Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integral Equations of the Volterra Type: Mathematical Methodology and Illustrative Application to Nuclear Engineering

This work presents the general mathematical frameworks of the “First and Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integral Equations of Volterra Type” designated as the 1st-FASAM-NIE-V and the 2nd-FASAM-NIE-V methodologies, respectively. Using a single large-scale (adjoint) computation, the 1st-FASAM-NIE-V enables the most efficient computation of the exact expressions of all first-order sensitivities of the decoder response to the feature functions and also with respect to the optimal values of the NIE-net’s parameters/weights after the respective NIE-Volterra-net was optimized to represent the underlying physical system. The computation of all second-order sensitivities with respect to the feature functions using the 2nd-FASAM-NIE-V requires as many large-scale computations as there are first-order sensitivities of the decoder response with respect to the feature functions. Subsequently, the second-order sensitivities of the decoder response with respect to the primary model parameters are obtained trivially by applying the “chain-rule of differentiation” to the second-order sensitivities with respect to the feature functions. The application of the 1st-FASAM-NIE-V and the 2nd-FASAM-NIE-V methodologies is illustrated by using a well-known model for neutron slowing down in a homogeneous hydrogenous medium, which yields tractable closed-form exact explicit expressions for all quantities of interest, including the various adjoint sensitivity functions and first- and second-order sensitivities of the decoder response with respect to all feature functions and also primary model parameters.