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Control Barrier Functions (CBFs) that provide formal safety guarantees havebeen widely used for safety-critical systems. However, it is non-trivial todesign a CBF. Utilizing neural networks as CBFs has shown great success, but itnecessitates their certification as CBFs. In this work, we leverage boundpropagation techniques and the Branch-and-Bound scheme to efficiently verifythat a neural network satisfies the conditions to be a CBF over the continuousstate space. To accelerate training, we further present a framework that embedsthe verification scheme into the training loop to synthesize and verify aneural CBF simultaneously. In particular, we employ the verification scheme toidentify partitions of the state space that are not guaranteed to satisfy theCBF conditions and expand the training dataset by incorporating additional datafrom these partitions. The neural network is then optimized using the augmenteddataset to meet the CBF conditions. We show that for a non-linearcontrol-affine system, our framework can efficiently certify a neural networkas a CBF and render a larger safe set than state-of-the-art neural CBF works.We further employ our learned neural CBF to derive a safe controller toillustrate the practical use of our framework.

Simultaneous Synthesis and Verification of Neural Control Barrier Functions through Branch-and-Bound Verification-in-the-loop Training