, Dnn Modeling of Partial Differential Equations with Incomplete Data. Available at SSRN: https://ssrn.com/abstract=4377952 or http://dx.doi.org/10.2139/ssrn.4377952

Victor Churchill

The Ohio State University ( email )

Yuan Chen

The Ohio State University ( email )

Zhongshu Xu

The Ohio State University ( email )

Dongbin Xiu (Contact Author)

The Ohio State University ( email )

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