# Zermelo's problem: Optimal point-to-point navigation in 2D turbulent flows using Reinforcement Learning

## TL;DR

This paper demonstrates that Reinforcement Learning can effectively solve Zermelo's problem in 2D turbulent flows, providing robust navigation strategies that outperform traditional optimal control methods in dynamic and noisy environments.

## Contribution

The study introduces an Actor-Critic RL approach for Zermelo's problem in turbulent flows, showing its robustness and superiority over analytical solutions in practical scenarios.

## Key findings

- RL solutions are more robust to initial condition changes and noise.
- RL outperforms analytical ON strategies in dynamic flow conditions.
- RL effectively exploits flow properties for navigation with low steering speed.

## Abstract

To find the path that minimizes the time to navigate between two given points in a fluid flow is known as Zermelo's problem. Here, we investigate it by using a Reinforcement Learning (RL) approach for the case of a vessel which has a slip velocity with fixed intensity, Vs , but variable direction and navigating in a 2D turbulent sea. We show that an Actor-Critic RL algorithm is able to find quasi-optimal solutions for both time-independent and chaotically evolving flow configurations. For the frozen case, we also compared the results with strategies obtained analytically from continuous Optimal Navigation (ON) protocols. We show that for our application, ON solutions are unstable for the typical duration of the navigation process, and are therefore not useful in practice. On the other hand, RL solutions are much more robust with respect to small changes in the initial conditions and to external noise, even when V s is much smaller than the maximum flow velocity. Furthermore, we show how the RL approach is able to take advantage of the flow properties in order to reach the target, especially when the steering speed is small.