{"id":62042,"date":"2025-04-07T07:57:07","date_gmt":"2025-04-07T07:57:07","guid":{"rendered":""},"modified":"2025-04-07T07:57:07","modified_gmt":"2025-04-07T07:57:07","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%87-62042","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%87-62042\/","title":{"rendered":"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u0628\u0648\u062f \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0622\u0645\u0648\u0632\u0634 Pinn Bayesian \u062f\u0631 \u062d\u0644 PDE \u0647\u0627\u06cc \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 \u0628\u0627 \u0633\u0631 \u0648 \u0635\u062f\u0627"},"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>\nImprovement of Bayesian PINN Training Convergence in Solving Multi-scale PDEs with Noise<\/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 \u0628\u0647\u0628\u0648\u062f \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0622\u0645\u0648\u0632\u0634 Pinn Bayesian \u062f\u0631 \u062d\u0644 PDE \u0647\u0627\u06cc \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 \u0628\u0627 \u0633\u0631 \u0648 \u0635\u062f\u0627<\/td>\n<\/tr>\n
\u0646\u0648\u06cc\u0633\u0646\u062f\u06af\u0627\u0646 <\/td>\nYilong Hou, Xi’an Li, Jinran Wu<\/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>\n9<\/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>\nMachine Learning,Numerical Analysis,\u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 , \u062a\u062d\u0644\u06cc\u0644 \u0639\u062f\u062f\u06cc ,<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a <\/td>\nSubmitted 17 August, 2024; originally announced August 2024.<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u0627\u0631\u0633\u0627\u0644 \u0634\u062f\u0647 17 \u0627\u0648\u062a 2024 \u061b\u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0627\u0648\u062a 2024 \u0627\u0639\u0644\u0627\u0645 \u0634\u062f.<\/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

Bayesian Physics Informed Neural Networks (BPINN) have received considerable attention for inferring differential equations’ system states and physical parameters according to noisy observations. However, in practice, Hamiltonian Monte Carlo (HMC) used to estimate the internal parameters of BPINN often encounters troubles, including poor performance and awful convergence for a given step size used to adjust the momentum of those parameters. To improve the efficacy of HMC convergence for the BPINN method and extend its application scope to multi-scale partial differential equations (PDE), we developed a robust multi-scale Bayesian PINN (dubbed MBPINN) method by integrating multi-scale deep neural networks (MscaleDNN) and Bayesian inference. In this newly proposed MBPINN method, we reframe HMC with Stochastic Gradient Descent (SGD) to ensure the most “likely” estimation is always provided, and we configure its solver as a Fourier feature mapping-induced MscaleDNN. The MBPINN method offers several key advantages: (1) it is more robust than HMC, (2) it incurs less computational cost than HMC, and (3) it is more flexible for complex problems. We demonstrate the applicability and performance of the proposed method through general Poisson and multi-scale elliptic problems in one- to three-dimensional spaces. Our findings indicate that the proposed method can avoid HMC failures and provide valid results. Additionally, our method can handle complex PDE and produce comparable results for general PDE. These findings suggest that our proposed approach has excellent potential for physics-informed machine learning for parameter estimation and solution recovery in the case of ill-posed problems.<\/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

