mlsanityData to 14 Oct 2026

Neural Inverse Operators for Solving PDE Inverse Problems

arXiv 2301.11167 ↗First posted 26 Jan 2023cs.LG math-ph math.APcited by 65

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Abstract (arXiv)

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

Key facts. Cited by 65 papers in the map (data through 14 Oct 2026). Builds on 10 of its 37 references that are in the map. First posted on arXiv 26 Jan 2023.

Builds on

10 of 37 references.

References · 10

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  4. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  5. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  6. 2022Message Passing Neural PDE Solverscited by 386
  7. 2022Learning Operators with Coupled Attentioncited by 90
  8. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28
  9. 2022Variable-Input Deep Operator Networkscited by 25
  10. 2019Deep learning architectures for nonlinear operator functions and nonlinear inverse problemscited by 17

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 6; 2024: 22; 2025: 19; 2026: 18.

65 papers.

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  1. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
  2. 2024Poseidon: Efficient Foundation Models for PDEscited by 200
  3. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  4. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  5. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 25
  6. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  7. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 19
  8. 2024Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discoverycited by 18
  9. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  10. 2024Mixtures of Neural Operators Reduce Active Complexity in Operator Learningcited by 16
  11. 2023Deciphering and integrating invariants for neural operator learning with various physical mechanismscited by 15
  12. 2024Generating synthetic data for neural operatorscited by 11
38 more
  1. 2025Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEscited by 9
  2. 2024BiLO: Bilevel Local Operator Learning for PDE Inverse Problemscited by 8
  3. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 8
  4. 2023A Domain-adaptive Physics-informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Mediacited by 7
  5. 2024Learning Hidden Physics and System Parameters with Deep Operator Networkscited by 7
  6. 2025Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programscited by 7
  7. 2025Operator Learning at Machine Precisioncited by 7
  8. 2023Operator learning for hyperbolic partial differential equationscited by 5
  9. 2024Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformationscited by 5
  10. 2024Neural Operators Can Play Dynamic Stackelberg Gamescited by 5
  11. 2024Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurementscited by 4
  12. 2024Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order modelscited by 4
  13. 2024Disentangled Representation Learning for Parametric Partial Differential Equationscited by 3
  14. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 3
  15. 2025Physics-informed sensor coverage through structure preserving machine learningcited by 3
  16. 2024ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equationcited by 2
  17. 2025Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discoverycited by 2
  18. 2025OpenBreastUS: Benchmarking Neural Operators for Wave Imaging Using Breast Ultrasound Computed Tomographycited by 2
  19. 2025Bayesian BiLO: Bilevel Local Operator Learning for Efficient Uncertainty Quantification of Bayesian PDE Inverse Problems with Low-Rank Adaptationcited by 2
  20. 2025A unified physics-informed generative operator framework for general inverse problemscited by 2
  21. 2026PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problemscited by 2
  22. 2024Physics-aware deep learning framework for the limited aperture inverse obstacle scattering problemcited by 1
  23. 2025An Attention-based Spatio-Temporal Neural Operator for Evolving Physicscited by 1
  24. 2025Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Losscited by 1
  25. 2025Graph Neural Regularizers for PDE Inverse Problemscited by 1
  26. 2026Inverse Neural Operator for ODE Parameter Optimizationcited by 1
  27. 2026A Deep Risk Estimator for Known Operator Learningcited by 1
  28. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 1
  29. 2026Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalizationcited by 1
  30. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1
  31. 2026GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problemscited by 1
  32. 2026PosteriorBench: From Point Estimates to Posterior Matching in Evaluating Generative Inverse Solverscited by 1
  33. 2024State estimation of urban air pollution with statistical, physical, and super-learning graph modelscited by 0
  34. 2024Physics-Informed Machine Learning On Polar Ice: A Surveycited by 0
  35. 2024A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical resultscited by 0
  36. 2024Deep Optimal Sensor Placement for Black Box Stochastic Simulationscited by 0
  37. 2024Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order modelscited by 0
  38. 2024DeepF-fNet: a physics-informed neural network for vibration isolation optimizationcited by 0

50 most cited of 65 All 65 (JSON)