# Only Project Once: Projection-Adaptive Loss for Exact Constraint Satisfaction

arXiv: https://arxiv.org/abs/2610.04572 · HTML: https://mlsanity.com/paper/2610.04572/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 35 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 18 of 18)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [LTX-Video: Realtime Video Latent Diffusion](https://mlsanity.com/paper/2501.00103/) (2024)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [pandapower - an Open Source Python Tool for Convenient Modeling, Analysis and Optimization of Electric Power Systems](https://mlsanity.com/paper/1709.06743/) (2017)
- [ACE: A fast, skillful learned global atmospheric model for climate prediction](https://mlsanity.com/paper/2310.02074/) (2023)
- [DeepOPF: A Feasibility-Optimized Deep Neural Network Approach for AC Optimal Power Flow Problems](https://mlsanity.com/paper/2007.01002/) (2020)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [FSNet: Feasibility-Seeking Neural Network for Constrained Optimization with Guarantees](https://mlsanity.com/paper/2506.00362/) (2025)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)
- [Modeling Design and Control Problems Involving Neural Network Surrogates](https://mlsanity.com/paper/2111.10489/) (2021)
- [FloorSet -- a VLSI Floorplanning Dataset with Design Constraints of Real-World SoCs](https://mlsanity.com/paper/2405.05480/) (2024)
- [SnareNet: Flexible Repair Layers for Neural Networks with Hard Constraints](https://mlsanity.com/paper/2602.09317/) (2026)
- [Pretrained Video Models as Differentiable Physics Simulators for Urban Wind Flows](https://mlsanity.com/paper/2603.21210/) (2026)

## Abstract (arXiv)

Precise constraint satisfaction is a prerequisite to deploying learned models in many areas, motivating methods that repair raw neural predictions with a repair procedure. Current methods unroll multiple repair steps in training and softly penalize constraint violations that remain after the unroll. This is compute- and memory-intensive, lacks robustness when the repair fails to converge, and surrenders most of the constraint satisfaction work to the repair. Our central finding is that, contrary to common practice, a single detached projection step suffices in training. We accomplish this with a Projection-Adaptive Loss (PAL), which uses the constraint residual after this single step to adaptively weigh constraint penalties on the raw prediction. In experiments, PAL is the only method that retains virtually perfect feasibility on extremely nonlinear constraints, and matches or outperforms current methods on synthetic and engineering benchmarks. Because it only requires a single detached projection step, PAL trains 2.5x faster than the canonical repair-based method (DC3) on its own ACOPF benchmark. PAL can also be trained when constraints are expensive to evaluate (e.g., via neural surrogates), a setting where current unrolled methods are memory-intractable.