{"id":62018,"date":"2025-04-07T07:54:19","date_gmt":"2025-04-07T07:54:19","guid":{"rendered":""},"modified":"2025-04-07T07:54:19","modified_gmt":"2025-04-07T07:54:19","slug":"%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-62018","status":"publish","type":"product","link":"https:\/\/express24.ir\/d\/product\/%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-62018\/","title":{"rendered":"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc \u0641\u0631\u062f\u0648\u0644\u0645"},"content":{"rendered":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nFredholm Neural Networks<\/td>\n<\/tr>\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc \u0641\u0631\u062f\u0648\u0644\u0645<\/td>\n<\/tr>\n
\u0646\u0648\u06cc\u0633\u0646\u062f\u06af\u0627\u0646 <\/td>\nKyriakos Georgiou, Constantinos Siettos, Athanasios N. Yannacopoulos<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nPDF<\/td>\n<\/tr>\n
\u0632\u0628\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u062a\u062d\u0648\u06cc\u0644\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u062a\u0631\u062c\u0645\u0647 \u0634\u062f\u0647 <\/td>\n\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\n<\/tr>\n
\u0646\u062d\u0648\u0647 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 <\/td>\n\u062f\u0648 \u062a\u0627 \u0633\u0647 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634 (\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u062f\u0627\u0646\u0644\u0648\u062f\u06cc)<\/td>\n<\/tr>\n
\u062a\u0639\u062f\u0627\u062f \u0635\u0641\u062d\u0627\u062a<\/td>\n22<\/td>\n<\/tr>\n
\u0644\u06cc\u0646\u06a9 \u062f\u0627\u0646\u0644\u0648\u062f \u0631\u0627\u06cc\u06af\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc<\/td>\n\u062f\u0627\u0646\u0644\u0648\u062f \u0645\u0642\u0627\u0644\u0647<\/a><\/td>\n<\/tr>\n
\u062f\u0633\u062a\u0647 \u0628\u0646\u062f\u06cc \u0645\u0648\u0636\u0648\u0639\u0627\u062a <\/td>\nNumerical Analysis,Dynamical Systems,\u062a\u062c\u0632\u06cc\u0647 \u0648 \u062a\u062d\u0644\u06cc\u0644 \u0639\u062f\u062f\u06cc , \u0633\u06cc\u0633\u062a\u0645 \u0647\u0627\u06cc \u062f\u06cc\u0646\u0627\u0645\u06cc\u06a9\u06cc ,<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a <\/td>\nSubmitted 20 August, 2024; v1 submitted 18 August, 2024; originally announced August 2024. , Comments: 22 pages, 11 figures Replacement reason: corrected licence to arXiv.org perpetual, non-exclusive license , MSC Class: 65R20 (Primary); 68T07; 45B05; 65N38 (Secondary)<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n20 \u0627\u0648\u062a 2024 \u0627\u0631\u0633\u0627\u0644 \u0634\u062f.V1 \u0627\u0631\u0633\u0627\u0644 \u0634\u062f\u0647 18 \u0627\u0648\u062a 2024 \u061b\u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0627\u0648\u062a 2024. \u060c \u0646\u0638\u0631\u0627\u062a: 22 \u0635\u0641\u062d\u0647 \u060c 11 \u0631\u0642\u0645 \u062f\u0644\u06cc\u0644 \u062c\u0627\u06cc\u06af\u0632\u06cc\u0646\u06cc: \u0645\u062c\u0648\u0632 \u0627\u0635\u0644\u0627\u062d \u0634\u062f\u0647 \u0628\u0647 Arxiv.org \u062f\u0627\u0626\u0645\u06cc \u060c \u0645\u062c\u0648\u0632 \u063a\u06cc\u0631 \u0627\u0646\u062d\u0635\u0627\u0631\u06cc \u060c \u06a9\u0644\u0627\u0633 MSC: 65R20 (\u0627\u0648\u0644\u06cc\u0647) \u061b68T07 \u061b45B05 \u061b65N38 (\u062b\u0627\u0646\u0648\u06cc\u0647)<\/td>\n<\/tr>\n
