{"id":6880,"date":"2025-09-16T07:03:45","date_gmt":"2025-09-16T07:03:45","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/09\/16\/2509-11070\/"},"modified":"2025-09-16T07:03:45","modified_gmt":"2025-09-16T07:03:45","slug":"2509-11070","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/09\/16\/2509-11070\/","title":{"rendered":"Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning"},"content":{"rendered":"<p>    Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n    <!-- no image --><br \/>\n \t<BR><br \/>\n<BR><\/BR><\/p>\n<div>arXiv:2509.11070v1 Announce Type: new<br \/>\nAbstract: We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which admit discrete spectral decompositions, and (ii) diagonal kernels of the form $K(x,x&#8217;)=k(x,x&#8217;)T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical $K=kI$ paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish dimension-free polynomial convergence rates, demonstrating that nonlinear operator learning can overcome the curse of dimensionality. The use of general operator-valued kernels further allows us to derive rates for intrinsically nonlinear operator learning, going beyond the linear-type behavior inherent in diagonal constructions of $K=kI$. Importantly, this framework accommodates a wide range of operator learning tasks, ranging from integral operators such as Fredholm operators to architectures based on encoder-decoder representations. Moreover, we validate its effectiveness through numerical experiments on the two-dimensional Navier-Stokes equations.<\/div>\n<p> \t<BR><br \/>\n <BR><\/BR><br \/>\n    Jia-Qi Yang, Lei Shi<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n<a href=\"https:\/\/arxiv.org\/abs\/2509.11070\">Go to original source<\/a><br \/>\n \t<BR><br \/>\n <BR><\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning arXiv:2509.11070v1 Announce Type: new Abstract: We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which admit discrete spectral decompositions, and (ii) diagonal kernels of the form $K(x,x&#8217;)=k(x,x&#8217;)T$, where $k$ [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,450,1172,451,190,112,191],"tags":[924,199,1529],"class_list":["post-6880","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-cs-na","category-math-fa","category-math-na","category-math-st","category-stat-ml","category-stat-th","tag-framework","tag-learning","tag-operator"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6880"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=6880"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6880\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=6880"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=6880"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=6880"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}