mlsanityData to 14 Oct 2026

Solving PDE-constrained Control Problems Using Operator Learning

arXiv 2111.04941 ↗First posted 9 Nov 2021math.OC cs.AI cs.LGcited by 42

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Abstract (arXiv)

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.

Key facts. Cited by 42 papers in the map (data through 14 Oct 2026). Builds on 21 of its 47 references that are in the map. First posted on arXiv 9 Nov 2021.

Builds on

21 of 47 references.

References · 21

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9 more
  1. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 6; 2023: 8; 2024: 13; 2025: 8; 2026: 7.

42 papers.

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  1. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
  2. 2022Clifford Neural Layers for PDE Modelingcited by 92
  3. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  4. 2023Super-Resolution Neural Operatorcited by 49
  5. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 45
  6. 2024Wavelet Diffusion Neural Operatorcited by 29
  7. 2023HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetworkcited by 26
  8. 2022Physics-informed neural networks for PDE-constrained optimization and controlcited by 19
  9. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 19
  10. 2023Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delaycited by 13
  11. 2023Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspectivecited by 11
  12. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
30 more
  1. 2022Nonparametric learning of kernels in nonlocal operatorscited by 10
  2. 2022Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equationscited by 8
  3. 2024Neural Network Approaches for Parameterized Optimal Controlcited by 8
  4. 2024CL-DiffPhyCon: Closed-loop Diffusion Control of Complex Physical Systemscited by 8
  5. 2024Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular gridscited by 7
  6. 2024On Bellman equations for continuous-time policy evaluation I: discretization and approximationcited by 7
  7. 2025Deep Operator Networks for Bayesian Parameter Estimation in PDEscited by 6
  8. 2024Separable Operator Networkscited by 5
  9. 2023Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problemscited by 4
  10. 2024Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order modelscited by 4
  11. 2024Learning To Solve Differential Equation Constrained Optimization Problemscited by 3
  12. 2025NeurOp-Diff:Continuous Remote Sensing Image Super-Resolution via Neural Operator Diffusioncited by 3
  13. 2025From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Controlcited by 3
  14. 2025Deep Neural ODE Operator Networks for PDEscited by 3
  15. 2024Universal Approximation of Operators with Transformers and Neural Integral Operatorscited by 2
  16. 2024Multi-modal Policies with Physics-informed Representations in Complex Fluid Environmentscited by 1
  17. 2025Accelerating PDE-Constrained Optimization by the Derivative of Neural Operatorscited by 1
  18. 2025Learning to Solve Optimization Problems Constrained with Partial Differential Equationscited by 1
  19. 2026Physically Accurate Differentiable Inverse Rendering for Radio Frequency Digital Twincited by 1
  20. 2026Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalizationcited by 1
  21. 2026On the training of physics-informed neural operators for solving parametric partial differential equationscited by 1
  22. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  23. 2024A Straightforward Gradient-Based Approach for High-Tc Superconductor Design: Leveraging Domain Knowledge via Adaptive Constraintscited by 0
  24. 2024Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order modelscited by 0
  25. 2025Sample-Efficient Diffusion-based Control of Complex Physics Systemscited by 0
  26. 2025Employing Deep Neural Operators for PDE control by decoupling training and optimizationcited by 0
  27. 2026Provable imitation learning for control of instability in partially-observed Vlasov--Poisson equationscited by 0
  28. 2026Real-time optimal control with shallow recurrent decoder networkscited by 0
  29. 2026Toward Goal-Agnostic Joint-Embedding Predictive Control of Partial Differential Equationscited by 0
  30. 2026STCO: Conditional Neural Operators for Time-Dependent PDEscited by 0