{"id":3396,"date":"2026-06-22T05:40:42","date_gmt":"2026-06-22T03:40:42","guid":{"rendered":"https:\/\/neuraldesigner.com\/blog\/learning-problem-in-nn\/"},"modified":"2026-07-28T11:54:18","modified_gmt":"2026-07-28T09:54:18","slug":"learning-problem-in-nn","status":"publish","type":"blog","link":"https:\/\/www.neuraldesigner.com\/blog\/learning-problem-in-nn\/","title":{"rendered":"Mathematics of artificial intelligence: the learning problem in neural networks"},"content":{"rendered":"<style>.ndb{width:100vw;margin-left:calc(50% - 50vw);background:#eeeeee;padding:22px 24px 14px;font-family:\"Outfit\",\"Roboto\",Arial,sans-serif;color:#1b2635}.ndb *{box-sizing:border-box}.ndb a{text-decoration:none}.ndb-wrap{width:min(100%,1200px);margin:0 auto}.ndb-lead{font-size:18px;line-height:1.6;color:#3a4a5a;font-weight:300;margin:0 0 22px}.ndb-lead a{color:#2d799f;font-weight:600}.ndb-toc{list-style:none;display:flex;flex-wrap:wrap;gap:10px;justify-content:center;margin:0 0 42px;padding:0}.ndb-toc a{padding:10px 18px;background:#f2f2f2;border-radius:22px;color:#12354b!important;font-size:14px;font-weight:600;box-shadow:-6px -6px 12px rgba(255,255,255,.9),6px 6px 12px rgba(30,83,116,.10)}.ndb-toc a:hover{color:#2d799f!important}.ndb-card{margin:0 0 26px;padding:32px 40px;background:#f5f6f7;border-radius:20px;box-shadow:-12px -12px 24px rgba(255,255,255,.9),12px 12px 24px rgba(30,83,116,.10);scroll-margin-top:90px}.ndb-card h2{margin:0 0 20px;padding-bottom:12px;color:#001233;font-size:24px;font-weight:700;position:relative;border-bottom:1px solid #dbe5ec}.ndb-card h2:after{content:\"\";position:absolute;left:0;bottom:-1px;width:62px;height:3px;background:#56a1c8;border-radius:2px}.ndb-card p{margin:0 0 14px;font-size:16.5px;line-height:1.62;color:#33424f}.ndb-card a{color:#2d799f;font-weight:500}.ndb-card ul{margin:0 0 14px;padding-left:22px}.ndb-card li{margin:5px 0;font-size:16px;line-height:1.5;color:#33424f}.ndb-card img:not([src$=\".svg\"]):not([data-src$=\".svg\"]){display:block;width:auto;max-width:min(560px,100%);height:auto;margin:22px auto;border-radius:12px;box-shadow:0 12px 28px rgba(0,18,51,.12)}.ndb-card img[src$=\".svg\"],.ndb-card img[data-src$=\".svg\"]{display:inline-block;max-width:24px;height:auto;margin:0 6px -4px 0}.ndb-card table{border-collapse:separate;border-spacing:0;width:auto;max-width:100%;margin:22px 0;font-size:15px;background:#fbfcfd;border-radius:12px;overflow:hidden;box-shadow:0 10px 24px rgba(0,18,51,.08)}.ndb-card th,.ndb-card td{padding:11px 16px;border-bottom:1px solid #e6ecf0;text-align:left;vertical-align:top}.ndb-card thead th{background:#12354b!important;color:#fff!important;font-weight:600;text-align:center}.ndb-card tbody th{background:#e9f1f6!important;color:#12354b!important;font-weight:600}.ndb-card td{background:#fbfcfd!important;color:#33424f!important}.ndb-card pre{margin:22px 0;padding:22px 24px;background:#0b1830!important;color:#e6eef5!important;border-radius:14px;overflow-x:auto;font-family:Consolas,Menlo,monospace;font-size:12.5px;line-height:1.5;white-space:pre}.ndb-card--accent{background:#e9f2f8}.ndb-eq{text-align:center;margin:22px 0;overflow-x:auto}.ndb-card iframe{max-width:100%;border-radius:12px;margin:8px 0}.ndb-card .mjx-chtml.MJXc-display{overflow-x:auto;overflow-y:hidden;max-width:100%;padding:2px 0 8px}.ndb-card .mjx-chtml.MathJax_CHTML{font-size:18px!important}@media(max-width:820px){.ndb-card{padding:26px 22px}.ndb-card h2{font-size:21px}}@media(max-width:640px){.ndb{padding:12px 14px}}<\/style><div class=\"ndb\"><div class=\"ndb-wrap\"><div class=\"ndb-lead\"><p><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\">The mathematics behind Artificial Intelligence can be challenging to understand.&nbsp;<\/span><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\">It is essential to comprehend them to apply machine learning algorithms properly. The interpretation of the results also requires an understanding of mathematics and statistics.&nbsp;<\/span><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\">This post explains the learning problem of neural networks from a mathematical point of view. In particular, from the perspective of functional analysis and variational calculus.<\/span><\/p>\n<ol>\n<li><a href=\"#Preliminaries\">Preliminaries<\/a>.<\/li>\n<li><a href=\"#LearningProblem\">Learning problem<\/a>.<\/li>\n<li><a href=\"#Conclusions\">Conclusions<\/a>.<\/li>\n<li><a href=\"#Video\">Video<\/a>.<\/li>\n<\/ol>\n<\/div><ul class=\"ndb-toc\"><li><a href=\"#Preliminaries\">Preliminaries<\/a><\/li><li><a href=\"#LearningProblem\">Learning problem<\/a><\/li><li><a href=\"#Conclusions\">Conclusions<\/a><\/li><li><a href=\"#Video\">Video<\/a><\/li><\/ul><div class=\"ndb-card\" id=\"Preliminaries\"><h2>Preliminaries<\/h2><p>First, we will define some mathematical concepts.