{"id":62937,"date":"2025-04-13T14:09:36","date_gmt":"2025-04-13T14:09:36","guid":{"rendered":""},"modified":"2025-04-13T14:09:36","modified_gmt":"2025-04-13T14:09:36","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%87-62937","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%87-62937\/","title":{"rendered":"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u062a\u0635\u0627\u062f\u0641\u06cc \u0633\u0631\u06cc\u0639\u062a\u0631 \u0628\u0627 \u062a\u0627\u062e\u06cc\u0631\u0647\u0627\u06cc \u062f\u0644\u062e\u0648\u0627\u0647 \u0627\u0632 \u0637\u0631\u06cc\u0642 \u0645\u06cc\u0646\u06cc \u062f\u0633\u062a\u0647 \u0628\u0646\u062f\u06cc \u0646\u0627\u0647\u0645\u0632\u0645\u0627\u0646"},"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>\nWasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions<\/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\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u062a\u0635\u0627\u062f\u0641\u06cc \u0633\u0631\u06cc\u0639\u062a\u0631 \u0628\u0627 \u062a\u0627\u062e\u06cc\u0631\u0647\u0627\u06cc \u062f\u0644\u062e\u0648\u0627\u0647 \u0627\u0632 \u0637\u0631\u06cc\u0642 \u0645\u06cc\u0646\u06cc \u062f\u0633\u062a\u0647 \u0628\u0646\u062f\u06cc \u0646\u0627\u0647\u0645\u0632\u0645\u0627\u0646<\/td>\n<\/tr>\n
\u0646\u0648\u06cc\u0633\u0646\u062f\u06af\u0627\u0646 <\/td>\nRichard Duong, Viktor Stein, Robert Beinert, Johannes Hertrich, Gabriele Steidl<\/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>\n44<\/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>\nAnalysis of PDEs,Machine Learning,\u062a\u062c\u0632\u06cc\u0647 \u0648 \u062a\u062d\u0644\u06cc\u0644 PDE , \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 ,<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a <\/td>\nSubmitted 14 August, 2024; originally announced August 2024. , Comments: 44 pages, 21 figures, comments welcome! , MSC Class: 49Q22 (Primary); 46N10; 35B99 (Secondary)<\/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 \u062f\u0631 14 \u0627\u0648\u062a 2024 \u061b\u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0627\u0648\u062a 2024 \u0627\u0639\u0644\u0627\u0645 \u0634\u062f \u060c \u0646\u0638\u0631\u0627\u062a: 44 \u0635\u0641\u062d\u0647 \u060c 21 \u0634\u06a9\u0644 \u060c \u0646\u0638\u0631\u0627\u062a \u062e\u0648\u0634 \u0622\u0645\u062f\u06cc\u062f!\u060c \u06a9\u0644\u0627\u0633 MSC: 49Q22 (\u0627\u0648\u0644\u06cc\u0647) \u061b46N10 \u061b35B99 (\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>
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\n NASA ADS<\/a>
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\n Google Scholar<\/a>
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\n Semantic Scholar<\/a>
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\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

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\\mathcal F_\u03bd:= \\text{MMD}_K^2(\\cdot, \u03bd)$ towards given target measures $\u03bd$ on the real line, where we focus on the negative distance kernel $K(x,y) := -|x-y|$. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone $\\mathcal C(0,1) \\subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\\mathcal F_\u03bd$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\u03bd$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $\\mathcal C(0,1)$. For certain $\\mathcal F_\u03bd$-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme and demonstrate differences to the explicit Euler scheme, which is easier to compute, but comes with limited convergence guarantees.<\/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