\u0641\u06cc\u0632\u06cc\u06a9 \u0628\u06cc\u0632\u06cc \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc (BPINN) \u0628\u0631\u0627\u06cc \u0627\u0633\u062a\u0646\u0628\u0627\u0637 \u062d\u0627\u0644\u062a \u0647\u0627\u06cc \u0633\u06cc\u0633\u062a\u0645 \u0645\u0639\u0627\u062f\u0644\u0627\u062a \u062f\u06cc\u0641\u0631\u0627\u0646\u0633\u06cc\u0644 \u0648 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0641\u06cc\u0632\u06cc\u06a9\u06cc \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0645\u0634\u0627\u0647\u062f\u0627\u062a \u067e\u0631 \u0633\u0631 \u0648 \u0635\u062f\u0627 \u060c \u062a\u0648\u062c\u0647 \u0642\u0627\u0628\u0644 \u062a\u0648\u062c\u0647\u06cc \u0631\u0627 \u0628\u0647 \u062e\u0648\u062f \u062c\u0644\u0628 \u06a9\u0631\u062f\u0647 \u0627\u0646\u062f.\u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644 \u060c \u062f\u0631 \u0639\u0645\u0644 \u060c Hamiltonian Monte Carlo (HMC) \u0628\u0631\u0627\u06cc \u0628\u0631\u0622\u0648\u0631\u062f \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u062f\u0627\u062e\u0644\u06cc BPINN \u0627\u063a\u0644\u0628 \u0628\u0627 \u0645\u0634\u06a9\u0644\u0627\u062a \u0631\u0648\u0628\u0631\u0648 \u0645\u06cc \u0634\u0648\u062f \u060c \u0627\u0632 \u062c\u0645\u0644\u0647 \u0639\u0645\u0644\u06a9\u0631\u062f \u0636\u0639\u06cc\u0641 \u0648 \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0627\u0641\u062a\u0636\u0627\u062d \u0628\u0631\u0627\u06cc \u0627\u0646\u062f\u0627\u0632\u0647 \u0645\u0631\u062d\u0644\u0647 \u0645\u0639\u06cc\u0646 \u06a9\u0647 \u0628\u0631\u0627\u06cc \u062a\u0646\u0638\u06cc\u0645 \u062d\u0631\u06a9\u062a \u0622\u0646 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.\u0628\u0631\u0627\u06cc \u0628\u0647\u0628\u0648\u062f \u0627\u062b\u0631\u0628\u062e\u0634\u06cc \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc HMC \u0628\u0631\u0627\u06cc \u0631\u0648\u0634 BPINN \u0648 \u06af\u0633\u062a\u0631\u0634 \u062f\u0627\u0645\u0646\u0647 \u06a9\u0627\u0631\u0628\u0631\u062f \u0622\u0646 \u0628\u0647 \u0645\u0639\u0627\u062f\u0644\u0627\u062a \u062f\u06cc\u0641\u0631\u0627\u0646\u0633\u06cc\u0644 \u062c\u0632\u0626\u06cc \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 (PDE) \u060c \u0645\u0627 \u0628\u0627 \u0627\u062f\u063a\u0627\u0645 \u0634\u0628\u06a9\u0647 \u0647\u0627\u06cc \u0639\u0635\u0628\u06cc \u0639\u0645\u06cc\u0642 \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 (Dubbed MBPINN) \u06cc\u06a9 \u0631\u0648\u0634 \u0642\u0648\u06cc \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 \u0628\u06cc\u0632\u06cc (Dubbed MBPINN) \u062a\u0647\u06cc\u0647 \u06a9\u0631\u062f\u06cc\u0645 (\u0645\u0627 \u06cc\u06a9 \u0631\u0648\u0634 \u0642\u0648\u06cc \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 Bayesian (Dubbed MBPINN) \u0631\u0627 \u062a\u0647\u06cc\u0647 \u06a9\u0631\u062f\u06cc\u0645.MSCALEDNN) \u0648 \u0627\u0633\u062a\u0646\u0628\u0627\u0637 \u0628\u06cc\u0632\u06cc.\u062f\u0631 \u0627\u06cc\u0646 \u0631\u0648\u0634 \u062a\u0627\u0632\u0647 \u067e\u06cc\u0634\u0646\u0647\u0627\u062f\u06cc MBPINN \u060c \u0645\u0627 HMC \u0631\u0627 \u0628\u0627 \u0646\u0632\u0648\u0644 \u0634\u06cc\u0628 \u062a\u0635\u0627\u062f\u0641\u06cc (SGD) \u062a\u063a\u06cc\u06cc\u0631 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u062a\u0627 \u0627\u0637\u0645\u06cc\u0646\u0627\u0646 \u062d\u0627\u0635\u0644 \u0634\u0648\u062f \u06a9\u0647 “\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u064b” \u062a\u062e\u0645\u06cc\u0646 \u0647\u0645\u06cc\u0634\u0647 \u0627\u0631\u0627\u0626\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a \u060c \u0648 \u0645\u0627 \u062d\u0644 \u06a9\u0646\u0646\u062f\u0647 \u0622\u0646 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 MSSCALEDNN \u0646\u0627\u0634\u06cc \u0627\u0632 \u0646\u0642\u0634\u0647 \u0628\u0631\u062f\u0627\u0631\u06cc \u0627\u0632 \u0648\u06cc\u0698\u06af\u06cc \u0647\u0627\u06cc \u0641\u0648\u0631\u06cc\u0647 \u067e\u06cc\u06a9\u0631\u0628\u0646\u062f\u06cc \u0645\u06cc \u06a9\u0646\u06cc\u0645.\u0631\u0648\u0634 MBPINN \u0686\u0646\u062f\u06cc\u0646 \u0645\u0632\u06cc\u062a \u06a9\u0644\u06cc\u062f\u06cc \u0631\u0627 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u062f: (1) \u0627\u0632 HMC \u0642\u0648\u06cc \u062a\u0631 \u0627\u0633\u062a \u060c (2) \u0647\u0632\u06cc\u0646\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc \u06a9\u0645\u062a\u0631\u06cc \u0646\u0633\u0628\u062a \u0628\u0647 HMC \u0645\u062a\u062d\u0645\u0644 \u0645\u06cc \u0634\u0648\u062f \u060c \u0648 (3) \u0628\u0631\u0627\u06cc \u0645\u0634\u06a9\u0644\u0627\u062a \u067e\u06cc\u0686\u06cc\u062f\u0647 \u0627\u0646\u0639\u0637\u0627\u0641 \u067e\u0630\u06cc\u0631 \u062a\u0631 \u0627\u0633\u062a.