PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equations

TL;DR

This paper introduces the first reinforcement learning benchmark environment for boundary control of PDEs, including three foundational problems, enabling easier development and comparison of data-driven PDE control methods.

Contribution

It provides a user-friendly, open-source benchmark platform for learning-based PDE control, including three PDE problems and initial RL algorithms, lowering barriers for research in this area.

Findings

RL algorithms achieved stability on benchmark PDEs

Benchmark environment is accessible and well-documented

Model-free RL methods are feasible for PDE control

Abstract

Over the last decade, data-driven methods have surged in popularity, emerging as valuable tools for control theory. As such, neural network approximations of control feedback laws, system dynamics, and even Lyapunov functions have attracted growing attention. With the ascent of learning based control, the need for accurate, fast, and easy-to-use benchmarks has increased. In this work, we present the first learning-based environment for boundary control of PDEs. In our benchmark, we introduce three foundational PDE problems - a 1D transport PDE, a 1D reaction-diffusion PDE, and a 2D Navier-Stokes PDE - whose solvers are bundled in an user-friendly reinforcement learning gym. With this gym, we then present the first set of model-free, reinforcement learning algorithms for solving this series of benchmark problems, achieving stability, although at a higher cost compared to model-based PDE backstepping. With the set of benchmark environments and detailed examples, this work significantly lowers the barrier to entry for learning-based PDE control - a topic largely unexplored by the data-driven control community. The entire benchmark is available on Github along with detailed documentation and the presented reinforcement learning models are open sourced.