{"id":23,"date":"2025-02-12T22:11:37","date_gmt":"2025-02-12T21:11:37","guid":{"rendered":"http:\/\/boissiz.cluster029.hosting.ovh.net\/?p=23"},"modified":"2026-08-25T15:39:43","modified_gmt":"2026-08-25T13:39:43","slug":"perceptron","status":"publish","type":"post","link":"https:\/\/www.laboratoiredesidees.fr\/index.php\/2025\/02\/12\/perceptron\/","title":{"rendered":"Neurone formel – Perceptron de Rosenblatt"},"content":{"rendered":"\n

Commen\u00e7ons par pr\u00e9senter le neurone formel:<\/p>\n\n\n\n

import matplotlib.pyplot as plt\nimport networkx as nx\n\nCr\u00e9ation du graphe\n\nG = nx.DiGraph()\n\nAjout des n\u0153uds\n\nG.add_node(\"x1\", pos=(0, 1))\nG.add_node(\"x2\", pos=(0, -1))\nG.add_node(\"z\", pos=(1, 0))\nG.add_node(\"a\", pos=(2, 0))\nG.add_node(\"y_pred\", pos=(3, 0))\nG.add_node(\"-1\", pos=(1, 1))\n\nAjout des ar\u00eates avec les labels\n\nG.add_edge(\"x1\", \"z\", weight=\"w1\")\nG.add_edge(\"x2\", \"z\", weight=\"w2\")\nG.add_edge(\"z\", \"a\")\nG.add_edge(\"a\", \"y_pred\")\nG.add_edge(\"-1\", \"z\")\n\nPositions des n\u0153uds\n\npos = nx.get_node_attributes(G, 'pos')\n\nDessin du graphe\n\nplt.figure(figsize=(8, 6))\nnx.draw(G, pos, with_labels=True, node_size=3000, node_color=\"white\", edgecolors=\"black\", font_size=15)\n\nDessin des labels sur les ar\u00eates\n\nedge_labels = {('x1', 'z'): 'w1', ('x2', 'z'): 'w2', ('z', 'a'): 'z', ('a', 'y_pred'): 'a', ('-1', 'z'): 'b'}\nnx.draw_networkx_edge_labels(G, pos, edge_labels=edge_labels, font_color='red', font_size=15)\n\nAffichage de la formule en bas\n\nplt.text(1.5, -1, r'$z = w_1 x_1 + w_2 x_2 - b$', fontsize=15, ha='center')\n\nplt.axis('off')\nplt.show()<\/code><\/pre>\n\n\n\n
Code en python pour r\u00e9aliser ce sch\u00e9ma:<\/p>\n\n\n\n
\n
\"\"<\/figure>\n<\/figure>\n\n\n\n
Cette repr\u00e9sentation math\u00e9matique pens\u00e9e en 1943 par  Warren McCulloch<\/a> et Walter Pitts<\/a> est le premier mod\u00e8le math\u00e9matique du neurone biologique.<\/p>\n\n\n\n
Les poids w_i<\/span>sont aussi appel\u00e9s poids synaptique par r\u00e9f\u00e9rence au cerveau humain.<\/p>\n\n\n\n
Ce mod\u00e8le permet de r\u00e9aliser des op\u00e9rations de classifications lin\u00e9aire (fronti\u00e8re de s\u00e9paration une droite, ou un plan ou bien plus largement un hyperplan en lorsqu’il s’agit de prendre en compte plus de 3 dimensions).<\/p>\n\n\n\n
La fonction d’activation utilis\u00e9 initialement \u00e9tait une variante de la fonction dite de Heavyside , la fonction signe :<\/p>\n\n\n\n
<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
Cela permet de fournir en sortie deux r\u00e9sultats: 1 ou -1 et donc d’identifier deux classes diff\u00e9rentes.<\/p>\n\n\n\n
Il s’agit alors de trouver les poids synaptique permettant d’effectuer correctement la classification en trouvant une fronti\u00e8re de d\u00e9cision, exemple ici pour une porte OR:<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
Ce neurone formel a ses limites en particuliers il ne peut effectuer qu’une classification lin\u00e9aire et n’est donc pas capable de repr\u00e9senter une porte XOR (ou exclusif)<\/p>\n\n\n\n
