{"id":24371,"date":"2026-10-17T16:19:36","date_gmt":"2026-10-17T14:19:36","guid":{"rendered":"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/approximation-model\/"},"modified":"2026-10-17T16:27:47","modified_gmt":"2026-10-17T14:27:47","slug":"approximation-model","status":"publish","type":"learning","link":"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/approximation-model\/","title":{"rendered":"Model types: Approximation (or function regression)"},"content":{"rendered":"<style id=\"nd-tutorial-series-css\">.ndts{max-width:1000px;margin:0 auto 26px;padding:18px 20px;border:1px solid #d8e2ef;border-radius:14px;background:#f7fafc;font-family:\"Outfit\",\"Roboto\",Arial,sans-serif}.ndts-meta{color:#2d799f;font-size:14px;font-weight:800;letter-spacing:.06em;text-transform:uppercase}.ndts-intro{margin:8px 0 14px;color:#405361;line-height:1.5}.ndts-progress{display:grid;grid-template-columns:repeat(7,minmax(0,1fr));gap:6px;margin:0;padding:0;list-style:none}.ndts-progress a{display:flex;align-items:center;justify-content:center;min-height:38px;padding:6px;border:1px solid #cbd9e2;border-radius:7px;background:#fff;color:#285c78!important;text-decoration:none!important;font-size:13px;font-weight:700;text-align:center}.ndts-progress a[aria-current=\"step\"]{border-color:#2d799f;background:#2d799f;color:#fff!important}.ndts-controls{display:flex;justify-content:space-between;align-items:center;gap:10px;margin-top:14px;flex-wrap:wrap}.ndts-controls a{color:#1261a0!important;font-weight:700;text-decoration:none!important}.ndts-bottom{max-width:1000px;margin:26px auto;padding:18px 20px;border-top:1px solid #d8e2ef;font-family:\"Outfit\",\"Roboto\",Arial,sans-serif}.ndts-finish{margin:0 0 18px;padding:22px;border:1px solid #d8e2ef;border-radius:12px;background:#f6f9fc}.ndts-btn{display:inline-block;margin:6px 8px 0 0;padding:11px 15px;border-radius:7px;background:#1261a0;color:#fff!important;text-decoration:none!important;font-weight:700}@media(max-width:760px){.ndts-progress{grid-template-columns:repeat(2,minmax(0,1fr))}}<\/style><nav class=\"ndts\" aria-label=\"Machine learning tutorial progress\"><div class=\"ndts-meta\">Machine learning tutorial \u00b7 Chapter 1 of 7<\/div><p class=\"ndts-intro\">This page is part of Chapter 1, <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-networks-applications\/\">Model types<\/a>. Follow the sequence or use the progress links to open a specific topic. <a href=\"https:\/\/www.neuraldesigner.com\/learning\/machine-learning-tutorial\/\">Tutorial overview<\/a>.<\/p><ol class=\"ndts-progress\"><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-networks-applications\/\" aria-current=\"step\" data-tutorial-nav=\"chapter-1\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'1',link_url:this.href});}\">1. Model types<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\/\" data-tutorial-nav=\"chapter-2\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'2',link_url:this.href});}\">2. Data set<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\/\" data-tutorial-nav=\"chapter-3\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'3',link_url:this.href});}\">3. Neural network<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy\/\" data-tutorial-nav=\"chapter-4\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'4',link_url:this.href});}\">4. Training strategy<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/model-selection\/\" data-tutorial-nav=\"chapter-5\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'5',link_url:this.href});}\">5. Model selection<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/testing-analysis\/\" data-tutorial-nav=\"chapter-6\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'6',link_url:this.href});}\">6. Testing analysis<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/model-deployment\/\" data-tutorial-nav=\"chapter-7\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'progress',tutorial_chapter:'7',link_url:this.href});}\">7. Model deployment<\/a><\/li><\/ol><div class=\"ndts-controls\"><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-networks-applications\/\" data-tutorial-nav=\"chapter\">\u2190 Model types<\/a><a href=\"https:\/\/www.neuraldesigner.com\/learning\/machine-learning-tutorial\/\">Overview<\/a><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/classification-model\/\" data-tutorial-nav=\"next\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'next',tutorial_chapter:'1',link_url:this.href});}\">1.2. Classification \u2192<\/a><\/div><\/nav><style>.ndg{width:100vw;margin-left:calc(50% - 