\u0627\u0637\u0644\u0627\u0639\u0627\u062a \u0628\u06cc\u0634\u062a\u0631 \u0627\u0632 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u062f\u0631 \u067e\u0627\u06cc\u06af\u0627\u0647 \u0647\u0627\u06cc \u0639\u0644\u0645\u06cc <\/td>\n\n INSPIRE HEP<\/a>
\n
\n NASA ADS<\/a>
\n
\n Google Scholar<\/a>
\n
\n Semantic Scholar<\/a>
\n
\n arXiv<\/a><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n

\u0686\u06a9\u06cc\u062f\u0647<\/h2>\n

Within the family of explainable machine-learning, we present Fredholm neural networks (Fredholm NNs), deep neural networks (DNNs) which replicate fixed point iterations for the solution of linear and nonlinear Fredholm Integral Equations (FIE) of the second kind. Applications of FIEs include the solution of ordinary, as well as partial differential equations (ODEs, PDEs) and many more. We first prove that Fredholm NNs provide accurate solutions. We then provide insight into the values of the hyperparameters and trainable\/explainable weights and biases of the DNN, by directly connecting their values to the underlying mathematical theory. For our illustrations, we use Fredholm NNs to solve both linear and nonlinear problems, including elliptic PDEs and boundary value problems. We show that the proposed scheme achieves significant numerical approximation accuracy across both the domain and boundary. The proposed methodology provides insight into the connection between neural networks and classical numerical methods, and we posit that it can have applications in fields such as Uncertainty Quantification (UQ) and explainable artificial intelligence (XAI). Thus, we believe that it will trigger further advances in the intersection between scientific machine learning and numerical analysis.<\/p>\n

\u0686\u06a9\u06cc\u062f\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc (\u062a\u0631\u062c\u0645\u0647 \u0645\u0627\u0634\u06cc\u0646\u06cc)<\/h2>\n

\u062f\u0631 \u062f\u0627\u062e\u0644 \u062e\u0627\u0646\u0648\u0627\u062f\u0647 \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 \u0642\u0627\u0628\u0644 \u062a\u0648\u0636\u06cc\u062d \u060c \u0645\u0627 \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc Fredholm (Fredholm NNS) \u060c \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc \u0639\u0645\u06cc\u0642 (DNN) \u0631\u0627 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u06a9\u0647 \u062a\u06a9\u0631\u0627\u0631\u0647\u0627\u06cc \u0646\u0642\u0637\u0647 \u062b\u0627\u0628\u062a \u0631\u0627 \u0628\u0631\u0627\u06cc \u062d\u0644 \u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0646\u062a\u06af\u0631\u0627\u0644 \u062e\u0637\u06cc \u0648 \u063a\u06cc\u0631\u062e\u0637\u06cc \u0641\u0631\u062f\u0648\u0644\u0645 (FIE) \u0627\u0632 \u0646\u0648\u0639 \u062f\u0648\u0645 \u062a\u06a9\u0631\u0627\u0631 \u0645\u06cc \u06a9\u0646\u0646\u062f.\u0628\u0631\u0646\u0627\u0645\u0647 \u0647\u0627\u06cc \u06a9\u0627\u0631\u0628\u0631\u062f\u06cc FIE \u0634\u0627\u0645\u0644 \u062d\u0644 \u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0645\u0639\u0645\u0648\u0644\u06cc \u0648 \u0647\u0645\u0686\u0646\u06cc\u0646 \u062f\u06cc\u0641\u0631\u0627\u0646\u0633\u06cc\u0644 \u062c\u0632\u0626\u06cc (ODE \u060c PDE) \u0648 \u0645\u0648\u0627\u0631\u062f \u062f\u06cc\u06af\u0631 \u0627\u0633\u062a.\u0645\u0627 \u0627\u0628\u062a\u062f\u0627 \u062b\u0627\u0628\u062a \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u06a9\u0647 NNS Fredholm \u0631\u0627\u0647 \u062d\u0644 \u0647\u0627\u06cc \u062f\u0642\u06cc\u0642\u06cc \u0631\u0627 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u062f.