mlsanityData to 7 Oct 2026

Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems

arXiv 1904.05417 ↗First posted 10 Apr 2019cs.LG stat.MLcited by 31

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Abstract (arXiv)

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. The method is mesh free and can be easily applied to an arbitrary regular domain. We focus on 2D second order elliptical system with non-constant coefficients, with application to Electrical Impedance Tomography.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Apr 2019.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 14; 2023: 6; 2024: 2; 2025: 4; 2026: 2.

31 papers.

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  1. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  2. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  3. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
  4. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  5. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  6. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
  7. 2025Walrus: A Cross-Domain Foundation Model for Continuum Dynamicscited by 26
  8. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22
  9. 2022Guiding continuous operator learning through Physics-based boundary constraintscited by 21
  10. 2022A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurementscited by 19
  11. 2024Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturingcited by 14
  12. 2024Operator SVD with Neural Networks via Nested Low-Rank Approximationcited by 12
19 more
  1. 2022A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEscited by 11
  2. 2023KoopmanLab: machine learning for solving complex physics equationscited by 9
  3. 2022Non-equispaced Fourier Neural Solvers for PDEscited by 8
  4. 2025The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operatorscited by 8
  5. 2026Representation Learning for Spatiotemporal Physical Systemscited by 8
  6. 2022Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equationscited by 6
  7. 2022Adaptive physics-informed neural operator for coarse-grained non-equilibrium flowscited by 6
  8. 2022Separable PINN: Mitigating the Curse of Dimensionality in Physics-Informed Neural Networkscited by 5
  9. 2025SciML Agents: Write the Solver, Not the Solutioncited by 5
  10. 2022Unsupervised Learning of the Total Variation Flowcited by 3
  11. 2021Survey of Deep Learning Methods for Inverse Problemscited by 1
  12. 2022STONet: A Neural-Operator-Driven Spatio-temporal Networkcited by 1
  13. 2022NLP Inspired Training Mechanics For Modeling Transient Dynamicscited by 1
  14. 2023Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networkscited by 1
  15. 2023GIT-Net: Generalized Integral Transform for Operator Learningcited by 1
  16. 2025A Differentiable Framework for Full and Phaseless Data Inversion Using Neural Implicit Contrast-Source Representationcited by 1
  17. 2026Unlearning Noise in PINNs: A Selective Pruning Framework for PDE Inverse Problemscited by 1
  18. 2022Faster Optimization on Sparse Graphs via Neural Reparametrizationcited by 0
  19. 2023A Spectral Approach for Learning Spatiotemporal Neural Differential Equationscited by 0