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

arXiv: https://arxiv.org/abs/2602.12273 · HTML: https://mlsanity.com/paper/2602.12273/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 77 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Model-Based Deep Learning](https://mlsanity.com/paper/2012.08405/) (2020)
- [PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programming](https://mlsanity.com/paper/2406.01908/) (2024)
- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)
- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)
- [The Hard-Constraint PINNs for Interface Optimal Control Problems](https://mlsanity.com/paper/2308.06709/) (2023)
- [Prox-PINNs: A Deep Learning Algorithmic Framework for Elliptic Variational Inequalities](https://mlsanity.com/paper/2505.14430/) (2025)
- [Stability and Error Analysis for Optimization and Generalized Equations](https://mlsanity.com/paper/1903.08754/) (2019)
- [An Operator Learning Approach to Nonsmooth Optimal Control of Nonlinear PDEs](https://mlsanity.com/paper/2409.14417/) (2024)
- [Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems](https://mlsanity.com/paper/1407.1144/) (2014)
- [Learned iterative networks: An operator learning perspective](https://mlsanity.com/paper/2512.08444/) (2025)
- [Mathematical Opportunities in Digital Twins (MATH-DT)](https://mlsanity.com/paper/2402.10326/) (2024)

## Built on by (most cited first, 1 of 1)

- [FNOangle θ: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems](https://mlsanity.com/paper/2604.05187/) (2026)

## Abstract (arXiv)

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.