mlsanityData to 7 Oct 2026

Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems

arXiv 2209.02772 ↗First posted 6 Sep 2022stat.ML cs.LG physics.comp-phcited by 25

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Abstract (arXiv)

Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and differentiable map between the parametric input and the branch-net output. This allows us to construct accurate approximations of the full posterior, irrespective of the number of observations and the magnitude of the observation noise, without any need for additional forward solves nor for cumbersome, iterative sampling procedures. We demonstrate the efficacy and accuracy of the proposed methodology in the context of inverse problems for three benchmarks: an anti-derivative equation, reaction-diffusion dynamics and flow through porous media.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). Builds on 18 of its 39 references that are in the map. First posted on arXiv 6 Sep 2022.

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18 of 39 references.

References · 18

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2016Density estimation using Real NVPcited by 2,029
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  6. 2018Invertible Residual Networkscited by 364
  7. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  8. 2018Analyzing Inverse Problems with Invertible Neural Networkscited by 239
  9. 2017Solving ill-posed inverse problems using iterative deep neural networkscited by 145
  10. 2022Learning Operators with Coupled Attentioncited by 88
  11. 2018A Stein variational Newton methodcited by 49
  12. 2021Momentum Residual Neural Networkscited by 36
6 more
  1. 2018Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identificationcited by 26
  2. 2019Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systemscited by 25
  3. 2021Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equationscited by 15
  4. 2021Physics-aware, probabilistic model order reduction with guaranteed stabilitycited by 12
  5. 2017Bayesian model and dimension reduction for uncertainty propagation: applications in random mediacited by 4
  6. 2015Multimodal, high-dimensional, model-based, Bayesian inverse problems with applications in biomechanicscited by 3

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Citing papers per year: 2023: 6; 2024: 8; 2025: 7; 2026: 4.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  2. 2023Monte Carlo guided Diffusion for Bayesian linear inverse problemscited by 65
  3. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 18
  4. 2025Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEscited by 9
  5. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  6. 2024BiLO: Bilevel Local Operator Learning for PDE Inverse Problemscited by 8
  7. 2023Efficient Bayesian inference using physics-informed invertible neural networks for inverse problemscited by 6
  8. 2024Generative Learning for Forecasting the Dynamics of Complex Systemscited by 6
  9. 2024Learning Hidden Physics and System Parameters with Deep Operator Networkscited by 6
  10. 2023Learning to solve Bayesian inverse problems: An amortized variational inference approach using Gaussian and Flow guidescited by 5
  11. 2024Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastographycited by 5
  12. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
13 more
  1. 2024Resolution invariant deep operator network for PDEs with complex geometriescited by 4
  2. 2023Bayesian ECG reconstruction using denoising diffusion generative modelscited by 2
  3. 2024Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learningcited by 2
  4. 2024Physics-aware deep learning framework for the limited aperture inverse obstacle scattering problemcited by 1
  5. 2025Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Latercited by 1
  6. 2026Inverse Neural Operator for ODE Parameter Optimizationcited by 1
  7. 2025Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Datacited by 0
  8. 2025How Regularization Terms Make Invertible Neural Networks Bayesian Point Estimatorscited by 0
  9. 2025Learning Generalizable Neural Operators for Inverse Problemscited by 0
  10. 2025DeepONet-accelerated Bayesian inversion for moving boundary problemscited by 0
  11. 2026GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Mediacited by 0
  12. 2026Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Fieldcited by 0
  13. 2026GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problemscited by 0