mlsanityData to 14 Oct 2026

Solving Inverse Physics Problems with Score Matching

arXiv 2301.10250 ↗First posted 24 Jan 2023cs.LG physics.data-ancited by 16

On the map: Diffusion models → Inverse problems and image restoration

Abstract (arXiv)

We propose to solve inverse problems involving the temporal evolution of physics systems by leveraging recent advances from diffusion models. Our method moves the system's current state backward in time step by step by combining an approximate inverse physics simulator and a learned correction function. A central insight of our work is that training the learned correction with a single-step loss is equivalent to a score matching objective, while recursively predicting longer parts of the trajectory during training relates to maximum likelihood training of a corresponding probability flow. We highlight the advantages of our algorithm compared to standard denoising score matching and implicit score matching, as well as fully learned baselines for a wide range of inverse physics problems. The resulting inverse solver has excellent accuracy and temporal stability and, in contrast to other learned inverse solvers, allows for sampling the posterior of the solutions.

Key facts. Cited by 16 papers in the map (data through 14 Oct 2026). Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models. Builds on 43 of its 65 references that are in the map. First posted on arXiv 24 Jan 2023.

Builds on

43 of 65 references.

Closest ancestors · 3

Same direction, newest first

  1. 2022Diffusion Posterior Sampling for General Noisy Inverse Problems29 Sep
  2. 2022Improving Diffusion Models for Inverse Problems using Manifold Constraints2 Jun
  3. 2021Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction9 Dec

Other references · 40

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  3. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  5. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,255
  6. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  7. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  8. 2018Neural Ordinary Differential Equationscited by 3,779
  9. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  10. 2022Video Diffusion Modelscited by 2,305
  11. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  12. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
28 more
  1. 2020Learning Mesh-Based Simulation with Graph Networkscited by 868
  2. 2016Automatic chemical design using a data-driven continuous representation of moleculescited by 765
  3. 2021Maximum Likelihood Training of Score-Based Diffusion Modelscited by 672
  4. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 622
  5. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 521
  6. 2021Machine learning accelerated computational fluid dynamicscited by 471
  7. 2022Message Passing Neural PDE Solverscited by 386
  8. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  9. 2022Diffusion Probabilistic Modeling for Video Generationcited by 238
  10. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
  11. 2021SNIPS: Solving Noisy Inverse Problems Stochasticallycited by 191
  12. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matchingcited by 183
  13. 2022Generative Modelling With Inverse Heat Dissipationcited by 167
  14. 2021Neural SDEs as Infinite-Dimensional GANscited by 161
  15. 2022Diffusion Models for Video Prediction and Infillingcited by 135
  16. 2020Learning to Control PDEs with Differentiable Physicscited by 132
  17. 2016Accelerating Eulerian Fluid Simulation With Convolutional Networkscited by 130
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  19. 2021Diffusion Normalizing Flowcited by 77
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  21. 2021Physics-based Deep Learningcited by 68
  22. 2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 47
  23. 2022Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizonscited by 39
  24. 2020Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systemscited by 14
  25. 2022Contact Points Discovery for Soft-Body Manipulations with Differentiable Physicscited by 14
  26. 2020Denoising Score-Matching for Uncertainty Quantification in Inverse Problemscited by 11
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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 7; 2025: 4; 2026: 4.

16 papers.

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  1. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 71
  2. 2025PDE-Transformer: Efficient and Versatile Transformers for Physics Simulationscited by 32
  3. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 28
  4. 2025Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraintscited by 28
  5. 2024DiffPhyCon: A Generative Approach to Control Complex Physical Systemscited by 17
  6. 2024When are dynamical systems learned from time series data statistically accurate?cited by 15
  7. 2025Machine learning for modelling unstructured grid data in computational physics: a reviewcited by 12
  8. 2024Parametric model reduction of mean-field and stochastic systems via higher-order action matchingcited by 9
  9. 2024Disentangled Representation Learning for Parametric Partial Differential Equationscited by 3
  10. 2024Flow Matching for Posterior Inference with Simulator Feedbackcited by 2
  11. 2024An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problemscited by 1
  12. 2026Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutionscited by 1
4 more
  1. 2025Rigid Body Adversarial Attackscited by 0
  2. 2026(HB-ARFM) History-Bootstrapped Flow Matching for Inverse Boiling Reconstructioncited by 0
  3. 2026Generative Diffusion Surrogates with Analytical Variance Schedulecited by 0
  4. 2026Characterizing and Mitigating Distributional Bias in Physics-Informed Generative Modelscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Decoupled Diffusion Sampling for Inverse Problems on Function Spaces
  2. 2025Solving Diffusion Inverse Problems with Restart Posterior Sampling
  3. 2026Beyond Accuracy: Evaluating Posterior Fidelity of Diffusion Inverse Solvers
  4. 2025Unifying and extending Diffusion Models through PDEs for solving Inverse Problems
  5. 2025ReGuidance: A Simple Diffusion Wrapper for Boosting Sample Quality on Hard Inverse Problems
  6. 2024Enhancing and Accelerating Diffusion-Based Inverse Problem Solving through Measurements Optimization
  7. 2025InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences
  8. 2025Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems