Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations
This work presents the “First-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations” (1st-FASAM-NIE-Fredholm) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations” (2nd-FASAM-NIE-Fredholm). I...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/1/14 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1841549178646298624 |
|---|---|
| author | Dan Gabriel Cacuci |
| author_facet | Dan Gabriel Cacuci |
| author_sort | Dan Gabriel Cacuci |
| collection | DOAJ |
| description | This work presents the “First-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations” (1st-FASAM-NIE-Fredholm) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations” (2nd-FASAM-NIE-Fredholm). It is shown that the 1st-FASAM-NIE-Fredholm methodology enables the efficient computation of exactly determined first-order sensitivities of decoder response with respect to the optimized NIE-parameters, requiring a single “large-scale” computation for solving the First-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIE-net. The 2nd-FASAM-NIE-Fredholm methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights involved in the NIE’s decoder, hidden layers, and encoder, requiring only as many “large-scale” computations as there are first-order sensitivities with respect to the feature functions. The application of both the 1st-FASAM-NIE-Fredholm and the 2nd-FASAM-NIE-Fredholm methodologies is illustrated by considering a system of nonlinear Fredholm-type NIE that admits analytical solutions, thereby facilitating the verification of the expressions obtained for the first- and second-order sensitivities of NIE-decoder responses with respect to the model parameters (weights) that characterize the respective NIE-net. |
| format | Article |
| id | doaj-art-d01412dbafb44242ab641fd3dbee7cc2 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-d01412dbafb44242ab641fd3dbee7cc22025-01-10T13:17:58ZengMDPI AGMathematics2227-73902024-12-011311410.3390/math13010014Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral EquationsDan Gabriel Cacuci0Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USAThis work presents the “First-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations” (1st-FASAM-NIE-Fredholm) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations” (2nd-FASAM-NIE-Fredholm). It is shown that the 1st-FASAM-NIE-Fredholm methodology enables the efficient computation of exactly determined first-order sensitivities of decoder response with respect to the optimized NIE-parameters, requiring a single “large-scale” computation for solving the First-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIE-net. The 2nd-FASAM-NIE-Fredholm methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights involved in the NIE’s decoder, hidden layers, and encoder, requiring only as many “large-scale” computations as there are first-order sensitivities with respect to the feature functions. The application of both the 1st-FASAM-NIE-Fredholm and the 2nd-FASAM-NIE-Fredholm methodologies is illustrated by considering a system of nonlinear Fredholm-type NIE that admits analytical solutions, thereby facilitating the verification of the expressions obtained for the first- and second-order sensitivities of NIE-decoder responses with respect to the model parameters (weights) that characterize the respective NIE-net.https://www.mdpi.com/2227-7390/13/1/14neural integral equationsnonlinear Fredholm-type neural integral equationsfirst-order features adjoint sensitivity analysis methodologysecond-order features adjoint sensitivity analysis methodology |
| spellingShingle | Dan Gabriel Cacuci Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations Mathematics neural integral equations nonlinear Fredholm-type neural integral equations first-order features adjoint sensitivity analysis methodology second-order features adjoint sensitivity analysis methodology |
| title | Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations |
| title_full | Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations |
| title_fullStr | Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations |
| title_full_unstemmed | Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations |
| title_short | Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Fredholm-Type Neural Integral Equations |
| title_sort | introducing the second order features adjoint sensitivity analysis methodology for fredholm type neural integral equations |
| topic | neural integral equations nonlinear Fredholm-type neural integral equations first-order features adjoint sensitivity analysis methodology second-order features adjoint sensitivity analysis methodology |
| url | https://www.mdpi.com/2227-7390/13/1/14 |
| work_keys_str_mv | AT dangabrielcacuci introducingthesecondorderfeaturesadjointsensitivityanalysismethodologyforfredholmtypeneuralintegralequations |