# Learning to Solve Optimization Problems Constrained with Partial Differential Equations

arXiv: https://arxiv.org/abs/2509.24573 · HTML: https://mlsanity.com/paper/2509.24573/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Progressive construction of a parametric reduced-order model for PDE-constrained optimization](https://mlsanity.com/paper/1407.7618/) (2014)
- [Optimal Control Operator Perspective and a Neural Adaptive Spectral Method](https://mlsanity.com/paper/2412.12469/) (2024)

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

- [PDE-Constrained Optimization for Neural Image Segmentation with Physics Priors](https://mlsanity.com/paper/2602.01069/) (2026)

## Abstract (arXiv)

Partial differential equation (PDE)-constrained optimization arises in many scientific and engineering domains, such as energy systems, fluid dynamics and material design. In these problems, the decision variables (e.g., control inputs or design parameters) are tightly coupled with the PDE state variables, and the feasible set is implicitly defined by the governing PDE constraints. This coupling makes the problems computationally demanding, as it requires handling high dimensional discretization and dynamic constraints. To address these challenges, this paper introduces a learning-based framework that integrates a dynamic predictor with an optimization surrogate. The dynamic predictor, a novel time-discrete Neural Operator (Lu et al.), efficiently approximate system trajectories governed by PDE dynamics, while the optimization surrogate leverages proxy optimizer techniques (Kotary et al.) to approximate the associated optimal decisions. This dual-network design enables real-time approximation of optimal strategies while explicitly capturing the coupling between decisions and PDE dynamics. We validate the proposed approach on benchmark PDE-constrained optimization tasks inlacing Burgers' equation, heat equation and voltage regulation, and demonstrate that it achieves solution quality comparable to classical control-based algorithms, such as the Direct Method and Model Predictive Control (MPC), while providing up to four orders of magnitude improvement in computational speed.