# Integral Equations and Machine Learning

arXiv: https://arxiv.org/abs/1712.06115 · HTML: https://mlsanity.com/paper/1712.06115/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2017.

## Built on by (most cited first, 3 of 3)

- [Real-time Neural Radiance Caching for Path Tracing](https://mlsanity.com/paper/2106.12372/) (2021)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Lattice-based Deep Neural Networks: Regularity and Tailored Regularization](https://mlsanity.com/paper/2603.02809/) (2026)

## Abstract (arXiv)

As both light transport simulation and reinforcement learning are ruled by the same Fredholm integral equation of the second kind, reinforcement learning techniques may be used for photorealistic image synthesis: Efficiency may be dramatically improved by guiding light transport paths by an approximate solution of the integral equation that is learned during rendering. In the light of the recent advances in reinforcement learning for playing games, we investigate the representation of an approximate solution of an integral equation by artificial neural networks and derive a loss function for that purpose. The resulting Monte Carlo and quasi-Monte Carlo methods train neural networks with standard information instead of linear information and naturally are able to generate an arbitrary number of training samples. The methods are demonstrated for applications in light transport simulation.