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

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.

Reverse Flow Matching: A Unified Framework for Online Reinforcement Learning with Diffusion and Flow Policies

Diffusion and flow policies are gaining prominence in online reinforcement learning (RL) due to their expressive power, yet training them efficiently remains a critical challenge. A fundamental difficulty in online RL is the lack of direct samples from the target distribution; instead, the target is an unnormalized Boltzmann distribution defined by the Q-function. To address this, two seemingly distinct families of methods have been proposed for diffusion policies: a noise-expectation family, which utilizes a weighted average of noise as the training target, and a gradient-expectation family, which employs a weighted average of Q-function gradients. Yet, it remains unclear how these objectives relate formally or if they can be synthesized into a more general formulation. In this paper, we propose a unified framework, reverse flow matching (RFM), which rigorously addresses the problem of training diffusion and flow models without direct target samples. By adopting a reverse inferential perspective, we formulate the training target as a posterior mean estimation problem given an intermediate noisy sample. Crucially, we introduce Langevin Stein operators to construct zero-mean control variates, deriving a general class of estimators that effectively reduce importance sampling variance. We show that existing noise-expectation and gradient-expectation methods are two specific instances within this broader class. This unified view yields two key advancements: it extends the capability of targeting Boltzmann distributions from diffusion to flow policies, and enables the principled combination of Q-value and Q-gradient information to derive an optimal, minimum-variance estimator, thereby improving training efficiency and stability. We instantiate RFM to train a flow policy in online RL, and demonstrate improved performance on continuous-control benchmarks compared to diffusion policy baselines.