Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model

Sunwoo Hwang; Seongwon Lee; Hyung Ju Hwang; Sunwoo Hwang; Seongwon Lee; Hyung Ju Hwang

doi:10.3934/mbe.2021421

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 8524-8534. doi: 10.3934/mbe.2021421

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Research article

Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model

1.
Department of Mathematics Pohang University of Science and Technology Pohang, Republic of Korea
2.
Innovation Center for Industrial Mathematics, National Institute for Mathematical Sciences, Daejeon, Republic of Korea

Received: 05 July 2021 Accepted: 22 September 2021 Published: 29 September 2021

We consider the mathematical model of chemotaxis introduced by Patlak, Keller, and Segel. Aggregation and progression waves are present everywhere in the population dynamics of chemotactic cells. Aggregation originates from the chemotaxis of mobile cells, where cells are attracted to migrate to higher concentrations of the chemical signal region produced by themselves. The neural net can be used to find the approximate solution of the PDE. We proved that the error, the difference between the actual value and the predicted value, is bound to a constant multiple of the loss we are learning. Also, the Neural Net approximation can be easily applied to the inverse problem. It was confirmed that even when the coefficient of the PDE equation was unknown, prediction with high accuracy was achieved.
- differential equation,
- approximated solution,
- artificial neural networks,
- Patlak-Keller-Segel equation
Citation: Sunwoo Hwang, Seongwon Lee, Hyung Ju Hwang. Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 8524-8534. doi: 10.3934/mbe.2021421

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Abstract

We consider the mathematical model of chemotaxis introduced by Patlak, Keller, and Segel. Aggregation and progression waves are present everywhere in the population dynamics of chemotactic cells. Aggregation originates from the chemotaxis of mobile cells, where cells are attracted to migrate to higher concentrations of the chemical signal region produced by themselves. The neural net can be used to find the approximate solution of the PDE. We proved that the error, the difference between the actual value and the predicted value, is bound to a constant multiple of the loss we are learning. Also, the Neural Net approximation can be easily applied to the inverse problem. It was confirmed that even when the coefficient of the PDE equation was unknown, prediction with high accuracy was achieved.

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