mlsanityData to 13 Oct 2026

Safe PDE Boundary Control with Neural Operators

arXiv 2411.15643 ↗First posted 23 Nov 2024eess.SY cs.LG cs.NEcited by 1

Not on the map. Why?

Abstract (arXiv)

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.

Key facts. Cited by 1 paper in the map (data through 13 Oct 2026). Builds on 52 of its 80 references that are in the map. First posted on arXiv 23 Nov 2024.

Builds on

52 of 80 references.

References · 52

  1. 2017Proximal Policy Optimization Algorithmscited by 14,592
  2. 2018Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actorcited by 4,656
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,857
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,989
  5. 2017Constrained Policy Optimizationcited by 931
  6. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 766
  7. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 728
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 522
  9. 2021Machine learning accelerated computational fluid dynamicscited by 465
  10. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 345
  11. 2019Control Barrier Functions: Theory and Applicationscited by 334
  12. 2022Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methodscited by 148
38 more
  1. 2020Neural Lyapunov Controlcited by 142
  2. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  3. 2020Learning to Walk in the Real World with Minimal Human Effortcited by 103
  4. 2021Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functionscited by 91
  5. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 83
  6. 2020DeepReach: A Deep Learning Approach to High-Dimensional Reachabilitycited by 81
  7. 2023The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systemscited by 79
  8. 2020NeuralSim: Augmenting Differentiable Simulators with Neural Networkscited by 78
  9. 2022Constrained Variational Policy Optimization for Safe Reinforcement Learningcited by 70
  10. 2023Datasets and Benchmarks for Offline Safe Reinforcement Learningcited by 59
  11. 2023Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamicscited by 56
  12. 2021Optimal control of PDEs using physics-informed neural networkscited by 51
  13. 2021Robust Control Barrier-Value Functions for Safety-Critical Controlcited by 47
  14. 2020Learning Stability Certificates from Datacited by 45
  15. 2021Learning Dissipative Dynamics in Chaotic Systemscited by 41
  16. 2023How to Train Your Neural Control Barrier Function: Learning Safety Filters for Complex Input-Constrained Systemscited by 40
  17. 2022Safe Control Under Input Limits with Neural Control Barrier Functionscited by 27
  18. 2023Neural Operators for Bypassing Gain and Control Computations in PDE Backsteppingcited by 27
  19. 2020Learning Hybrid Control Barrier Functions from Datacited by 24
  20. 2024DiffPhyCon: A Generative Approach to Control Complex Physical Systemscited by 17
  21. 2023Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEscited by 16
  22. 2022Safety Guarantees for Neural Network Dynamic Systems via Stochastic Barrier Functionscited by 14
  23. 2023Exact Verification of ReLU Neural Control Barrier Functionscited by 14
  24. 2024Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagationcited by 12
  25. 2022Persistently Feasible Robust Safe Control by Safety Index Synthesis and Convex Semi-Infinite Programmingcited by 11
  26. 2024NeuralFluid: Neural Fluidic System Design and Control with Differentiable Simulationcited by 10
  27. 2021Safe Control Synthesis via Input Constrained Control Barrier Functionscited by 8
  28. 2024PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equationscited by 8
  29. 2024Parametric PDE Control with Deep Reinforcement Learning and Differentiable L0-Sparse Polynomial Policiescited by 7
  30. 2024Scalable Synthesis of Formally Verified Neural Value Function for Hamilton-Jacobi Reachability Analysiscited by 7
  31. 2020Sufficient Conditions for Feasibility of Optimal Control Problems Using Control Barrier Functionscited by 6
  32. 2022Learning a Better Control Barrier Functioncited by 5
  33. 2024Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuationcited by 4
  34. 2021High Order Control Lyapunov-Barrier Functions for Temporal Logic Specificationscited by 3
  35. 2023Neural Differentiable Integral Control Barrier Functions for Unknown Nonlinear Systems with Input Constraintscited by 3
  36. 2024Robust Tracking Control with Neural Network Dynamic Models under Input Perturbationscited by 2
  37. 2020Characterization of Domain of Fixed-time Stability under Control Input Constraintscited by 1
  38. 2021Safe PDE Backstepping QP Control with High Relative Degree CBFs: Stefan Model with Actuator Dynamicscited by 1

50 most cited of 52 All 52 (JSON)

Built on by

Cited by year (2026 to 13 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026STCO: Conditional Neural Operators for Time-Dependent PDEscited by 0