# Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs

arXiv: https://arxiv.org/abs/1911.02109 · HTML: https://mlsanity.com/paper/1911.02109/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2019.

## Built on by (most cited first, 28 of 28)

- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [Reproducing Activation Function for Deep Learning](https://mlsanity.com/paper/2101.04844/) (2021)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method](https://mlsanity.com/paper/2111.01356/) (2021)
- [Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient](https://mlsanity.com/paper/2506.04375/) (2025)
- [Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law](https://mlsanity.com/paper/2105.11627/) (2021)
- [Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networks](https://mlsanity.com/paper/2205.13283/) (2022)
- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)
- [A Shallow Ritz Method for Elliptic Problems with Singular Sources](https://mlsanity.com/paper/2107.12013/) (2021)
- [First Order System Least Squares Neural Networks](https://mlsanity.com/paper/2409.20264/) (2024)
- [r-Adaptive Deep Learning Method for Solving Partial Differential Equations](https://mlsanity.com/paper/2210.10900/) (2022)
- [Self-adaptive deep neural network: Numerical approximation to functions and PDEs](https://mlsanity.com/paper/2109.02839/) (2021)
- [Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operator](https://mlsanity.com/paper/2110.10895/) (2021)
- [A variational neural network approach for glacier modelling with nonlinear rheology](https://mlsanity.com/paper/2209.02088/) (2022)
- [A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems](https://mlsanity.com/paper/2502.19893/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time](https://mlsanity.com/paper/2303.12245/) (2023)
- [Astral: training physics-informed neural networks with error majorants](https://mlsanity.com/paper/2406.02645/) (2024)
- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)
- [A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Media](https://mlsanity.com/paper/2603.11250/) (2026)
- [Characterizing and Mitigating the Difficulty in Training Physics-informed Artificial Neural Networks under Pointwise Constraints](https://mlsanity.com/paper/2206.09321/) (2022)
- [Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networks](https://mlsanity.com/paper/2406.06350/) (2024)
- [Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problems](https://mlsanity.com/paper/2407.01496/) (2024)
- [Global error estimators for parametric monotone nonlinearities and neural approximations](https://mlsanity.com/paper/2610.05767/) (2026)

## Abstract (arXiv)

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.