50vw);background:#eeeeee;padding:22px 24px 12px;font-family:\"Outfit\",\"Roboto\",Arial,sans-serif;color:#1b2635}.ndg *{box-sizing:border-box}.ndg a{text-decoration:none}.ndg-wrap{width:min(100%,1000px);margin:0 auto}.ndg-lead{font-size:19px;line-height:1.6;color:#3a4a5a;font-weight:300;margin:0 0 22px}.ndg-lead a{color:#2d799f;font-weight:600}.ndg-herofig{margin:0 0 30px;text-align:center}.ndg-herofig img{max-width:min(540px,100%);height:auto}.ndg-eyebrow{margin:0 0 14px;text-align:center;color:#2d799f;font-size:13px;font-weight:800;letter-spacing:.14em;text-transform:uppercase}.ndg-toc{list-style:none;display:grid;grid-template-columns:repeat(3,minmax(0,1fr));gap:12px;margin:0 0 48px;padding:0}.ndg-toc a{display:flex;align-items:center;gap:10px;height:100%;padding:14px 16px;background:#f2f2f2;border-radius:12px;color:#12354b!important;font-size:15px;font-weight:600;line-height:1.25;box-shadow:-8px -8px 16px rgba(255,255,255,.9),8px 8px 16px rgba(30,83,116,.10)}.ndg-toc a:hover{color:#2d799f!important}.ndg-step{display:grid;grid-template-columns:72px 1fr;gap:26px;margin:0 0 34px;padding:34px 38px;background:#f2f2f2;border-radius:22px;box-shadow:-14px -14px 28px rgba(255,255,255,.92),14px 14px 28px rgba(30,83,116,.12);scroll-margin-top:90px}.ndg-ic{width:72px;height:72px;border-radius:50%;background:#e7f1f7;display:flex;align-items:center;justify-content:center;flex:0 0 72px}.ndg-ic img{width:38px!important;height:38px!important;object-fit:contain;margin:0!important;box-shadow:none!important;border-radius:0!important}.ndg-step__body{min-width:0}.ndg-step__body h2{display:flex!important;align-items:center!important;min-height:72px;margin:0 0 16px!important;padding:0!important;color:#001233;font-size:25px;font-weight:700;line-height:1.2}.ndg-step__body h3{margin:26px 0 12px;color:#12354b;font-size:19px;font-weight:600}.ndg-step__body p{margin:0 0 16px;font-size:17px;line-height:1.62;color:#33424f}.ndg-step__body a{color:#2d799f;font-weight:500}.ndg-step__body ul{margin:0 0 16px;padding-left:22px}.ndg-step__body li{margin:7px 0;font-size:16px;line-height:1.55;color:#33424f}.ndg-step__body img{display:block!important;width:auto!important;max-width:min(560px,100%)!important;height:auto!important;margin:22px auto!important;border-radius:12px!important;box-shadow:0 12px 28px rgba(0,18,51,.12)!important}.ndg-step__body figure{margin:22px auto!important;max-width:min(620px,100%)}.ndg-step__body figcaption{text-align:center;color:#6a7884;font-size:13px;margin-top:8px}.ndg-nav{display:flex;justify-content:space-between;gap:16px;margin:12px 0 8px;flex-wrap:wrap}.ndg-nav a{display:inline-flex;align-items:center;padding:13px 24px;border-radius:10px;background:#f2f2f2;color:#2d799f!important;font-weight:800;font-size:15px;box-shadow:-8px -8px 16px rgba(255,255,255,.92),8px 8px 16px rgba(30,83,116,.10)}.ndg-nav a:hover{color:#1f5f80!important}@media(max-width:820px){.ndg-toc{grid-template-columns:1fr}.ndg-step{grid-template-columns:1fr;gap:16px;padding:26px 22px}.ndg-ic{width:56px;height:56px;flex:0 0 56px}.ndg-step__body h2{min-height:0;font-size:22px}}@media(max-width:640px){.ndg{padding:12px 14px}}<\/style><div class=\"ndg\"><div class=\"ndg-wrap\"><div class=\"ndg-step\" id=\"Approximation\"><div class=\"ndg-ic\"><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/approximation.svg\" width=\"44\" height=\"44\" alt=\"\" \/><\/div><div class=\"ndg-step__body\"><h2>1.1. Approximation (or function regression)<\/h2><p>In an approximation problem, a neural network learns how one or more continuous target variables depend on a set of input variables. No formula for that relationship is available beforehand; the model has to infer it from examples.<\/p>\n<p>The examples come from a <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\/\">data set<\/a>. Each <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\/#Samples\">sample<\/a> pairs the values of the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\/#InputVariables\">input variables<\/a> with the values measured for the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\/#TargetVariables\">target variables<\/a>. During training, the targets play the role of the correct answers against which the outputs of the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\/\">neural network<\/a> are compared.<\/p>\n<p>Once trained, the model returns an estimate of the targets for any new combination of inputs. In the simplest case there is a single input, <em>x<\/em>, and a single target, <em>y<\/em>, and the model is a curve that runs through the cloud of points, as the following figure shows.