\u0645\u0627 \u062a\u0648\u0636\u06cc\u062d\u0627\u062a \u06a9\u0627\u0645\u0644\u06cc \u0627\u0632 \u062c\u0631\u06cc\u0627\u0646 \u0634\u06cc\u0628 Wasserstein \u0628\u0627 \u062d\u062f\u0627\u06a9\u062b\u0631 \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0627\u062e\u062a\u0644\u0627\u0641 (MMD) \u0639\u0645\u0644\u06a9\u0631\u062f\u0647\u0627\u06cc $ \\ Mathcal f_\u03bd: = \\ text {mmd} _k^2 (\\ cdot \u060c \u03bd) $ \u0628\u0647 \u0633\u0645\u062a \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u0647\u062f\u0641 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 $ \u03bd $ \u062f\u0631 \u062e\u0637 \u0648\u0627\u0642\u0639\u06cc \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u06cc\u0645.\u062c\u0627\u06cc\u06cc \u06a9\u0647 \u0645\u0627 \u0631\u0648\u06cc \u0647\u0633\u062a\u0647 \u0641\u0627\u0635\u0644\u0647 \u0645\u0646\u0641\u06cc $ k (x \u060c y) \u062a\u0645\u0631\u06a9\u0632 \u0645\u06cc \u06a9\u0646\u06cc\u0645: = -| x -y | $.\u062f\u0631 \u06cc\u06a9 \u0628\u0639\u062f \u060c \u0641\u0636\u0627\u06cc Wasserstein-2 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0628\u0647 \u0635\u0648\u0631\u062a \u0627\u06cc\u0632\u0648\u0645\u062a\u0631\u06cc\u06a9 \u062f\u0631 \u0645\u062e\u0631\u0648\u0637 $ \\ Mathcal C (0\u060c1) \\ \u0632\u06cc\u0631 \u0645\u062c\u0645\u0648\u0639\u0647 L_2 (0\u060c1) \u0627\u0632 \u062a\u0648\u0627\u0628\u0639 \u06a9\u0648\u0627\u0646\u062a\u06cc\u0644 \u06a9\u0647 \u0645\u0646\u062c\u0631 \u0628\u0647 \u062a\u0648\u0635\u06cc\u0641 \u062c\u0631\u06cc\u0627\u0646 \u0634\u06cc\u0628 Wasserstein \u0627\u0632 \u0637\u0631\u06cc\u0642 \u0645\u062d\u0644\u0648\u0644 \u06cc\u06a9 \u0645\u062d\u0644\u0648\u0644 \u06cc\u06a9 \u0645\u062d\u0644\u0648\u0644 \u0627\u0633\u062a \u060c \u062a\u0639\u0628\u06cc\u0647 \u06a9\u0646\u06cc\u062f.\u0645\u0634\u06a9\u0644 cauchy \u0645\u0631\u062a\u0628\u0637 \u0628\u0627 $ l_2 (0\u060c1) $.\u0628\u0631\u0627\u0633\u0627\u0633 \u0633\u0627\u062e\u062a \u06cc\u06a9 \u0647\u0645\u062a\u0627\u06cc \u0645\u0646\u0627\u0633\u0628 \u0627\u0632 $ \\ Mathcal f_\u03bd $ \u062f\u0631 $ l_2 (0\u060c1) $ \u0648 \u062a\u0642\u0633\u06cc\u0645 \u0622\u0646 \u060c \u0645\u0627 \u06cc\u06a9 \u0631\u0627\u0647 \u062d\u0644 \u0628\u0631\u0627\u06cc \u0645\u0634\u06a9\u0644 cauchy \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u06cc\u0645.\u0628\u0631\u0627\u06cc \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u0647\u062f\u0641 \u06af\u0633\u0633\u062a\u0647 $ \u03bd $ \u060c \u0627\u06cc\u0646 \u0645\u0646\u062c\u0631 \u0628\u0647 \u0641\u0631\u0645\u0648\u0644 \u0631\u0627\u0647 \u062d\u0644 \u062e\u0637\u06cc \u067e\u0631\u0627\u06a9\u0646\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.\u0645\u0627 \u0648\u06cc\u0698\u06af\u06cc \u0647\u0627\u06cc \u062b\u0627\u0628\u062a \u0648 \u0635\u0627\u0641 \u06a9\u0646\u0646\u062f\u0647 \u062c\u0631\u06cc\u0627\u0646 \u0631\u0627 \u062f\u0631 \u0632\u06cc\u0631 \u0645\u062c\u0645\u0648\u0639\u0647 \u0647\u0627\u06cc $ \\ Mathcal C (0\u060c1) $ \u062b\u0627\u0628\u062a \u0645\u06cc \u06a9\u0646\u06cc\u0645.\u0628\u0631\u0627\u06cc \u0628\u0631\u062e\u06cc \u0627\u0632 $ \\ Mathcal f_\u03bd $-\u062c\u0631\u06cc\u0627\u0646 \u0627\u06cc\u0646 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u0627\u0648\u0644\u06cc\u0647 \u0646\u0642\u0637\u0647 \u0641\u0648\u0631\u0627\u064b \u06a9\u0627\u0645\u0644\u0627\u064b \u0645\u062f\u0627\u0648\u0645 \u0645\u06cc \u0634\u0648\u0646\u062f \u0648 \u0628\u0627 \u06af\u0630\u0634\u062a \u0632\u0645\u0627\u0646 \u0686\u0646\u06cc\u0646 \u0645\u06cc \u0645\u0627\u0646\u0646\u062f.\u0633\u0631\u0627\u0646\u062c\u0627\u0645 \u060c \u0645\u0627 \u0631\u0641\u062a\u0627\u0631 \u062c\u0631\u06cc\u0627\u0646 \u0631\u0627 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0646\u0645\u0648\u0646\u0647 \u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0639\u062f\u062f\u06cc \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u06cc\u06a9 \u0637\u0631\u062d \u0627\u0648\u06cc\u0644\u0631 \u0636\u0645\u0646\u06cc \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u0648 \u062a\u0641\u0627\u0648\u062a \u0647\u0627\u06cc\u06cc \u0631\u0627 \u0628\u0627 \u0637\u0631\u062d \u0635\u0631\u06cc\u062d \u0627\u0648\u06cc\u0644\u0631 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u060c \u06a9\u0647 \u0645\u062d\u0627\u0633\u0628\u0647 \u0622\u0646 \u0622\u0633\u0627\u0646 \u062a\u0631 \u0627\u0633\u062a \u060c \u0627\u0645\u0627 \u0628\u0627 \u0636\u0645\u0627\u0646\u062a \u0647\u0627\u06cc \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0645\u062d\u062f\u0648\u062f \u0647\u0645\u0631\u0627\u0647 \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
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