# Solving Fredholm Integral Equations of the First Kind via Wasserstein Gradient Flows

arXiv: https://arxiv.org/abs/2209.09936 · HTML: https://mlsanity.com/paper/2209.09936/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2022.

## Built on by (most cited first, 4 of 4)

- [On the Wasserstein Gradient Flow Interpretation of Drifting Models](https://mlsanity.com/paper/2605.05118/) (2026)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Manually Selecting The Data Function for Supervised Learning of small datasets](https://mlsanity.com/paper/2303.03894/) (2023)
- [DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimation](https://mlsanity.com/paper/2408.02045/) (2024)

## Abstract (arXiv)

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean--Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.