<\/p>\n<h3>Function<\/h3>\n<p>A function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set.<br \/>$$\\begin{align*}<br \/>y \\colon X &amp;\\to Y\\\\<br \/>x &amp;\\mapsto y(x)<br \/>\\end{align*}$$<\/p>\n<h3>Function space<\/h3>\n<p>A function space is a normed vector space whose elements are functions. For example, $C^1(X, Y)$ is the space of continuously differentiable functions, or $P^3(X, Y)$ is the space of all polynomials of order 3.<\/p>\n<h3>Optimization problem<\/h3>\n<p>An optimization problem consists of finding a vector that minimizes a function.<\/p>\n<div class=\"ndb-eq\">$$Find \\ x^* \\in X \\ such \\ that \\ f(x) \\ is \\ minimum.$$<\/div>\n<p>For example, let&#8217;s look at the function $f(x)=x^2$.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_optimization.webp\" alt=\"Optimization problem for Mathematics of artificial intelligence\" width=\"600\" height=\"400\"><\/p>\n<p>In this case, the minimum is at $x=0$.<\/p>\n<h3>Functional<\/h3>\n<p>A functional correspondence assigns a number to each function belonging to some class.<br \/>$$\\begin{align*}<br \/>F \\colon V &amp;\\to R\\\\<br \/>y(x) &amp;\\mapsto F[y(x)]<br \/>\\end{align*}$$<\/p>\n<h3>Variational problem<\/h3>\n<p>A variational problem consists of finding a function that minimizes a functional.<\/p>\n<div class=\"ndb-eq\">$$Find \\ x^* \\in X \\ such \\ that \\ f(x) \\ is \\ minimum.$$<\/div>\n<p>For example, given two points $A$ and $B$, we consider the functional of length.<\/p>\n<p><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_variational.webp\" alt=\"Variational problem for Mathematics of artificial intelligence\" width=\"502\" height=\"338\"><\/p>\n<p>The function $y^*(x)$ that minimizes the length between the two points is a straight line.<\/p>\n<h3>Neural network<\/h3>\n<p>A neural network is a biologically inspired computational model consisting of a network architecture composed of artificial neurons. Neural networks are a powerful technique for solving approximation, classification, and forecasting problems.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_nn_example.webp\" alt=\"Neural network\"><\/p>\n<p>This diagram shows a neural network with four inputs and two outputs.<\/p>\n<\/div><div class=\"ndb-card\" id=\"LearningProblem\"><h2>Learning problem<\/h2><p>We can express the learning problem as finding a function that causes some functional to assume an extreme value. This functional, or loss index, defines the task the neural network must do and provides a measure of the quality of the representation that the network needs to learn. The choice of a suitable loss index depends on the particular application.<\/p>\n<p>Here is the activity diagram.<\/p>\n<p><img decoding=\"async\" class=\"alignnone\" style=\"width: 50%;\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_activity_diagram.webp\" alt=\"Activity diagram to understand mathematics of artificial intelligence. Fist we have to define function space (neural network). Next, we formulate de variational problem (loss index). And finally we solve the reduced function optimization problem (optimization algorithm).\" width=\"2389\" height=\"1652\"><\/p>\n<p>We will explain each of the steps one by one.<\/p>\n<h3 id=\"NeuralNetwork\">2.1. Neural network<\/h3>\n<p>A neural network spans a function space, which we will call $V$. It is the space of all the functions that the neural network can produce.<br \/>$$\\begin{align*}<br \/>y \\colon X \\subset R^n &amp;\\to Y \\subset R^m\\\\<br \/>x &amp;\\mapsto y(x;\\theta)<br \/>\\end{align*}$$<br \/>This structure contains a set of parameters. The training strategy will then adjust the parameters to perform specific tasks.<br \/>Each function depends on a vector of free parameters $theta$. The dimension of $V$ is the number of parameters in the neural network.<\/p>\n<p>For example, here, we have a neural network. It has a hidden layer with a hyperbolic tangent activation function and an output layer with a linear activation function. The dimension of the function space is 7, the number of parameters in this neural network.<\/p>\n<p><img decoding=\"async\" style=\"width: 50%;\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_nn.webp\" alt=\"Neural network\"><\/p>\n<p>The elements of the function space are of the following form.<br \/>$$y(x;theta)=b_1^{(2)}+w_{1,1}^{(2)}tanh(b_1^{(1)}+w_{1,1}^{(1)}x)+w_{2,1}^{(2)}tanh(b_2^{(1)}+w_{1,2}^{(1)}x)$$<\/p>\n<p>Here, we have different examples of elements of the function space.