# LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers

arXiv: https://arxiv.org/abs/2604.05374 · HTML: https://mlsanity.com/paper/2604.05374/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 21 references that are in the map.
- First posted on arXiv 7 Apr 2026.

## Builds on (most cited first, 16 of 16)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Learning Stability Certificates from Data](https://mlsanity.com/paper/2008.05952/) (2020)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [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)
- [End-to-End Learning to Warm-Start for Real-Time Quadratic Optimization](https://mlsanity.com/paper/2212.08260/) (2022)
- [Learning Hybrid Control Barrier Functions from Data](https://mlsanity.com/paper/2011.04112/) (2020)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Convergence rate analysis of several splitting schemes](https://mlsanity.com/paper/1406.4834/) (2014)
- [Data-Driven Control with Inherent Lyapunov Stability](https://mlsanity.com/paper/2303.03157/) (2023)
- [Learning Dissipative Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2410.00976/) (2024)
- [Learning Contraction Policies from Offline Data](https://mlsanity.com/paper/2112.05911/) (2021)
- [ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2512.01984/) (2025)
- [Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions](https://mlsanity.com/paper/2104.05445/) (2021)

## Built on by (most cited first, 4 of 4)

- [HardNet++: Nonlinear Constraint Enforcement in Neural Networks](https://mlsanity.com/paper/2604.19669/) (2026)
- [Provably Safe, Yet Scalable Reinforcement Learning](https://mlsanity.com/paper/2606.14536/) (2026)
- [DisjunctiveNet: Neural Symbolic Learning via Differentiable Convexified Optimization Layers](https://mlsanity.com/paper/2605.30456/) (2026)
- [SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization](https://mlsanity.com/paper/2609.33515/) (2026)

## Abstract (arXiv)

Linear matrix inequalities (LMIs) have played a central role in certifying stability, robustness, and forward invariance of dynamical systems. Despite rapid development in learning-based methods for control design and certificate synthesis, existing approaches often fail to preserve the hard matrix inequality constraints required for formal guarantees. We propose LMI-Net, an efficient and modular differentiable projection layer that enforces LMI constraints by construction. Our approach lifts the set defined by LMI constraints into the intersection of an affine equality constraint and the positive semidefinite cone, performs the forward pass via Douglas-Rachford splitting, and supports efficient backward propagation through implicit differentiation. We establish theoretical guarantees that the projection layer converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints. Evaluated on experiments including invariant ellipsoid synthesis and joint controller-and-certificate design for a family of disturbed linear systems, LMI-Net substantially improves feasibility over soft-constrained models under distribution shift while retaining fast inference speed, bridging semidefinite-program-based certification and modern learning techniques.