Safe PDE Boundary Control with Neural Operators

TL;DR

This work tackles safe control of PDE boundary dynamics under unknown equations by introducing a neural boundary control barrier function (BCBF) and a neural operator that maps boundary inputs to outputs. A quadratic programming safety filter leverages the learned transfer to minimally adjust the boundary input, enforcing finite-time convergence of the boundary output into a user-defined safe set. The approach is validated on challenging PDEs including 1D hyperbolic, 1D parabolic, and 2D Navier–Stokes cases, showing improved boundary feasibility and generally robust performance across vanilla and constrained RL baselines. The results demonstrate a plug-and-play safety mechanism that does not require exact PDE models, offering a practical pathway for safe data-driven PDE control, while highlighting limitations due to neural-operator model mismatch and the need for online deployment studies.

Abstract

The physical world dynamics are generally governed by underlying partial differential equations (PDEs) with unknown analytical forms in science and engineering problems. Neural network based data-driven approaches have been heavily studied in simulating and solving PDE problems in recent years, but it is still challenging to move forward from understanding to controlling the unknown PDE dynamics. PDE boundary control instantiates a simplified but important problem by only focusing on PDE boundary conditions as the control input and output. However, current model-free PDE controllers cannot ensure the boundary output satisfies some given user-specified safety constraint. To this end, we propose a safety filtering framework to guarantee the boundary output stays within the safe set for current model-free controllers. Specifically, we first introduce a neural boundary control barrier function (BCBF) to ensure the feasibility of the trajectory-wise constraint satisfaction of boundary output. Based on the neural operator modeling the transfer function from boundary control input to output trajectories, we show that the change in the BCBF depends linearly on the change in input boundary, so quadratic programming-based safety filtering can be done for pre-trained model-free controllers. Extensive experiments under challenging hyperbolic, parabolic and Navier-Stokes PDE dynamics environments validate the plug-and-play effectiveness of the proposed method by achieving better general performance and boundary constraint satisfaction compared to the vanilla and constrained model-free controller baselines. The code is available at https://github.com/intelligent-control-lab/safe-pde-control.

Figures (6)

• Figure 1: Overview of our safety filtering method for PDE boundary control with neural BCBF. Solid line arrows denote the safety filtering, while dashed ones denote the model training.
• Figure 2: Visualization of three state trajectories $u(x,t)$ (left, mid, right) for hyperbolic equation under PPO controller with and without safety filtering. Boundary control inputs $U(t)$ are in dashed lines, and boundary outputs $Y(t)$ are in solid lines. The boundary constraint $Y(t) < 1$ is in green.
• Figure 3: The reward and feasible rate under different filtering threshold $\eta$ in \ref{['eq:filter_threshold']} with BCBF $\phi(Y)$ (left) and $\phi(t,Y)$ (right) for PPO model in hyperbolic equation. Note that $\eta=0$ indicates the vanilla PPO model without safety filtering.
• Figure 4: Visualization of state $u(x,t)$ of hyperbolic equation under SAC controller with (in blue) and without (in red) filtering. Boundary control inputs $U(t)$ are in dashed lines and boundary output $Y(t)$ are in solid lines. The boundary constraint $Y(t) < 0$ is in green.
• Figure 5: Visualization of tracking performance with PPO and SAC models before and after filtering with $\phi(t,Y)$ at the end time step of the trajectory for Navier-Stokes equation.

Theorems & Definitions (11)

• definition 1: Boundary Feasibility for Finite-time Constraint Satisfaction
• theorem 1: Boundary Feasibility with Boundary Control Barrier Function
• proof
• theorem 2: Boundary Feasibility with Neural Operator
• remark 1
• definition 2: Boundary Feasibility for Finite-time Constraint Satisfaction
• definition 3: Neural operator for input-output boundary mapping
• theorem 3: Boundary Feasibility with Boundary Control Barrier Function
• proof
• theorem 4: Boundary Feasibility with Neural Operator