<\/p>\n<p><img loading=\"lazy\" loading=\"lazy\" decoding=\"async\" style=\"max-width: 600px;\" src=\"https:\/\/www.neuraldesigner.com\/images\/approximation.webp\" alt=\"Approximation problem\" width=\"600\" height=\"350\" \/><\/p>\n<p>Approximation, also called regression, is one of the most widely used model types. Some examples are the following:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/examples\/concrete-properties-assesment\/\">Model the strength of high-performance concretes<\/a>.<\/li>\n<li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/examples\/airfoil-self-noise-prediction\/\">Predict the noise generated by airfoil blades<\/a>.<\/li>\n<li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/examples\/yacht-hydrodynamics-modeling\/\">Predict the residuary resistance of sailing yachts<\/a>.<\/li>\n\n<li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/examples\/combined-cycle-power-plant\/\">Predict the electricity generated by combined cycle power plants<\/a>.<\/li>\n<li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/examples\/solar-power-generation\/\">Forecast the power generated by a solar plant<\/a>.<\/li>\n<li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/examples\/wine-quality-improvement\/\">Model wine preferences from physicochemical properties<\/a>.<\/li>\n<\/ul>\n<p>Measured targets are never exact. Each value combines a systematic part, which changes smoothly with the inputs, and a random part caused by sensor noise, unrecorded factors or rounding. In statistical terms, when errors are measured as squared differences, the best possible prediction for a given input is the mean of the target values that would be observed for it, and that mean is precisely the systematic part.<\/p>\n<p>A good model therefore follows the trend and ignores the scatter around it. A <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\/\">neural network<\/a> that bends to pass through every training point has also learned the noise, and its predictions for new samples get worse rather than better. The ability to predict well on samples that were not used for training is called generalization, and it is what ultimately decides how good an approximation model is. The <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/model-selection\/\">Model selection<\/a> chapter explains how to control it.<\/p>\n<\/div><\/div><p class=\"ndg-eyebrow\">All model types<\/p><ul class=\"ndg-toc\"><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/approximation-model\/\" aria-current=\"page\">1.1. Approximation (or function regression)<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/classification-model\/\">1.2. Classification (or pattern recognition)<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/forecasting-model\/\">1.3. Forecasting (or time series prediction)<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/anomaly-detection-model\/\">1.4. Anomaly detection<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/text-classification-model\/\">1.5. Text classification<\/a><\/li><li><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/image-classification-model\/\">1.6. Image classification model<\/a><\/li><\/ul><\/div><\/div><nav class=\"ndts-bottom\" aria-label=\"Tutorial chapter navigation\"><div class=\"ndts-controls\"><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-networks-applications\/\" data-tutorial-nav=\"previous\">\u2190 Model types<\/a><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-networks-applications\/\">All model types<\/a><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/classification-model\/\" data-tutorial-nav=\"next\" onclick=\"if(typeof gtag==='function'){gtag('event','tutorial_nav_click',{tutorial_action:'next',tutorial_chapter:'1',link_url:this.href});}\">1.2. Classification \u2192<\/a><\/div><\/nav>","protected":false},"author":122,"featured_media":2702,"template":"","categories":[30],"tags":[36],"class_list":["post-24371","learning","type-learning","status-publish","has-post-thumbnail","hentry","category-tutorials","tag-tutorials"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Approximation Models (Function Regression) - Neural Designer<\/title>\n<meta name=\"description\" content=\"Learn how approximation (regression) models estimate continuous targets from input variables, with examples and how to achieve good generalization.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/approximation-model\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Approximation Models (Function Regression) - 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