HypeRL: Parameter-Informed Reinforcement Learning for Parametric PDEs

2025-01-08

AI Summary

Traditional methods for optimal control of parametrized partial differential equations (PDEs) suffer from prohibitive computational cost and poor generalization in high-dimensional parameter spaces, as they require repeated PDE solves per parameter instance. To address this, we propose an end-to-end learnable, generalizable feedback control framework. Our approach innovatively embeds a hypernetwork within an Actor-Critic architecture, enabling dynamic generation of policy and value network weights conditioned on continuous parameters—thus explicitly modeling parametric dependence and facilitating efficient cross-parameter transfer. Crucially, the framework eliminates the need for retraining or solving PDEs for each new parameter, substantially improving sample efficiency and robustness to unseen parameters. We validate its strong generalization capability on two canonical PDE control tasks: 1D Kuramoto–Sivashinsky and 2D incompressible Navier–Stokes equations. The results establish a scalable, data-efficient deep reinforcement learning paradigm for parametrized PDE control.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Deep Neural Architectures and Foundation Models
• Intelligent Robots: Behavior Learning & Control

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this work, we devise a new, general-purpose reinforcement learning strategy for the optimal control of parametric partial differential equations (PDEs). Such problems frequently arise in applied sciences and engineering and entail a significant complexity when control and/or state variables are distributed in high-dimensional space or depend on varying parameters. Traditional numerical methods, relying on either iterative minimization algorithms or dynamic programming, while reliable, often become computationally infeasible. Indeed, in either way, the optimal control problem must be solved for each instance of the parameters, and this is out of reach when dealing with high-dimensional time-dependent and parametric PDEs. In this paper, we propose HypeRL, a deep reinforcement learning (DRL) framework to overcome the limitations shown by traditional methods. HypeRL aims at approximating the optimal control policy directly. Specifically, we employ an actor-critic DRL approach to learn an optimal feedback control strategy that can generalize across the range of variation of the parameters. To effectively learn such optimal control laws, encoding the parameter information into the DRL policy and value function neural networks (NNs) is essential. To do so, HypeRL uses two additional NNs, often called hypernetworks, to learn the weights and biases of the value function and the policy NNs. We validate the proposed approach on two PDE-constrained optimal control benchmarks, namely a 1D Kuramoto-Sivashinsky equation and a 2D Navier-Stokes equations, by showing that the knowledge of the PDE parameters and how this information is encoded, i.e., via a hypernetwork, is an essential ingredient for learning parameter-dependent control policies that can generalize effectively to unseen scenarios and for improving the sample efficiency of such policies.

Problem

• Parameterized Complex Equations
• Partial Differential Equations
• High-dimensional Computation

Innovation

• HyperRL
• Parameterized PDE Control
• Actor-Critic Algorithm