First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations: Mathematical Framework and Illustrative Application to the Nordheim–Fuchs Reactor Safety Model

Abstract

This work introduces the mathematical framework of the novel “First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations” (1st-CASAM-NODE) which yields exact expressions for the first-order sensitivities of NODE decoder responses to the NODE parameters, including encoder initial conditions, while enabling the most efficient computation of these sensitivities. The application of the 1st-CASAM-NODE is illustrated by using the Nordheim–Fuchs reactor dynamics/safety phenomenological model, which is representative of physical systems that would be modeled by NODE while admitting exact analytical solutions for all quantities of interest (hidden states, decoder outputs, sensitivities with respect to all parameters and initial conditions, etc.). This work also lays the foundation for the ongoing work on conceiving the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations” (2nd-CASAM-NODE) which aims at yielding exact expressions for the second-order sensitivities of NODE decoder responses to the NODE parameters and initial conditions while enabling the most efficient computation of these sensitivities.