TNODEV: Toolbox for Neural ODE Verification

We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points. Our method is then tested on two paradigmatic physical models.

Reachability analysis of neural networks, which seeks to compute or bound the set of outputs attainable over a given input domain, is central to certifying safety and robustness in learning-enabled physical systems. Since exact reachable set computation is generally intractable, existing methods typically rely on tractable overapproximations. Examining the state of the art for smooth, twice-differentiable networks, we observe that existing approaches exploit at most second-order information and do not systematically leverage higher-order information. In this work, we introduce \textsc{HiTaB}, a novel verification framework that exploits second-order smoothness through both the Hessian, $\nabla^2 f$, and its Lipschitz constant, $L_{\nabla^2 f}$. We further develop a unified hierarchy of zeroth-, first-, and second-order bounds, together with precise conditions under which higher-order approximations yield provable improvements. Our main technical contribution is a compositional procedure for efficiently bounding $L_{\nabla^2 f}$ in deep neural networks via layerwise propagation of curvature bounds. We extend the framework to both $\ell_2$- and $\ell_\infty$-constrained input sets and show how it can be integrated into branch-and-bound verification pipelines. To our knowledge, this is the first practical reachability analysis framework for smooth neural networks that systematically exploits Lipschitz continuity of curvature, leading to tighter and more informative safety certificates.

Learning-based methods for synthesizing controllers have gained popularity due to their high expressiveness and strong empirical performance. However, in safety-critical scenarios such as autonomous driving, robotics, and power systems, empirical performance alone is insufficient, and formal verification of controller properties such as stability and safety is highly desirable. Unfortunately, many prior verification approaches are either tied to specific structural assumptions on the system or the certificate, making them difficult to transfer across settings, or suffer from poor scalability on higher-dimensional neural network systems. In this tutorial, we present a unified framework that aims to mitigate this gap via bridging control with the state-of-the-art neural network verifier $α,\!β$-CROWN (alpha-beta-CROWN). At its core, $α,\!β$-CROWN is a general-purpose bounding engine for nonlinear functions represented as computation graphs: given an input domain, it can produce certified bounds and explicit linear relaxation of the nonlinear function. These certified bounds are useful on their own for tasks such as reachability analysis, and they also provide the foundation for more complex routines that perform satisfiability checking and optimization. More specifically, many control problems reduce to verifying real-valued inequalities over a state domain (e.g., Lyapunov theory). Consequently, $α,\!β$-CROWN enables scalable verification of such conditions by computing tight bounds and recursively partitioning and pruning subdomains based on the bounds. Thanks to GPU parallelization, this pipeline demonstrates superior scalability on verification and optimization problems that are challenging for traditional approaches. In this tutorial, we discuss the basics of $α,\!β$-CROWN and introduce its application to various control-related tasks.