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Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs

Yongcun Song, Xiaoming Yuan, Hangrui Yue, Tianyou Zeng

TL;DR

The paper tackles the challenge of real-time solution of parameterized nonsmooth optimal control problems governed by linear PDEs. It introduces iUzawa-Net, an optimization-informed neural network that unrolls an inexact Uzawa method and replaces PDE solves with learnable surrogates for the state operator and preconditioners, enabling fast inference after a single offline training phase. The authors establish universal approximation properties and asymptotic $\varepsilon$-optimality for the layer outputs, and provide weight-tying results under mild regularity, ensuring robust performance with reduced model complexity. Numerical experiments on elliptic and parabolic problems demonstrate high accuracy with few layers, strong generalization including zero-shot super-resolution, and substantial speedups over classical solvers, highlighting the practical impact of optimization-informed neural networks for PDE-constrained optimization.

Abstract

We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic $\varepsilon$-optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.

Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs

TL;DR

The paper tackles the challenge of real-time solution of parameterized nonsmooth optimal control problems governed by linear PDEs. It introduces iUzawa-Net, an optimization-informed neural network that unrolls an inexact Uzawa method and replaces PDE solves with learnable surrogates for the state operator and preconditioners, enabling fast inference after a single offline training phase. The authors establish universal approximation properties and asymptotic -optimality for the layer outputs, and provide weight-tying results under mild regularity, ensuring robust performance with reduced model complexity. Numerical experiments on elliptic and parabolic problems demonstrate high accuracy with few layers, strong generalization including zero-shot super-resolution, and substantial speedups over classical solvers, highlighting the practical impact of optimization-informed neural networks for PDE-constrained optimization.

Abstract

We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic -optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.
Paper Structure (32 sections, 25 theorems, 195 equations, 6 figures, 9 tables)

This paper contains 32 sections, 25 theorems, 195 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Classical Numerical Methods for Problem (\ref{['eq:opt-ctrl']})
  4. Physics-Informed Neural Networks and Operator Learning for Problem (\ref{['eq:opt-ctrl']})
  5. Optimization-Informed Neural Networks
  6. Methodology
  7. An inexact Uzawa method for (\ref{['eq:opt-ctrl']})
  8. iUzawa-Net: an overview
  9. Theoretical results of the iUzawa-Net
  10. Contributions
  11. Organization
  12. iUzawa-Net: Modules Design and Training
  13. Neural Network Architecture for Sk and Ak
  14. Neural Network Architecture for QSk
  15. Neural Network Architecture for QAk
  16. ...and 17 more sections

Key Result

Proposition 2.1

Let $\kappa \in L^2(\mathbb{T}^d; \mathbb{R}^{m \times m})$ and $W \in \mathbb{R}^{m \times m}$ for some positive integer $m$. For any $v \in L^2(\mathbb{T}^d; \mathbb{R}^m)$, define the linear operator $\mathcal{K}: L^2(\mathbb{T}^d; \mathbb{R}^m) \to L^2(\mathbb{T}^d; \mathbb{R}^m)$ by If $W = V^\top V$ for some $V \in \mathbb{R}^{m \times m}$ and $\kappa$ takes the form for some $\phi \in L^2

Figures (6)

  • Figure 2.1: The neural network architecture of $\mathcal{Q}_S^k$.
  • Figure 2.2: The neural network architecture of $\mathcal{Q}_A^k$.
  • Figure 2.3: An overview of the of the iUzawa-Net\ref{['eq:inexact-uzawa-duf']}.
  • Figure 6.1: Computed optimal control for a single instance of \ref{['eq:elliptic-opt-ctrl']} with $m=64$.
  • Figure 6.2: Computed optimal control for a single instance of \ref{['eq:elliptic-opt-ctrl-illcond']} with $m=64$.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4: c.f. pinkus1999approximation
  • Proposition 3.5: c.f. kovachki2021universal
  • Lemma 3.6
  • Proposition 3.7
  • ...and 40 more