Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
This is an incremental approach for computational mathematics and PDE solving, offering a domain-specific method that may benefit researchers in numerical analysis and machine learning.
The paper tackles solving elliptic PDEs by proposing an unsupervised deep learning method using neural networks with least-squares functionals as loss functions, specifically applying the first-order system least-squares (FOSLS) functional to one-dimensional second-order elliptic PDEs, but no concrete numerical results or performance metrics are provided.
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine parameters of the deep neural network. There are various least-squares functionals for a partial differential equation. This paper focuses on the so-called first-order system least-squares (FOSLS) functional studied in [3], which is based on a first-order system of scalar second-order elliptic PDEs. Numerical results for second-order elliptic PDEs in one dimension are presented.