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The Evolving Neural Network Method for Scalar Hyperbolic Conservation Laws

thesis
posted on 2024-04-10, 20:21 authored by Brooke E HejnalBrooke E Hejnal

This thesis introduces the evolving neural network method for solving scalar hyperbolic conservation laws. This method uses neural networks to compute solutions with an optimal moving mesh that evolves with the solution over time. The motivation for this method was to produce solutions with high accuracy near shocks while reducing the overall computational cost. The evolving neural network method first approximates initial data with a neural network producing a continuous piecewise linear approximation. Then, the neural network representation is evolved in time according to a combination of characteristics and a finite volume-type method.

It is shown numerically and theoretically that the evolving neural network method out performs traditional fixed-mesh methods with respect to computational cost. Numerical results for benchmark test problems including Burgers’ equation and the Buckley-Leverett equation demonstrate that this method can accurately capture shocks and rarefaction waves with a minimal number of mesh points.

Funding

DMS-2110571

History

Degree Type

  • Doctor of Philosophy

Department

  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Zhiqiang Cai

Additional Committee Member 2

Min Chen

Additional Committee Member 3

Min Liu

Additional Committee Member 4

Xiangxiong Zhang

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