mlsanityData to 14 Oct 2026

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

arXiv 1911.02109 ↗First posted 5 Nov 2019cs.LG cs.NA math.NAcited by 28

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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.

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

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Citing papers per year: 2021: 10; 2022: 5; 2023: 2; 2024: 4; 2025: 5; 2026: 2.

28 papers.

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  1. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  2. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 26
  3. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  4. 2021A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problemscited by 20
  5. 2021Reproducing Activation Function for Deep Learningcited by 19
  6. 2021Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equationcited by 16
  7. 2022Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approachcited by 15
  8. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  9. 2025Solving engineering eigenvalue problems with neural networks using the Rayleigh quotientcited by 8
  10. 2021Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Lawcited by 7
  11. 2022Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networkscited by 7
  12. 2025Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimationcited by 7
16 more
  1. 2021A Shallow Ritz Method for Elliptic Problems with Singular Sourcescited by 6
  2. 2024First Order System Least Squares Neural Networkscited by 5
  3. 2022r-Adaptive Deep Learning Method for Solving Partial Differential Equationscited by 3
  4. 2021Self-adaptive deep neural network: Numerical approximation to functions and PDEscited by 2
  5. 2021Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operatorcited by 2
  6. 2022A variational neural network approach for glacier modelling with nonlinear rheologycited by 2
  7. 2025A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problemscited by 2
  8. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 2
  9. 2023Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Timecited by 1
  10. 2024Astral: training physics-informed neural networks with error majorantscited by 1
  11. 2025Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force methodcited by 1
  12. 2026A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Mediacited by 1
  13. 2022Characterizing and Mitigating the Difficulty in Training Physics-informed Artificial Neural Networks under Pointwise Constraintscited by 0
  14. 2024Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networkscited by 0
  15. 2024Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problemscited by 0
  16. 2026Global error estimators for parametric monotone nonlinearities and neural approximationscited by 0