# Learning to Solve PDE-constrained Inverse Problems with Graph Networks

arXiv: https://arxiv.org/abs/2206.00711 · HTML: https://mlsanity.com/paper/2206.00711/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 62 references that are in the map.
- First posted on arXiv 1 Jun 2022.

## Builds on (most cited first, 30 of 34)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Pivotal Tuning for Latent-based Editing of Real Images](https://mlsanity.com/paper/2106.05744/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework](https://mlsanity.com/paper/2005.01463/) (2020)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021)
- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)

## Built on by (most cited first, 30 of 37)

- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [BENO: Boundary-embedded Neural Operators for Elliptic PDEs](https://mlsanity.com/paper/2401.09323/) (2024)
- [Learning Controllable Adaptive Simulation for Multi-resolution Physics](https://mlsanity.com/paper/2305.01122/) (2023)
- [Compositional Generative Inverse Design](https://mlsanity.com/paper/2401.13171/) (2024)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators](https://mlsanity.com/paper/2305.20053/) (2023)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [DiffPhyCon: A Generative Approach to Control Complex Physical Systems](https://mlsanity.com/paper/2407.06494/) (2024)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [A Learnable Prior Improves Inverse Tumor Growth Modeling](https://mlsanity.com/paper/2403.04500/) (2024)
- [Random Grid Neural Processes for Parametric Partial Differential Equations](https://mlsanity.com/paper/2301.11040/) (2023)
- [Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution](https://mlsanity.com/paper/2402.08383/) (2024)
- [Sheaf4Rec: Sheaf Neural Networks for Graph-based Recommender Systems](https://mlsanity.com/paper/2304.09097/) (2023)
- [Inverse analysis of granular flows using differentiable graph neural network simulator](https://mlsanity.com/paper/2401.13695/) (2024)
- [A Neural Material Point Method for Particle-based Emulation](https://mlsanity.com/paper/2408.15753/) (2024)
- [Advancing Generalization in PINNs through Latent-Space Representations](https://mlsanity.com/paper/2411.19125/) (2024)
- [Graph Network Surrogate Model for Subsurface Flow Optimization](https://mlsanity.com/paper/2312.08625/) (2023)
- [MaNGO - Adaptable Graph Network Simulators via Meta-Learning](https://mlsanity.com/paper/2510.05874/) (2025)
- [3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization](https://mlsanity.com/paper/2512.08987/) (2025)
- [Physics-informed neural network for seismic wave inversion in layered semi-infinite domain](https://mlsanity.com/paper/2305.05150/) (2023)
- [ReMiDi: Reconstruction of Microstructure Using a Differentiable Diffusion MRI Simulator](https://mlsanity.com/paper/2502.01988/) (2025)
- [Numerical PDE solvers outperform neural PDE solvers](https://mlsanity.com/paper/2507.21269/) (2025)
- [ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance](https://mlsanity.com/paper/2601.20227/) (2026)
- [Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations](https://mlsanity.com/paper/2605.13790/) (2026)
- [Inverse Neural Operator for ODE Parameter Optimization](https://mlsanity.com/paper/2603.11854/) (2026)

## 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