# ShardNet: Training Neural Controllers with Hard, Non-Convex Constraints

arXiv: https://arxiv.org/abs/2606.30935 · HTML: https://mlsanity.com/paper/2606.30935/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 50 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Builds on (most cited first, 30 of 38)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [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)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Verification of Deep Convolutional Neural Networks Using ImageStars](https://mlsanity.com/paper/2004.05511/) (2020)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Safety and Liveness Guarantees through Reach-Avoid Reinforcement Learning](https://mlsanity.com/paper/2112.12288/) (2021)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [Learning Safe, Generalizable Perception-based Hybrid Control with Certificates](https://mlsanity.com/paper/2201.00932/) (2022)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints](https://mlsanity.com/paper/2507.08124/) (2025)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Scalable Synthesis of Formally Verified Neural Value Function for Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2407.20532/) (2024)
- [Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systems](https://mlsanity.com/paper/2210.08339/) (2022)
- [Constrained Feedforward Neural Network Training via Reachability Analysis](https://mlsanity.com/paper/2107.07696/) (2021)
- [POLICEd RL: Learning Closed-Loop Robot Control Policies with Provable Satisfaction of Hard Constraints](https://mlsanity.com/paper/2403.13297/) (2024)
- [Verifiable Safety Q-Filters via Hamilton-Jacobi Reachability and Multiplicative Q-Networks](https://mlsanity.com/paper/2506.15693/) (2025)
- [Goal-Reaching Trajectory Design Near Danger with Piecewise Affine Reach-avoid Computation](https://mlsanity.com/paper/2402.15604/) (2024)
- [Guaranteed Reach-Avoid for Black-Box Systems through Narrow Gaps via Neural Network Reachability](https://mlsanity.com/paper/2409.13195/) (2024)
- [State-action control barrier functions: Imposing safety on learning-based control with low online computational costs](https://mlsanity.com/paper/2312.11255/) (2023)

## Abstract (arXiv)

While neural network control policies are powerful, their deployment on safety critical systems depends on ensuring that they obey strict constraints. Existing work often treats safety as a metric to optimize for, which competes with other performance objectives, if training converges at all. Instead, we introduce ShardNet, a neural network architecture that strictly enforces unions of polyhedral constraints by construction, using a differentiable projection layer parameterized by a classification network. The key insight is to embed safety into the neural network's structure, allowing performance to be optimized independently because formal safety guarantees are always given. In contrast with existing neural architectures that can only enforce simple convex constraints, ShardNet enables the first safe-by-construction synthesis of forward-invariant neural network controllers on closed-loop systems where safety constraints are expressed as nonconvex unions of polyhedras or learned value function level sets. To support this, we also introduce a technique to verify and train such value functions correctly as rectified linear unit (ReLU) networks, which has not previously been possible. On double integrator benchmarks drawn from the literature, ShardNet policies maintain 100% safety on verified sets and achieves significantly lower objective loss compared to existing formal methods. Furthermore, our value function training technique also produces safe sets more than 3 times larger than existing verification approaches.