# Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations

arXiv: https://arxiv.org/abs/2512.23829 · HTML: https://mlsanity.com/paper/2512.23829/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 60 references that are in the map.
- First posted on arXiv 29 Dec 2025.

## Builds on (most cited first, 27 of 27)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [A characterization of proximity operators](https://mlsanity.com/paper/1807.04014/) (2018)
- [Deep Relaxation: partial differential equations for optimizing deep neural networks](https://mlsanity.com/paper/1704.04932/) (2017)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere](https://mlsanity.com/paper/1605.01799/) (2016)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [A Hamilton-Jacobi-based Proximal Operator](https://mlsanity.com/paper/2211.12997/) (2022)
- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning](https://mlsanity.com/paper/2404.08809/) (2024)
- [On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations](https://mlsanity.com/paper/1906.09502/) (2019)
- [A Plug-and-Play Image Registration Network](https://mlsanity.com/paper/2310.04297/) (2023)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations](https://mlsanity.com/paper/2501.19351/) (2025)
- [Connecting Hamilton--Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some non-convex priors](https://mlsanity.com/paper/2104.11285/) (2021)
- [On Hamilton-Jacobi PDEs and image denoising models with certain non-additive noise](https://mlsanity.com/paper/2105.13997/) (2021)
- [Plug-and-Play Image Restoration with Flow Matching: A Continuous Viewpoint](https://mlsanity.com/paper/2512.04283/) (2025)
- [Initial Data Identification in Conservation Laws and Hamilton-Jacobi Equations](https://mlsanity.com/paper/1903.06448/) (2019)
- [Optimal Human Navigation in Steep Terrain: a Hamilton-Jacobi-Bellman Approach](https://mlsanity.com/paper/1805.04973/) (2018)
- [An initial condition reconstruction in Hamilton-Jacobi equations](https://mlsanity.com/paper/2001.00864/) (2020)
- [A Primal-dual hybrid gradient method for solving optimal control problems and the corresponding Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2403.02468/) (2024)

## Abstract (arXiv)

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they encode priors and yield efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., plug-and-play methods with learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferentials of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton--Jacobi partial differential equations (HJ PDEs) to develop deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present numerical results in dimensions up to 64, where the recovered prior is evaluated in a single forward pass.