Parametric PDE Control with Deep Reinforcement Learning and Differentiable L0-Sparse Polynomial Policies

TL;DR

This paper introduces a novel sparse control policy framework for parametric PDEs using deep reinforcement learning combined with differentiable L0 regularization, enhancing interpretability, robustness, and generalization over traditional DNN policies.

Contribution

The work presents a new sparse policy architecture that is compatible with various DRL algorithms, improving control of parametric PDEs with interpretable and robust policies that generalize well.

Findings

Outperforms baseline DNN-based DRL policies.

Enables derivation of interpretable control laws.

Generalizes to unseen PDE parameters without retraining.

Abstract

Optimal control of parametric partial differential equations (PDEs) is crucial in many applications in engineering and science. In recent years, the progress in scientific machine learning has opened up new frontiers for the control of parametric PDEs. In particular, deep reinforcement learning (DRL) has the potential to solve high-dimensional and complex control problems in a large variety of applications. Most DRL methods rely on deep neural network (DNN) control policies. However, for many dynamical systems, DNN-based control policies tend to be over-parametrized, which means they need large amounts of training data, show limited robustness, and lack interpretability. In this work, we leverage dictionary learning and differentiable L$_0$ regularization to learn sparse, robust, and interpretable control policies for parametric PDEs. Our sparse policy architecture is agnostic to the DRL method and can be used in different policy-gradient and actor-critic DRL algorithms without changing their policy-optimization procedure. We test our approach on the challenging tasks of controlling parametric Kuramoto-Sivashinsky and convection-diffusion-reaction PDEs. We show that our method (1) outperforms baseline DNN-based DRL policies, (2) allows for the derivation of interpretable equations of the learned optimal control laws, and (3) generalizes to unseen parameters of the PDE without retraining the policies.