Version 1
: Received: 14 October 2024 / Approved: 14 October 2024 / Online: 15 October 2024 (09:00:32 CEST)
How to cite:
Cacuci, D. G. Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations. I: Mathematical Framework. Preprints2024, 2024101110. https://doi.org/10.20944/preprints202410.1110.v1
Cacuci, D. G. Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations. I: Mathematical Framework. Preprints 2024, 2024101110. https://doi.org/10.20944/preprints202410.1110.v1
Cacuci, D. G. Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations. I: Mathematical Framework. Preprints2024, 2024101110. https://doi.org/10.20944/preprints202410.1110.v1
APA Style
Cacuci, D. G. (2024). Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations. I: Mathematical Framework. Preprints. https://doi.org/10.20944/preprints202410.1110.v1
Chicago/Turabian Style
Cacuci, D. G. 2024 "Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations. I: Mathematical Framework" Preprints. https://doi.org/10.20944/preprints202410.1110.v1
Abstract
This work introduces the mathematical framework of the novel “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations” (1st-FASAM-NODE). The 1st-FASAM-NODE methodology produces and computes most efficiently the exact expressions of all of the first-order sensitivities of NODE-decoder responses with respect to the parameters underlying the NODE’s decoder, hidden layers, and encoder, after having optimized the NODE-net to represent the physical system under consideration. Building on the 1st-FASAM-NODE, this work subsequently introduces the mathematical framework of the novel “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (2nd-FASAM-NODE).” The 2nd-FASAM-NODE methodology yields and computes most efficiently the exact expressions of the second-order sensitivities of NODE decoder-responses with respect to the NODE parameters. Since the physical system modeled by the NODE-net necessarily comprises imprecisely known parameters that stem from measurements and/or computations subject to uncertainties, the availability of the first-and second-order sensitivities of decoder responses to the parameters underlying the NODE-net are essential for performing sensitivity analysis and quantifying the uncertainties induced in the NODE-decoder responses by uncertainties in the underlying uncertain NODE-parameters.
Keywords
neural ordinary differential equations (NODE); first-order sensitivities of decoder responses; second-order sensitivities of decoder responses; first-level adjoint sensitivity systems; second-level adjoint sensitivity systems; features/functions of model parameters
Subject
Engineering, Safety, Risk, Reliability and Quality
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
