# DC3: A learning method for optimization with hard constraints

arXiv: https://arxiv.org/abs/2104.12225 · HTML: https://mlsanity.com/paper/2104.12225/

## Key facts

- Cited by 184 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 33 references that are in the map.
- First posted on arXiv 25 Apr 2021.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Tackling Climate Change with Machine Learning](https://mlsanity.com/paper/1906.05433/) (2019)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (2019)
- [Learning for Constrained Optimization: Identifying Optimal Active Constraint Sets](https://mlsanity.com/paper/1802.09639/) (2018)
- [Learning Warm-Start Points for AC Optimal Power Flow](https://mlsanity.com/paper/1905.08860/) (2019)
- [Achieving Conservation of Energy in Neural Network Emulators for Climate Modeling](https://mlsanity.com/paper/1906.06622/) (2019)
- [What game are we playing? End-to-end learning in normal and extensive form games](https://mlsanity.com/paper/1805.02777/) (2018)
- [Solution Refinement at Regular Points of Conic Problems](https://mlsanity.com/paper/1811.02157/) (2018)
- [Differentiable Submodular Maximization](https://mlsanity.com/paper/1803.01785/) (2018)
- [Smart-PGSim: Using Neural Network to Accelerate AC-OPF Power Grid Simulation](https://mlsanity.com/paper/2008.11827/) (2020)
- [First-order Methods Almost Always Avoid Saddle Points](https://mlsanity.com/paper/1710.07406/) (2017)

## Built on by (most cited first, 30 of 184)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [NIFTY: Neural Object Interaction Fields for Guided Human Motion Synthesis](https://mlsanity.com/paper/2307.07511/) (2023)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [On the Constrained Time-Series Generation Problem](https://mlsanity.com/paper/2307.01717/) (2023)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Deep Equilibrium Approaches to Diffusion Models](https://mlsanity.com/paper/2210.12867/) (2022)
- [Adversarial Robustness with Semi-Infinite Constrained Learning](https://mlsanity.com/paper/2110.15767/) (2021)
- [End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch](https://mlsanity.com/paper/2304.11726/) (2023)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Hard-Constrained Deep Learning for Climate Downscaling](https://mlsanity.com/paper/2208.05424/) (2022)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [Constraint-based graph network simulator](https://mlsanity.com/paper/2112.09161/) (2021)
- [Scaling physics-informed hard constraints with mixture-of-experts](https://mlsanity.com/paper/2402.13412/) (2024)
- [Physics-Informed Neural Networks for Minimising Worst-Case Violations in DC Optimal Power Flow](https://mlsanity.com/paper/2107.00465/) (2021)
- [Evaluating Model-free Reinforcement Learning toward Safety-critical Tasks](https://mlsanity.com/paper/2212.05727/) (2022)
- [SurCo: Learning Linear Surrogates For Combinatorial Nonlinear Optimization Problems](https://mlsanity.com/paper/2210.12547/) (2022)
- [Guiding continuous operator learning through Physics-based boundary constraints](https://mlsanity.com/paper/2212.07477/) (2022)
- [CGD: Constraint-Guided Diffusion Policies for UAV Trajectory Planning](https://mlsanity.com/paper/2405.01758/) (2024)
- [FSNet: Feasibility-Seeking Neural Network for Constrained Optimization with Guarantees](https://mlsanity.com/paper/2506.00362/) (2025)
- [DiffuSolve: Diffusion-based Solver for Non-convex Trajectory Optimization](https://mlsanity.com/paper/2403.05571/) (2024)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [Competitive Physics Informed Networks](https://mlsanity.com/paper/2204.11144/) (2022)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Deep Equilibrium Diffusion Restoration with Parallel Sampling](https://mlsanity.com/paper/2311.11600/) (2023)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [OpenForest: A data catalogue for machine learning in forest monitoring](https://mlsanity.com/paper/2311.00277/) (2023)

## Abstract (arXiv)

Large optimization problems with hard constraints arise in many settings, yet classical solvers are often prohibitively slow, motivating the use of deep networks as cheap "approximate solvers." Unfortunately, naive deep learning approaches typically cannot enforce the hard constraints of such problems, leading to infeasible solutions. In this work, we present Deep Constraint Completion and Correction (DC3), an algorithm to address this challenge. Specifically, this method enforces feasibility via a differentiable procedure, which implicitly completes partial solutions to satisfy equality constraints and unrolls gradient-based corrections to satisfy inequality constraints. We demonstrate the effectiveness of DC3 in both synthetic optimization tasks and the real-world setting of AC optimal power flow, where hard constraints encode the physics of the electrical grid. In both cases, DC3 achieves near-optimal objective values while preserving feasibility.