Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations—I: Mathematical Framework

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-or...

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Bibliographic Details
Published inProcesses Vol. 12; no. 12; p. 2660
Main Author Cacuci, Dan Gabriel
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.12.2024
Subjects
Analysis
Differential equations
Dynamical systems
Efficiency
Mathematics
Methodology
Methods
Neural networks
Nodes
Ordinary differential equations
Parameter sensitivity
Parameter uncertainty
Sensitivity analysis
System theory
Uncertainty analysis
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Summary: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 efficiently computes 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 is essential for performing sensitivity analysis and quantifying the uncertainties induced in the NODE-decoder responses by uncertainties in the underlying uncertain NODE-parameters.
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ISSN:2227-9717
2227-9717
DOI:10.3390/pr12122660