# Learning Principle of Least Action with Reinforcement Learning

arXiv: https://arxiv.org/abs/2011.11891 · HTML: https://mlsanity.com/paper/2011.11891/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

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

- [Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths](https://mlsanity.com/paper/2202.02750/) (2022)
- [KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss](https://mlsanity.com/paper/2102.11923/) (2021)
- [Policy Optimization Prefers The Path of Least Resistance](https://mlsanity.com/paper/2510.21853/) (2025)

## Abstract (arXiv)

Nature provides a way to understand physics with reinforcement learning since nature favors the economical way for an object to propagate. In the case of classical mechanics, nature favors the object to move along the path according to the integral of the Lagrangian, called the action S. We consider setting the reward/penalty as a function of S, so the agent could learn the physical trajectory of particles in various kinds of environments with reinforcement learning. In this work, we verified the idea by using a Q-Learning based algorithm on learning how light propagates in materials with different refraction indices, and show that the agent could recover the minimal-time path equivalent to the solution obtained by Snell's law or Fermat's Principle. We also discuss the similarity of our reinforcement learning approach to the path integral formalism.