  Control systems are indispensable for ensuring the safety of cyber-physical
systems (CPS), spanning various domains such as automobiles, airplanes, and
missiles. Safeguarding CPS necessitates runtime methodologies that continuously
monitor safety-critical conditions and respond in a verifiably safe manner. A
fundamental aspect of many safety approaches involves predicting the future
behavior of systems. However, achieving this requires accurate models that can
operate in real time. Motivated by DeepONets, we propose a novel strategy that
combines the inductive bias of B-splines with data-driven neural networks to
facilitate real-time predictions of CPS behavior. We introduce our hybrid
B-spline neural operator, establishing its capability as a universal
approximator and providing rigorous bounds on the approximation error. These
findings are applicable to a broad class of nonlinear autonomous systems and
are validated through experimentation on a controlled 6-degree-of-freedom (DOF)
quadrotor with a 12 dimensional state space. Furthermore, we conduct a
comparative analysis of different network architectures, specifically fully
connected networks (FCNN) and recurrent neural networks (RNN), to elucidate the
practical utility and trade-offs associated with each architecture in
real-world scenarios.
