A Deep Learning Galerkin Method for the Closed-Loop Geothermal System

TL;DR

This paper introduces a Deep Learning Galerkin Method (DGM) for modeling closed-loop geothermal systems, combining neural networks with PDE solving to improve accuracy and convergence in simulating geothermal heat exchange.

Contribution

It proposes a novel neural network-based Galerkin method tailored for coupled multi-physics PDEs in geothermal systems, demonstrating convergence and approximation capabilities.

Findings

Neural network solutions satisfy PDEs and boundary conditions.

Convergence of the loss function and neural network to the exact solution proved.

Numerical examples show effective approximation of geothermal heat exchange.

Abstract

There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). This article is to propose a Deep Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is a new coupled multi-physics PDEs and mainly consists of a framework of underground heat exchange pipelines to extract the geothermal heat from the geothermal reservoir. This method is a natural combination of Galerkin Method and machine learning with the solution approximated by a neural network instead of a linear combination of basis functions. We train the neural network by randomly sampling the spatiotemporal points and minimize loss function to satisfy the differential operators, initial condition, boundary and interface conditions. Moreover, the approximate ability of the neural network is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in L^2 norm under certain conditions. Finally, some numerical examples are carried out to demonstrate the approximation ability of the neural networks intuitively.