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In this work we explain and implement a method that uses an artificial neural network to solve differential equations numerically. The method was applied to a model of the flow of water in an open channel described by the Saint-Venant Equations (SVE). These equations constitute a system of partial differential equations. The method was implemented in Python using the libraries Numpy and Pytorch to manage matrix operations and the construction of the artificial neural network. The results of the method were compared with a common numerical method using RK1, where an average relative error of 4,05% was obtained. The results show that the proposed method has a promising performance in the resolution of partial differential equations, especially because of the versatility that it offers to define boundary conditions in complex geometries. The execution time was comparable to traditional methods, thanks to common performance enhancements developed for training artificial neural networks. Possible improvements for further research are mentioned.

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A Novel Deep Learning Method for Solving PDE’s Applied to a Shallow Water Problem

  • 会议论文

  • 发表日期：

    2025年

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  • Jose Palacios-García,
  • Julio Ibarra-Fiallo,
  • Sevando Espín-Torres

摘要

In this work we explain and implement a method that uses an artificial neural network to solve differential equations numerically. The method was applied to a model of the flow of water in an open channel described by the Saint-Venant Equations (SVE). These equations constitute a system of partial differential equations. The method was implemented in Python using the libraries Numpy and Pytorch to manage matrix operations and the construction of the artificial neural network. The results of the method were compared with a common numerical method using RK1, where an average relative error of 4,05% was obtained. The results show that the proposed method has a promising performance in the resolution of partial differential equations, especially because of the versatility that it offers to define boundary conditions in complex geometries. The execution time was comparable to traditional methods, thanks to common performance enhancements developed for training artificial neural networks. Possible improvements for further research are mentioned.