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_function_space.webp\" alt=\"Function space examples\"><\/p>\n<p>They both have the same structure but different values for the parameters.<\/p>\n<h3 id=\"LossIndex\">2.2. Loss index<\/h3>\n<p>The loss index is a functional that defines the task for the neural network and provides a measure of quality. The choice of the functional depends on the application of the neural network. Some examples are the mean squared error or the cross-entropy error. In both these cases, we have a dataset with $k$ points $(x_i,y_i)$, and we are looking for a function $y(x,theta)$ that will fit them.<\/p>\n<ul>\n<li>Mean squared error: $E[y(x;theta)] = frac{1}{k} sum_{i=1}^k (y(x_i;theta)-y_i)^2$<\/li>\n<li>Cross-entropy error: $E[y(x;theta)] = &#8211; frac{1}{k} sum_{i=1}^k y(x_i;theta) log(y_i)$<\/li>\n<\/ul>\n<p>We formulate the variational problem, which is the learning task of the neural network. We look for the function $y^*(x;theta^*) in V$ for which the functional $F[y(x;theta)]$ gives a minimum value. In general, we approximate the solution with direct methods.<\/p>\n<p>Let&#8217;s see an example of a variational problem. Consider the function space of the previous neural network. Find the element $y^*(x;theta)$ such that the mean squared error functional, $MSE$, is minimized. We have a series of points, and we are looking for a function that will fit them.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_variational_before.webp\" alt=\"Variational problem unsolved\"><\/p>\n<p>We have taken a random function $y(x;theta)=-x$ for now, the functional $MSE[y(x;theta)]=1.226$ is not minimized yet.<br \/>Next, we will study a method to solve this problem.<\/p>\n<h3 id=\"OptimizationAlgorithm\">2.3. Optimization algorithm<\/h3>\n<p>The objective functional $F$ has an objective function $f$ associated.<br \/>$$\\begin{align*}<br \/>f \\colon \\Theta \\subset R^d &amp;\\to R\\\\<br \/>\\theta &amp;\\mapsto f(\\theta)<br \/>\\end{align*}$$<\/p>\n<p>We can reduce the variational problem to a function optimization problem. We aim to find the vector of free parameters $theta^* in Theta$ that will make $f(theta)$ have its minimum value. With it, the objective functional achieves its minimum.<\/p>\n<p>The optimization algorithm will solve the reduced function optimization problem. There are many methods for the optimization process, such as the Quasi-Newton method or the gradient descent.<\/p>\n<p>Let&#8217;s continue the example from before. We have the same set of points and are looking for a function that will fit them. We have reduced the variational problem to a function optimization problem. To solve the function optimization problem, the optimization algorithm searches for a vector $theta^*$ that minimizes the mean squared error function.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/learning_problem_variational_after.webp\" alt=\"Variational problem solved\"><\/p>\n<p>After finding this vector $theta^*=(\u22121.16,\u22121.72, 0.03,2.14,\u22121.67,\u22121.72, 1.24)$, for which $mse(theta^*)=0.007$,<br \/>we obtain the function $y^*(x;theta^*)=\u22121.16\u22121.72tanh\u2061(0.03+2.14x)\u22121.67tanh(\u22121.72+1.24x)$ that minimizes the functional. This is the solution to the problem.<\/p>\n<\/div><div class=\"ndb-card ndb-card--accent\" id=\"Conclusions\"><h2>Conclusions<\/h2><p>We have explained neural networks from the perspective of variational calculus.<\/p>\n<p>The neural network spans a parameterized function space. In this space, we formulate the variational problem. We must minimize a functional to solve it. To do so, we reduce it to a function optimization problem. This is solved with an optimization algorithm, and we obtain the function we sought.<\/p>\n<\/div><div class=\"ndb-card\" id=\"Video\"><h2>Video<\/h2><p>Watch this video to understand the learning problem in neural networks better.<\/p>\n<p><iframe src=\"https:\/\/www.youtube.com\/embed\/qMVZ4qC7VbA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div><\/div><\/div>","protected":false},"author":13,"featured_media":2008,"template":"","categories":[],"tags":[36],"class_list":["post-3396","blog","type-blog","status-publish","has-post-thumbnail","hentry","tag-tutorials"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Mathematics of artificial intelligence: the learning problem in neural networks<\/title>\n<meta name=\"description\" content=\"Understand the mathematics of artificial intelligence from the perspective of functional analysis 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