# mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks

arXiv: https://arxiv.org/abs/2502.02434 · HTML: https://mlsanity.com/paper/2502.02434/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 4 Feb 2025.

## Builds on (most cited first, 21 of 21)

- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Transformers Can Do Bayesian Inference](https://mlsanity.com/paper/2112.10510/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation](https://mlsanity.com/paper/2305.18470/) (2023)
- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [CGD: Constraint-Guided Diffusion Policies for UAV Trajectory Planning](https://mlsanity.com/paper/2405.01758/) (2024)
- [Constrained Optimization to Train Neural Networks on Critical and Under-Represented Classes](https://mlsanity.com/paper/2102.12894/) (2021)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [Architecture-Preserving Provable Repair of Deep Neural Networks](https://mlsanity.com/paper/2304.03496/) (2023)
- [XLB: A differentiable massively parallel lattice Boltzmann library in Python](https://mlsanity.com/paper/2311.16080/) (2023)
- [Neural Fields with Hard Constraints of Arbitrary Differential Order](https://mlsanity.com/paper/2306.08943/) (2023)
- [POLICEd RL: Learning Closed-Loop Robot Control Policies with Provable Satisfaction of Hard Constraints](https://mlsanity.com/paper/2403.13297/) (2024)
- [Generative Optimization: A Perspective on AI-Enhanced Problem Solving in Engineering](https://mlsanity.com/paper/2412.13281/) (2024)
- [Fast and Accurate Bayesian Optimization with Pre-trained Transformers for Constrained Engineering Problems](https://mlsanity.com/paper/2404.04495/) (2024)
- [Learning to Provably Satisfy High Relative Degree Constraints for Black-Box Systems](https://mlsanity.com/paper/2407.20456/) (2024)

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

- [ShardNet: Training Neural Controllers with Hard, Non-Convex Constraints](https://mlsanity.com/paper/2606.30935/) (2026)

## Abstract (arXiv)

Deep neural networks are increasingly used in safety-critical domains such as robotics and scientific modeling, where strict adherence to output constraints is essential. Methods like POLICE, which are tailored for single convex regions, face challenges when extended to multiple disjoint regions, often leading to constraint violations or unwanted affine behavior across regions. This paper proposes mPOLICE, a new approach that generalizes POLICE to provably enforce affine constraints over multiple disjoint convex regions. At its core, mPOLICE assigns distinct neuron activation patterns to each constrained region, enabling localized affine behavior and avoiding unintended generalization. This is implemented through a layer-wise optimization of the network parameters. Additionally, we introduce a training algorithm that incorporates mPOLICE into conventional deep learning pipelines, balancing task-specific performance with constraint enforcement using periodic sign pattern enforcement. We validate the flexibility and effectiveness of mPOLICE through experiments across various applications, including safety-critical reinforcement learning, implicit 3D shape representation with geometric constraints, and fluid dynamics simulations with boundary condition enforcement. Importantly, mPOLICE incurs no runtime overhead during inference, making it a practical and reliable solution for constraint handling in deep neural networks.