{"id":2128,"date":"2026-04-21T01:51:50","date_gmt":"2026-04-21T01:51:50","guid":{"rendered":"https:\/\/aijaps.us\/?p=2128"},"modified":"2026-04-21T01:51:50","modified_gmt":"2026-04-21T01:51:50","slug":"the-approximation-properties-of-neural-networks-a-review","status":"publish","type":"post","link":"https:\/\/aijaps.us\/?p=2128","title":{"rendered":"The approximation properties of neural networks- A Review"},"content":{"rendered":"

Eman Jawad<\/strong><\/h3>\n

Assistant. Lecture\/ AL \u2013Furat Al-Awsat Technical University\/Iraq<\/strong><\/h3>\n

eman.naji@atu.edu.iq<\/strong><\/a><\/h3>\n

009647802428220<\/u><\/strong><\/h3>\nEman Jawad<\/a>\n","protected":false},"excerpt":{"rendered":"

Eman Jawad Assistant. Lecture\/ AL \u2013Furat Al-Awsat Technical University\/Iraq eman.naji@atu.edu.iq 009647802428220<\/p>\n","protected":false},"author":1,"featured_media":2130,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,45],"tags":[],"class_list":["post-2128","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","category-45"],"yoast_head":"\nThe approximation properties of neural networks- A Review - aijaps<\/title>\n<meta name=\"description\" content=\"Abstract: We review the study of the general theoretical framework to the in-depth analysis of the mechanisms of action of neural networks from researchers, specifically those based on the corrected linear module (ReLU). Neural networks are examined not only as software tools, but as a form of "nonlinear ramified approximation". Using complex measurement tools such as the Kolmogorov width and metric entropy, they proved that neural networks have a unique ability to "fill the void" in Banach spaces, which gives them a theoretical superiority in the efficiency of representation compared to traditional linear methods that are unable to keep up with the complexity of high-dimensional data.As well as characterizing the mathematical outputs of ReLU networks, where the researchers proved that they produce continuous segmented linear functions (CPWL). This characterization is the cornerstone of understanding how simple calculations within classes turn into models capable of simulating any continuous function. In addition to addressing the capabilities of shallow networks (single layer), stressing that although it has the property of "mass approximation", it suffers from the "Curse of dimensions" and requires a huge number of neurons to achieve acceptable accuracy in complex tasks.The researchers reached the peak of mathematical analysis in their research papers for the "power of depth" by demonstrating that increasing the number of layers allows the network to simulate complex arithmetic operations and polynomials with amazing efficiency. This depth enables the network to approximate difficult function classes such as Sobolev and Besov spaces with optimal approximation rates that are superior to conventional wavelets and polynomials. It also highlights the ability of deep networks to exploit the "self-similarity" of functions, which explains their impressive success in processing repetitive images and patterns.Others raise the issue of "stability". They explained that there is an inevitable trade-off between the network's ability to approximate and the stability of the algorithms used to train it. The more the network is able to fill the void and super-approximate, the greater the likelihood that the results will be unstable when minor changes occur in the entered data. This analysis puts an end to the ideal expectations, stressing that success in approximation does not necessarily mean easy access to the optimal solution in numerically stable ways, which opens the door to the need to balance the depth of the network and its trainability.Keywords : ReLU, Deep Neural networks, approximation\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/aijaps.us\/?p=2128\" \/>\n<meta property=\"og:locale\" content=\"ar_AR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The approximation properties of neural networks- A Review - aijaps\" \/>\n<meta property=\"og:description\" content=\"Abstract: We review the study of the general theoretical framework to the in-depth analysis of the mechanisms of action of neural networks from researchers, specifically those based on the corrected linear module (ReLU). Neural networks are examined not only as software tools, but as a form of "nonlinear ramified approximation". Using complex measurement tools such as the Kolmogorov width and metric entropy, they proved that neural networks have a unique ability to "fill the void" in Banach spaces, which gives them a theoretical superiority in the efficiency of representation compared to traditional linear methods that