mlsanityData to 7 Oct 2026

Learning to Solve PDE-constrained Inverse Problems with Graph Networks

arXiv 2206.00711 ↗First posted 1 Jun 2022cs.LGcited by 37

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Abstract (arXiv)

Learned graph neural networks (GNNs) have recently been established as fast and accurate alternatives for principled solvers in simulating the dynamics of physical systems. In many application domains across science and engineering, however, we are not only interested in a forward simulation but also in solving inverse problems with constraints defined by a partial differential equation (PDE). Here we explore GNNs to solve such PDE-constrained inverse problems. Given a sparse set of measurements, we are interested in recovering the initial condition or parameters of the PDE. We demonstrate that GNNs combined with autodecoder-style priors are well-suited for these tasks, achieving more accurate estimates of initial conditions or physical parameters than other learned approaches when applied to the wave equation or Navier-Stokes equations. We also demonstrate computational speedups of up to 90x using GNNs compared to principled solvers. Project page: https://cyanzhao42.github.io/LearnInverseProblem

Key facts. Cited by 37 papers in the map (data through 7 Oct 2026). Builds on 34 of its 62 references that are in the map. First posted on arXiv 1 Jun 2022.

Builds on

34 of 62 references.

References · 34

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22 more
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 8; 2024: 13; 2025: 6; 2026: 8.

37 papers.

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  1. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  2. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  3. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  4. 2022Learning to Accelerate Partial Differential Equations via Latent Global Evolutioncited by 38
  5. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  6. 2024BENO: Boundary-embedded Neural Operators for Elliptic PDEscited by 22
  7. 2023Learning Controllable Adaptive Simulation for Multi-resolution Physicscited by 20
  8. 2024Compositional Generative Inverse Designcited by 20
  9. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  10. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 18
  11. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 18
  12. 2024DiffPhyCon: A Generative Approach to Control Complex Physical Systemscited by 17
25 more
  1. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  2. 2025Mesh-Informed Neural Operator : A Transformer Generative Approachcited by 11
  3. 2024A Learnable Prior Improves Inverse Tumor Growth Modelingcited by 9
  4. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  5. 2024Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolutioncited by 8
  6. 2023Sheaf4Rec: Sheaf Neural Networks for Graph-based Recommender Systemscited by 7
  7. 2024Inverse analysis of granular flows using differentiable graph neural network simulatorcited by 5
  8. 2024A Neural Material Point Method for Particle-based Emulationcited by 5
  9. 2024Advancing Generalization in PINNs through Latent-Space Representationscited by 5
  10. 2023Graph Network Surrogate Model for Subsurface Flow Optimizationcited by 4
  11. 2025MaNGO - Adaptable Graph Network Simulators via Meta-Learningcited by 3
  12. 20253DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimizationcited by 3
  13. 2023Physics-informed neural network for seismic wave inversion in layered semi-infinite domaincited by 2
  14. 2025ReMiDi: Reconstruction of Microstructure Using a Differentiable Diffusion MRI Simulatorcited by 2
  15. 2025Numerical PDE solvers outperform neural PDE solverscited by 2
  16. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 1
  17. 2026Inverse Neural Operator for ODE Parameter Optimizationcited by 1
  18. 2026Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observationscited by 1
  19. 2026When Does AI for PDEs Yield Scientific Evidence?cited by 1
  20. 2024Neural Persistence Dynamicscited by 0
  21. 2025Optimization Landscapes Learned: Proxy Networks Boost Convergence in Physics-based Inverse Problemscited by 0
  22. 2026Enabling Structure-Only Initialization and Out-of-Distribution Generalization in GNN-based Molecular Dynamics Simulatorscited by 0
  23. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 0
  24. 2026Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomographycited by 0
  25. 2026Point Cloud Sequence Encoding for Material-conditioned Graph Network Simulatorscited by 0