\u0633\u067e\u0633 \u0628\u0627 \u0627\u062a\u0635\u0627\u0644 \u0645\u0633\u062a\u0642\u06cc\u0645 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0622\u0646\u0647\u0627 \u0628\u0647 \u0646\u0638\u0631\u06cc\u0647 \u0631\u06cc\u0627\u0636\u06cc \u0627\u0633\u0627\u0633\u06cc \u060c \u0645\u0627 \u0628\u06cc\u0646\u0634 \u062f\u0631 \u0645\u0648\u0631\u062f \u0645\u0642\u0627\u062f\u06cc\u0631 \u0647\u0627\u06cc\u067e\u0631\u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627 \u0648 \u0648\u0632\u0646 \u0647\u0627\u06cc \u0642\u0627\u0628\u0644 \u0622\u0645\u0648\u0632\u0634\/\u0642\u0627\u0628\u0644 \u062a\u0648\u0636\u06cc\u062d \u0648 \u062a\u0639\u0635\u0628 DNN \u0631\u0627 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u06cc\u0645.\u0628\u0631\u0627\u06cc \u062a\u0635\u0627\u0648\u06cc\u0631 \u0645\u0627 \u060c \u0645\u0627 \u0627\u0632 NNS Fredholm \u0628\u0631\u0627\u06cc \u062d\u0644 \u0645\u0634\u06a9\u0644\u0627\u062a \u062e\u0637\u06cc \u0648 \u063a\u06cc\u0631\u062e\u0637\u06cc \u060c \u0627\u0632 \u062c\u0645\u0644\u0647 PDE \u0647\u0627\u06cc \u0628\u06cc\u0636\u0648\u06cc \u0648 \u0645\u0634\u06a9\u0644\u0627\u062a \u0627\u0631\u0632\u0634 \u0645\u0631\u0632\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645.\u0645\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u06a9\u0647 \u0637\u0631\u062d \u067e\u06cc\u0634\u0646\u0647\u0627\u062f\u06cc \u0628\u0647 \u062f\u0642\u062a \u062a\u0642\u0631\u06cc\u0628 \u0639\u062f\u062f\u06cc \u0642\u0627\u0628\u0644 \u062a\u0648\u062c\u0647\u06cc \u062f\u0631 \u0647\u0631 \u062f\u0648 \u062f\u0627\u0645\u0646\u0647 \u0648 \u0645\u0631\u0632 \u062f\u0633\u062a \u0645\u06cc \u06cc\u0627\u0628\u062f.\u0631\u0648\u0634 \u067e\u06cc\u0634\u0646\u0647\u0627\u062f\u06cc \u0628\u06cc\u0646\u0634\u06cc \u062f\u0631 \u0645\u0648\u0631\u062f \u0627\u0631\u062a\u0628\u0627\u0637 \u0628\u06cc\u0646 \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc \u0648 \u0631\u0648\u0634\u0647\u0627\u06cc \u0639\u062f\u062f\u06cc \u06a9\u0644\u0627\u0633\u06cc\u06a9 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u062f \u060c \u0648 \u0645\u0627 \u0627\u0638\u0647\u0627\u0631 \u062f\u0627\u0631\u06cc\u0645 \u06a9\u0647 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u062f\u0631 \u0632\u0645\u06cc\u0646\u0647 \u0647\u0627\u06cc\u06cc \u0645\u0627\u0646\u0646\u062f \u06a9\u0645\u06cc\u062a \u0639\u062f\u0645 \u0627\u0637\u0645\u06cc\u0646\u0627\u0646 (UQ) \u0648 \u0647\u0648\u0634 \u0645\u0635\u0646\u0648\u0639\u06cc \u0642\u0627\u0628\u0644 \u062a\u0648\u0636\u06cc\u062d (XAI) \u06a9\u0627\u0631\u0628\u0631\u062f\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f.\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u060c \u0645\u0627 \u0645\u0639\u062a\u0642\u062f\u06cc\u0645 \u06a9\u0647 \u0627\u06cc\u0646 \u0627\u0645\u0631 \u0628\u0627\u0639\u062b \u067e\u06cc\u0634\u0631\u0641\u062a \u0628\u06cc\u0634\u062a\u0631 \u062f\u0631 \u062a\u0642\u0627\u0637\u0639 \u0628\u06cc\u0646 \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 \u0639\u0644\u0645\u06cc \u0648 \u062a\u062c\u0632\u06cc\u0647 \u0648 \u062a\u062d\u0644\u06cc\u0644 \u0639\u062f\u062f\u06cc \u062e\u0648\u0627\u0647\u062f \u0634\u062f.<\/p>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
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