# Solving PDE-constrained Control Problems Using Operator Learning

arXiv: https://arxiv.org/abs/2111.04941 · HTML: https://mlsanity.com/paper/2111.04941/

## 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 (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning Latent Dynamics for Planning from Pixels](https://mlsanity.com/paper/1811.04551/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images](https://mlsanity.com/paper/1506.07365/) (2015)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Benchmarking deep inverse models over time, and the neural-adjoint method](https://mlsanity.com/paper/2009.12919/) (2020)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach](https://mlsanity.com/paper/1911.09843/) (2019)
- [Deep Neural Network Approach to Forward-Inverse Problems](https://mlsanity.com/paper/1907.12925/) (2019)

## Built on by (most cited first, 30 of 42)

- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Super-Resolution Neural Operator](https://mlsanity.com/paper/2303.02584/) (2023)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators](https://mlsanity.com/paper/2305.20053/) (2023)
- [Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay](https://mlsanity.com/paper/2307.11436/) (2023)
- [Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective](https://mlsanity.com/paper/2310.06379/) (2023)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)
- [Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations](https://mlsanity.com/paper/2201.11967/) (2022)
- [Neural Network Approaches for Parameterized Optimal Control](https://mlsanity.com/paper/2402.10033/) (2024)
- [CL-DiffPhyCon: Closed-loop Diffusion Control of Complex Physical Systems](https://mlsanity.com/paper/2408.03124/) (2024)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)
- [On Bellman equations for continuous-time policy evaluation I: discretization and approximation](https://mlsanity.com/paper/2407.05966/) (2024)
- [Deep Operator Networks for Bayesian Parameter Estimation in PDEs](https://mlsanity.com/paper/2501.10684/) (2025)
- [Separable Operator Networks](https://mlsanity.com/paper/2407.11253/) (2024)
- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Learning To Solve Differential Equation Constrained Optimization Problems](https://mlsanity.com/paper/2410.01786/) (2024)
- [NeurOp-Diff:Continuous Remote Sensing Image Super-Resolution via Neural Operator Diffusion](https://mlsanity.com/paper/2501.09054/) (2025)
- [From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control](https://mlsanity.com/paper/2502.02205/) (2025)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [Universal Approximation of Operators with Transformers and Neural Integral Operators](https://mlsanity.com/paper/2409.00841/) (2024)
- [Multi-modal Policies with Physics-informed Representations in Complex Fluid Environments](https://mlsanity.com/paper/2410.15250/) (2024)
- [Accelerating PDE-Constrained Optimization by the Derivative of Neural Operators](https://mlsanity.com/paper/2506.13120/) (2025)
- [Learning to Solve Optimization Problems Constrained with Partial Differential Equations](https://mlsanity.com/paper/2509.24573/) (2025)

## 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.