Extracting Exact Lie Derivatives Without Backpropagation: A Dual Compiler for Neural Control Barrier Functions

Abstract

Deploying neural-network control barrier functions (CBFs) on embedded hardware requires evaluating the barrier value and its Lie derivatives along the system vector fields at every control cycle. The standard mechanism for exact gradient extraction, reverse-mode automatic differentiation, constructs a dynamic computational graph whose memory footprint grows with network depth and whose backward traversal obstructs the worst-case execution time analysis required for safety-critical certification. This paper presents a dual-algebraic compiler that extracts the exact barrier value and its Lie derivatives through forward network evaluation alone. Encoding the system state as the real part of a dual number and a target vector field as the dual part, we prove that every affine and componentwise-activation layer admits a dual extension that propagates the exact directional derivative alongside the activation, and that the composed dual-extended network evaluates the exact Jacobian--vector-field product with zero truncation error. We derive closed-form expressions for the dual-pass floating-point operation count and peak memory footprint, prove that the proposed algorithm eliminates dynamic graph allocation, and extend the framework to the second-order Lie derivatives required by relative-degree-two CBFs using hyper-dual arithmetic. An open-source ahead-of-time compiler translates trained neural CBFs into self-contained C++ headers that assemble the complete safety constraint on an ESP32-S3 microcontroller from a statically allocated buffer, with zero dynamic memory allocation and a sub-millisecond cycle budget that supports kilohertz-rate safety filters.