are unable to keep up with the complexity of high-dimensional data.As well as characterizing the mathematical outputs of ReLU networks, where the researchers proved that they produce continuous segmented linear functions (CPWL). This characterization is the cornerstone of understanding how simple calculations within classes turn into models capable of simulating any continuous function. In addition to addressing the capabilities of shallow networks (single layer), stressing that although it has the property of "mass approximation", it suffers from the "Curse of dimensions" and requires a huge number of neurons to achieve acceptable accuracy in complex tasks.The researchers reached the peak of mathematical analysis in their research papers for the "power of depth" by demonstrating that increasing the number of layers allows the network to simulate complex arithmetic operations and polynomials with amazing efficiency. This depth enables the network to approximate difficult function classes such as Sobolev and Besov spaces with optimal approximation rates that are superior to conventional wavelets and polynomials. It also highlights the ability of deep networks to exploit the "self-similarity" of functions, which explains their impressive success in processing repetitive images and patterns.Others raise the issue of "stability". They explained that there is an inevitable trade-off between the network's ability to approximate and the stability of the algorithms used to train it. The more the network is able to fill the void and super-approximate, the greater the likelihood that the results will be unstable when minor changes occur in the entered data. This analysis puts an end to the ideal expectations, stressing that success in approximation does not necessarily mean easy access to the optimal solution in numerically stable ways, which opens the door to the need to balance the depth of the network and its trainability.Keywords : ReLU, Deep Neural networks, approximation\" \/>\n<meta property=\"og:url\" content=\"https:\/\/aijaps.us\/?p=2128\" \/>\n<meta property=\"og:site_name\" content=\"aijaps\" \/>\n<meta property=\"article:published_time\" content=\"2026-04-21T01:51:50+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/aijaps.us\/wp-content\/uploads\/2026\/04\/6.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1000\" \/>\n\t<meta property=\"og:image:height\" content=\"429\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"aijaps.us\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u0643\u064f\u062a\u0628 \u0628\u0648\u0627\u0633\u0637\u0629\" \/>\n\t<meta name=\"twitter:data1\" content=\"aijaps.us\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u0648\u0642\u062a \u0627\u0644\u0642\u0631\u0627\u0621\u0629 \u0627\u0644\u0645\u064f\u0642\u062f\u0651\u0631\" \/>\n\t<meta name=\"twitter:data2\" content=\"\u062f\u0642\u064a\u0642\u0629 \u0648\u0627\u062d\u062f\u0629\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128\"},\"author\":{\"name\":\"aijaps.us\",\"@id\":\"https:\\\/\\\/aijaps.us\\\/#\\\/schema\\\/person\\\/e0e97926a8d995bc1e65a0f9ac22f991\"},\"headline\":\"The approximation properties of neural networks- A Review\",\"datePublished\":\"2026-04-21T01:51:50+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128\"},\"wordCount\":35,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/aijaps.us\\\/wp-content\\\/uploads\\\/2026\\\/04\\\/6.png\",\"articleSection\":[\"Uncategorized\",\"\u0627\u0635\u062f\u0627\u0631\u0627\u062a \u0627\u0644\u0628\u062d\u0648\u062b\"],\"inLanguage\":\"ar\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/aijaps.us\\\/?p=2128#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128\",\"url\":\"https:\\\/\\\/aijaps.us\\\/?p=2128\",\"name\":\"The approximation properties of neural networks- A Review - aijaps\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/aijaps.us\\\/?p=2128#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/aijaps.us\\\/wp-content\\\/uploads\\\/2026\\\/04\\\/6.png\",\"datePublished\":\"2026-04-21T01:51:50+00:00\",\"description\":\"Abstract: We review the study of the general theoretical framework to the in-depth analysis of the mechanisms of action of neural networks from researchers, specifically those based on the corrected linear module (ReLU). 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This analysis puts an end to the ideal expectations, stressing that success in approximation does not necessarily mean easy access to the optimal solution in numerically stable ways, which opens the door to the need to balance the depth of the network and its trainability.Keywords : ReLU, Deep Neural networks, 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