\u0645\u0627 \u06a9\u0627\u0631\u0628\u0631\u062f \u0648 \u0639\u0645\u0644\u06a9\u0631\u062f \u0631\u0648\u0634 \u067e\u06cc\u0634\u0646\u0647\u0627\u062f\u06cc \u0631\u0627 \u0627\u0632 \u0637\u0631\u06cc\u0642 \u0645\u0634\u06a9\u0644\u0627\u062a \u0628\u06cc\u0636\u0648\u06cc \u0639\u0645\u0648\u0645\u06cc \u067e\u0648\u0627\u0633\u0648\u0646 \u0648 \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 \u062f\u0631 \u0641\u0636\u0627\u0647\u0627\u06cc \u06cc\u06a9 \u062a\u0627 \u0633\u0647 \u0628\u0639\u062f\u06cc \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u06cc\u0645.\u06cc\u0627\u0641\u062a\u0647 \u0647\u0627\u06cc \u0645\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0631\u0648\u0634 \u067e\u06cc\u0634\u0646\u0647\u0627\u062f\u06cc \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0627\u0632 \u062e\u0631\u0627\u0628\u06cc HMC \u062c\u0644\u0648\u06af\u06cc\u0631\u06cc \u06a9\u0646\u062f \u0648 \u0646\u062a\u0627\u06cc\u062c \u0645\u0639\u062a\u0628\u0631\u06cc \u0627\u0631\u0627\u0626\u0647 \u062f\u0647\u062f.\u0639\u0644\u0627\u0648\u0647 \u0628\u0631 \u0627\u06cc\u0646 \u060c \u0631\u0648\u0634 \u0645\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f PDE \u067e\u06cc\u0686\u06cc\u062f\u0647 \u0631\u0627 \u0627\u062f\u0627\u0631\u0647 \u06a9\u0646\u062f \u0648 \u0646\u062a\u0627\u06cc\u062c \u0642\u0627\u0628\u0644 \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc \u0631\u0627 \u0628\u0631\u0627\u06cc PDE \u0639\u0645\u0648\u0645\u06cc \u062a\u0648\u0644\u06cc\u062f \u06a9\u0646\u062f.\u0627\u06cc\u0646 \u06cc\u0627\u0641\u062a\u0647 \u0647\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0631\u0648\u06cc\u06a9\u0631\u062f \u067e\u06cc\u0634\u0646\u0647\u0627\u062f\u06cc \u0645\u0627 \u0627\u0632 \u067e\u062a\u0627\u0646\u0633\u06cc\u0644 \u0639\u0627\u0644\u06cc \u0628\u0631\u0627\u06cc \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 \u0622\u06af\u0627\u0647\u06cc \u0627\u0632 \u0641\u06cc\u0632\u06cc\u06a9 \u0628\u0631\u0627\u06cc \u0628\u0631\u0622\u0648\u0631\u062f \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627 \u0648 \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u0631\u0627\u0647 \u062d\u0644 \u062f\u0631 \u0635\u0648\u0631\u062a \u0628\u0631\u0648\u0632 \u0645\u0634\u06a9\u0644\u0627\u062a \u0628\u062f\u0628\u062e\u062a\u06cc \u0628\u0631\u062e\u0648\u0631\u062f\u0627\u0631 \u0627\u0633\u062a.<\/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
\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","protected":false},"excerpt":{"rendered":"

\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc Improvement of Bayesian PINN Training Convergence in Solving Multi-scale PDEs with Noise \u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc […]<\/p>\n","protected":false},"featured_media":27,"comment_status":"open","ping_status":"open","template":"","meta":{"pmpro_default_level":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}}},"product_cat":[21],"product_tag":[],"class_list":{"0":"post-62042","1":"product","2":"type-product","3":"status-publish","4":"has-post-thumbnail","6":"product_cat-21","7":"pmpro-has-access","8":"desktop-align-left","9":"tablet-align-left","10":"mobile-align-left","12":"first","13":"instock","14":"downloadable","15":"shipping-taxable","16":"purchasable","17":"product-type-simple"},"yoast_head":"\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u0628\u0648\u062f \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0622\u0645\u0648\u0632\u0634 Pinn Bayesian \u062f\u0631 \u062d\u0644 PDE \u0647\u0627\u06cc \u0686\u0646\u062f \u0645\u0642\u06cc\u0627\u0633 \u0628\u0627 \u0633\u0631 \u0648 \u0635\u062f\u0627 - 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