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An Operator Learning Approach to Nonsmooth Optimal Control of Nonlinear PDEs

Yongcun Song, Xiaoming Yuan, Hangrui Yue, Tianyou Zeng

TL;DR

This work tackles nonsmooth optimal control problems with nonlinear PDE constraints by marrying primal-dual optimization with operator learning. The authors replace repeated PDE solves in the state and adjoint updates with pretrained neural surrogates for the state operator S and its adjoint, forming a mesh-free PDHG-OPL framework that can reuse networks across iterations and parameter changes. They instantiate the approach across three canonical problems—stationary Burgers, sparse bilinear parabolic control, and semilinear parabolic control—using DeepONet, MIONet, and Fourier Neural Operators, and demonstrate substantial speedups with accuracy comparable to traditional high-fidelity solvers. The results indicate strong potential for real-time control tasks and scalable solvers for nonsmooth, nonlinear PDE-constrained optimization, with future work focusing on convergence theory and extensions to higher-dimensional PDEs.

Abstract

Optimal control problems with nonsmooth objectives and nonlinear partial differential equation (PDE) constraints are challenging, mainly because of the underlying nonsmooth and nonconvex structures and the demanding computational cost for solving multiple high-dimensional and ill-conditioned systems after mesh-based discretization. To mitigate these challenges numerically, we propose an operator learning approach in combination with an effective primal-dual optimization idea which can decouple the treatment of the control and state variables so that each of the resulting iterations only requires solving two PDEs. Our main purpose is to construct neural surrogate models for the involved PDEs by operator learning, allowing the solution of a PDE to be obtained with only a forward pass of the neural network. The resulting algorithmic framework offers a hybrid approach that combines the efficiency and generalization of operator learning with the model-based nature and structure-friendly efficiency of primal-dual-based algorithms. The primal-dual-based operator learning approach offers numerical methods that are mesh-free, easy to implement, and adaptable to various optimal control problems with nonlinear PDEs. It is notable that the neural surrogate models can be reused across iterations and parameter settings, hence retraining of neural networks can be avoided and computational cost can be substantially alleviated. We affirmatively validate the efficiency of the primal-dual-based operator learning approach across a range of typical optimal control problems with nonlinear PDEs.

An Operator Learning Approach to Nonsmooth Optimal Control of Nonlinear PDEs

TL;DR

This work tackles nonsmooth optimal control problems with nonlinear PDE constraints by marrying primal-dual optimization with operator learning. The authors replace repeated PDE solves in the state and adjoint updates with pretrained neural surrogates for the state operator S and its adjoint, forming a mesh-free PDHG-OPL framework that can reuse networks across iterations and parameter changes. They instantiate the approach across three canonical problems—stationary Burgers, sparse bilinear parabolic control, and semilinear parabolic control—using DeepONet, MIONet, and Fourier Neural Operators, and demonstrate substantial speedups with accuracy comparable to traditional high-fidelity solvers. The results indicate strong potential for real-time control tasks and scalable solvers for nonsmooth, nonlinear PDE-constrained optimization, with future work focusing on convergence theory and extensions to higher-dimensional PDEs.

Abstract

Optimal control problems with nonsmooth objectives and nonlinear partial differential equation (PDE) constraints are challenging, mainly because of the underlying nonsmooth and nonconvex structures and the demanding computational cost for solving multiple high-dimensional and ill-conditioned systems after mesh-based discretization. To mitigate these challenges numerically, we propose an operator learning approach in combination with an effective primal-dual optimization idea which can decouple the treatment of the control and state variables so that each of the resulting iterations only requires solving two PDEs. Our main purpose is to construct neural surrogate models for the involved PDEs by operator learning, allowing the solution of a PDE to be obtained with only a forward pass of the neural network. The resulting algorithmic framework offers a hybrid approach that combines the efficiency and generalization of operator learning with the model-based nature and structure-friendly efficiency of primal-dual-based algorithms. The primal-dual-based operator learning approach offers numerical methods that are mesh-free, easy to implement, and adaptable to various optimal control problems with nonlinear PDEs. It is notable that the neural surrogate models can be reused across iterations and parameter settings, hence retraining of neural networks can be avoided and computational cost can be substantially alleviated. We affirmatively validate the efficiency of the primal-dual-based operator learning approach across a range of typical optimal control problems with nonlinear PDEs.
Paper Structure (23 sections, 6 theorems, 42 equations, 7 figures, 7 tables, 4 algorithms)

This paper contains 23 sections, 6 theorems, 42 equations, 7 figures, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Traditional numerical methods
  3. Deep learning methods for PDEs
  4. Deep learning methods for optimal control
  5. ADMM for (1.1)
  6. Primal-dual methods for (1.1)
  7. Motivation
  8. Organization
  9. A primal-dual-based operator learning approach to (\ref{['eq:basic-problem']})
  10. Optimal control of stationary Burgers equations
  11. Primal-dual decoupling for (3.1)-(3.2)
  12. DeepONet and MIONet
  13. Primal-dual-based operator learning method for (3.1)-(3.2)
  14. Numerical experiments
  15. Sparse bilinear control of parabolic equations
  16. ...and 8 more sections

Key Result

Proposition 3.1

Given the functionals $F$ and $G$ defined in eq:fun1 and any $\tau, \sigma > 0$, the proximal operators $\mathop{\mathrm{prox}}\nolimits_{\tau G}$ and $\mathop{\mathrm{prox}}\nolimits_{\sigma F^*}$ are given, respectively, by Here, $\mathcal{P}_{U_{ad}}: L^2(\Omega) \to L^2(\Omega)$ denotes the projection operator onto $U_{ad}$ defined by

Figures (7)

  • Figure 1: The network structure of DeepONet, adopted from lu2019deeponet.
  • Figure 1: The structure of the Fourier neural operator li2021fourier.
  • Figure 1: Computed optimal control and state for \ref{['eq:sp-prob']}--\ref{['eq:sp-equation']} with $\beta = 0.004$.
  • Figure 2: The structure of MIONet with two input functions $y$ and $p$. The symbol $\odot$ denotes the element-wise product.
  • Figure 2: Computed control and exact control for problem \ref{['eq:bilinparab-optctrl']}--\ref{['eq:bilinparab-equation']} with $\alpha=\beta=0.01$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Proof 1: Proof of \ref{['prop:burgers-adjoint']}
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 5.1