Probabilistic modelling lies at the heart of machine learning.

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems.

However, these methods typically reach superlinear convergence only when the stochastic Hessian noise diminishes, increasing per-iteration costs over time.

Consequently, our framework transforms the task of proving convergence rate into verifying the positive semidefiniteness of a specific integral kernel.

ReLU (T (v u) + b) = ReLU( Tv + b), where u = 0, that is, ReLU (T + b) is not injective. By injectivity of T, we finally get a = b . Remark 2. An example that satisfies (3.1) is the neural operator whose This construction is given by the combination of "Pairs of projections" discussed in Kato [2013, Section I.4.6] with the idea presented in [Puthawala et al., 2022b, Lemma 29]. R. We write operator null G by Thus, in both cases, H is injective. Remark 4. W e make the following observations using Theorem 1: Leaky ReLU is one of example that satisfies (ii) in Theorem 1. Puthawala et al. [2022a, Theorem 15] assumes that We first revisit layerwise injectivity and bijectivity in the case of the finite rank approximation.