Frank Rosenblatt mis au point un algorithme permettant de calculer les poids synaptique(voir wikipedia):<\/p>\n\n\n\n
\"\"
Algorithme de calcul des poids synaptique<\/figcaption><\/figure>\n\n\n\n
import numpy as np\n\ndef perceptron(X, y, max_iterations=1000):\n    \"\"\"\n    Impl\u00e9mente l'algorithme du perceptron avec un biais n\u00e9gatif.\n    :param X: Matrice de caract\u00e9ristiques (n \u00e9chantillons, d caract\u00e9ristiques)\n    :param y: Vecteur des labels (-1 ou 1)\n    :param max_iterations: Nombre maximal d'it\u00e9rations pour \u00e9viter une boucle infinie\n    :return: Vecteur de poids W\n    \"\"\"\n    n_samples, n_features = X.shape\n    print(f\"Number of samples: {n_samples}, Number of features: {n_features}\")\n    W = np.zeros(n_features + 1)  # Initialisation du vecteur de poids avec biais\n    print(\"Initial W:\", W)\n    \n    X = np.c_[-np.ones(n_samples), X]  # Ajout de la colonne de biais avec -1\n    print(\"Modified X:\", X)\n    \n    for _ in range(max_iterations):\n        error_count = 0\n        for i in range(n_samples):\n            if y[i] * np.dot(W, X[i]) <= 0:\n                W += y[i] * X[i]\n                error_count += 1\n\n        if error_count == 0:\n            break  # Arr\u00eate si aucune erreur\n    \n    return W\n\n# Exemple d'utilisation\ndata = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])\nlabels = np.array([-1, 1, 1, 1])  # Labels pour la porte OR\n\nW_final = perceptron(data, labels)\nprint(\"Poids finaux:\", W_final)\n<\/code><\/pre>\n\n\n\n
Poids finaux: [1. 2. 2.]<\/strong><\/p>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
Commen\u00e7ons par pr\u00e9senter le neurone formel: Code en python pour r\u00e9aliser ce sch\u00e9ma: Cette repr\u00e9sentation math\u00e9matique pens\u00e9e en 1943 par  Warren McCulloch et Walter Pitts est le premier mod\u00e8le math\u00e9matique du neurone biologique. Les poids sont aussi appel\u00e9s poids synaptique par r\u00e9f\u00e9rence au cerveau humain. Ce mod\u00e8le permet de r\u00e9aliser des op\u00e9rations de classifications lin\u00e9aire (fronti\u00e8re de<\/p>\n
\n\t\t\t\t Read More <\/span> <\/i><\/a>\n\t\t\t<\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","footnotes":""},"categories":[2],"tags":[],"class_list":["post-23","post","type-post","status-publish","format-standard","hentry","category-intelligence-artificielle"],"yoast_head":"\nNeurone formel - Perceptron de Rosenblatt - Laboratoire des id\u00e9es<\/title>\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.laboratoiredesidees.fr\/index.php\/2025\/02\/12\/perceptron\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Neurone formel - Perceptron de Rosenblatt - Laboratoire des id\u00e9es\" \/>\n<meta property=\"og:description\" content=\"Commen\u00e7ons par pr\u00e9senter le neurone formel: Code en python pour r\u00e9aliser ce sch\u00e9ma: Cette repr\u00e9sentation math\u00e9matique pens\u00e9e en 1943 par  Warren McCulloch et Walter Pitts est le premier mod\u00e8le math\u00e9matique du neurone biologique. Les poids sont aussi appel\u00e9s poids synaptique par r\u00e9f\u00e9rence au cerveau humain. 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