# ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection

arXiv: https://arxiv.org/abs/2502.06774 · HTML: https://mlsanity.com/paper/2502.06774/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 61 references that are in the map.
- First posted on arXiv 10 Feb 2025.

## Builds on (most cited first, 30 of 32)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [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)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [AI in Finance: Challenges, Techniques and Opportunities](https://mlsanity.com/paper/2107.09051/) (2021)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [ContactNets: Learning Discontinuous Contact Dynamics with Smooth, Implicit Representations](https://mlsanity.com/paper/2009.11193/) (2020)
- [Multi-Label Classification Neural Networks with Hard Logical Constraints](https://mlsanity.com/paper/2103.13427/) (2021)
- [Global Deterministic Optimization with Artificial Neural Networks Embedded](https://mlsanity.com/paper/1801.07114/) (2018)
- [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)
- [Physics-Informed Neural Networks with Hard Linear Equality Constraints](https://mlsanity.com/paper/2402.07251/) (2024)
- [Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method](https://mlsanity.com/paper/2012.06148/) (2020)
- [How Realistic Is Your Synthetic Data? Constraining Deep Generative Models for Tabular Data](https://mlsanity.com/paper/2402.04823/) (2024)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [LinSATNet: The Positive Linear Satisfiability Neural Networks](https://mlsanity.com/paper/2407.13917/) (2024)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Architecture-Preserving Provable Repair of Deep Neural Networks](https://mlsanity.com/paper/2304.03496/) (2023)
- [Obey validity limits of data-driven models](https://mlsanity.com/paper/2010.03405/) (2020)
- [Sample-Specific Output Constraints for Neural Networks](https://mlsanity.com/paper/2003.10258/) (2020)
- [Learning To Solve Differential Equation Constrained Optimization Problems](https://mlsanity.com/paper/2410.01786/) (2024)

## Built on by (most cited first, 17 of 17)

- [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)
- [Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints](https://mlsanity.com/paper/2507.08124/) (2025)
- [Enforcing Hard Linear Constraints in Deep Learning Models with Decision Rules](https://mlsanity.com/paper/2505.13858/) (2025)
- [Molecular Machine Learning in Chemical Process Design](https://mlsanity.com/paper/2508.20527/) (2025)
- [Physics-Constrained Machine Learning for Chemical Engineering](https://mlsanity.com/paper/2508.20649/) (2025)
- [DAE-HardNet: A Physics Constrained Neural Network Enforcing Differential-Algebraic Hard Constraints](https://mlsanity.com/paper/2512.05881/) (2025)
- [ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2512.01984/) (2025)
- [HardNet++: Nonlinear Constraint Enforcement in Neural Networks](https://mlsanity.com/paper/2604.19669/) (2026)
- [DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables](https://mlsanity.com/paper/2606.05247/) (2026)
- [Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs](https://mlsanity.com/paper/2603.29237/) (2026)
- [NLPOpt-Net: A Learning Method for Nonlinear Optimization with Feasibility Guarantees](https://mlsanity.com/paper/2605.00260/) (2026)
- [DisjunctiveNet: Neural Symbolic Learning via Differentiable Convexified Optimization Layers](https://mlsanity.com/paper/2605.30456/) (2026)
- [PL-KKT-hPINN: Enforcing Nonlinear Equality Constraints on Neural Networks via Piecewise-Linear Projection](https://mlsanity.com/paper/2606.10682/) (2026)
- [ShardNet: Training Neural Controllers with Hard, Non-Convex Constraints](https://mlsanity.com/paper/2606.30935/) (2026)
- [Only Project Once: Projection-Adaptive Loss for Exact Constraint Satisfaction](https://mlsanity.com/paper/2610.04572/) (2026)
- [ORACLE: Optimizer-Relative Alignment for Constrained LEarning](https://mlsanity.com/paper/2610.09040/) (2026)

## Abstract (arXiv)

Ensuring neural networks adhere to domain-specific constraints is crucial for addressing safety and trustworthiness while also enhancing inference accuracy. Despite the nonlinear nature of most real-world tasks, the majority of existing methods are limited to affine (equality) or convex (inequality) constraints. We introduce ENFORCE, a neural network architecture that uses an adaptive projection module (AdaNP) to enforce nonlinear equality and inequality constraints in the predictions up to a specified tolerance varepsilon, and exactly in the affine-in-y case. For affine constraint sets, we prove that the associated projection mapping is non-expansive (1-Lipschitz), ensuring stable gradient propagation. For nonlinear constraints, we establish local convergence analysis under standard regularity conditions. We evaluate ENFORCE on multiple tasks, including function fitting, real-world engineering case studies, and learning optimization problems. For the latter, we introduce a class of scalable optimization problems as a benchmark for nonlinear constrained learning. In the benchmarks, the predictions of our architecture satisfy nonlinear equality and inequality constraints up to a prescribed tolerance varepsilon, while maintaining scalability with tractable computational complexity at training and inference time.