optimiz-rs/examples/notebooks/14_mckean_vlasov.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bab453dd",
   "metadata": {},
   "source": [
    "# 14 — Mean-reverting McKean–Vlasov\n",
    "\n",
    "Doc page: [mckean_vlasov.rst](../../docs/source/algorithms/mckean_vlasov.rst).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ef5d758f",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-12T14:05:47.635814Z",
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     "shell.execute_reply": "2026-05-12T14:05:48.204534Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from optimizr import _core as opt\n",
    "plt.rcParams['figure.figsize'] = (8.5, 4.5)\n",
    "plt.rcParams['figure.dpi'] = 110\n",
    "plt.rcParams['axes.grid'] = True\n",
    "plt.rcParams['grid.alpha'] = 0.3\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2a30664",
   "metadata": {},
   "source": [
    "## Cellule 1 — Conservation de la moyenne\n",
    "\n",
    "**Théorème (McKean 1966).** Pour la dynamique de champ moyen\n",
    "$$dX_t^i = \\theta\\,(\\bar X_t - X_t^i)\\,dt + \\sigma\\,dW_t^i,\n",
    "   \\qquad \\bar X_t = \\frac{1}{N}\\sum_j X_t^j,$$\n",
    "la moyenne empirique est conservée en espérance,\n",
    "$\\mathbb{E}[\\bar X_t] = \\bar X_0$.\n",
    "\n",
    "**Équation pivot.** $\\mathbb{E}[\\bar X_t] = \\bar X_0$ pour tout $t$.\n",
    "\n",
    "**Démonstration.** Sommer l'EDS sur $i$ : la dérive interne s'annule\n",
    "(moyenne — individu).  $\\square$\n",
    "\n",
    "**Ce que la cellule vérifie.** `mean_reverting_mckean_vlasov` préserve\n",
    "la moyenne et concentre la variance.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "abd6c348",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-12T14:05:48.211278Z",
     "iopub.status.busy": "2026-05-12T14:05:48.210791Z",
     "iopub.status.idle": "2026-05-12T14:05:48.950554Z",
     "shell.execute_reply": "2026-05-12T14:05:48.949277Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n_steps stocké : 201 (snapshots)\n",
      "n_particles    : 500\n",
      "Moyenne initiale : -8.527e-17\n",
      "Moyenne finale   : -1.148e-02\n",
      "Variance init    : 0.335\n",
      "Variance finale  : 0.039\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 1320x440 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N, T, n_steps = 500, 1.0, 200\n",
    "theta, sigma = 1.5, 0.3\n",
    "x0 = np.linspace(-1.0, 1.0, N).tolist()  # mean = 0\n",
    "\n",
    "res = opt.mean_reverting_mckean_vlasov(x0, theta, sigma, n_steps, T, 42)\n",
    "n_t = res['n_steps']\n",
    "n_part = res['n_particles']\n",
    "paths = np.array(res['paths_flat']).reshape(n_t, n_part)\n",
    "ts = np.array(res['time_grid'])\n",
    "\n",
    "mean = paths.mean(axis=1)\n",
    "var = paths.var(axis=1)\n",
    "print(f\"n_steps stocké : {n_t} (snapshots)\")\n",
    "print(f\"n_particles    : {n_part}\")\n",
    "print(f\"Moyenne initiale : {mean[0]:.3e}\")\n",
    "print(f\"Moyenne finale   : {mean[-1]:.3e}\")\n",
    "print(f\"Variance init    : {var[0]:.3f}\")\n",
    "print(f\"Variance finale  : {var[-1]:.3f}\")\n",
    "\n",
    "fig, axes = plt.subplots(1, 2, figsize=(12, 4))\n",
    "for i in range(0, n_part, 25):\n",
    "    axes[0].plot(ts, paths[:, i], alpha=0.4, lw=0.7)\n",
    "axes[0].plot(ts, mean, 'k-', lw=2, label='moyenne empirique')\n",
    "axes[0].set_xlabel('t'); axes[0].set_ylabel(r'$X_t^i$')\n",
    "axes[0].set_title(\"Trajectoires McKean–Vlasov\")\n",
    "axes[0].legend()\n",
    "axes[1].plot(ts, var, lw=2, color='C2')\n",
    "axes[1].set_xlabel('t'); axes[1].set_ylabel(r'Var($X_t$)')\n",
    "axes[1].set_title(\"Variance empirique\")\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f7da718",
   "metadata": {},
   "source": [
    "**Résultat attendu.** $|\\bar X_t - 0|$ très petit, variance qui\n",
    "décroît vers une borne stationnaire.\n",
    "\n",
    "**Lecture du graphique.** Faisceau qui se contracte autour de la\n",
    "moyenne ; variance qui diminue.\n",
    "\n",
    "**Conclusion.** Le solveur reproduit la concentration mean-field.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2f69220",
   "metadata": {},
   "source": [
    "## Cellule 2 — Exemple concret : dynamique d'opinion polarisée\n",
    "\n",
    "**Modèle.** Modèle de DeGroot continu : opinions initialement\n",
    "bimodales (deux groupes opposés).  La dynamique de champ moyen\n",
    "détruit progressivement la polarisation.\n",
    "\n",
    "**Équation pivot.**\n",
    "$$dX_t^i = \\theta\\,(\\bar X_t - X_t^i)\\,dt + \\sigma\\,dW_t^i.$$\n",
    "\n",
    "**Ce que la cellule vérifie.** Une distribution initiale bimodale\n",
    "fusionne en distribution unimodale centrée sur $\\bar X_0$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a9afa9b3",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-12T14:05:48.954428Z",
     "iopub.status.busy": "2026-05-12T14:05:48.954151Z",
     "iopub.status.idle": "2026-05-12T14:05:49.953468Z",
     "shell.execute_reply": "2026-05-12T14:05:49.952156Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variance t=0   : 1.033\n",
      "Variance t=mid : 0.025\n",
      "Variance t=T   : 0.007\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 1430x418 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rng = np.random.default_rng(7)\n",
    "N = 600\n",
    "half = N // 2\n",
    "x0 = np.concatenate([\n",
    "    rng.normal(-1.0, 0.2, half),\n",
    "    rng.normal(+1.0, 0.2, N - half),\n",
    "]).tolist()\n",
    "\n",
    "res = opt.mean_reverting_mckean_vlasov(x0, theta=2.0, sigma=0.15,\n",
    "                                        n_steps=400, t_horizon=2.0, seed=11)\n",
    "n_t = res['n_steps']\n",
    "n_part = res['n_particles']\n",
    "paths = np.array(res['paths_flat']).reshape(n_t, n_part)\n",
    "\n",
    "mid = n_t // 2\n",
    "print(f\"Variance t=0   : {paths[0].var():.3f}\")\n",
    "print(f\"Variance t=mid : {paths[mid].var():.3f}\")\n",
    "print(f\"Variance t=T   : {paths[-1].var():.3f}\")\n",
    "\n",
    "fig, axes = plt.subplots(1, 3, figsize=(13, 3.8))\n",
    "for ax, idx, label in zip(axes, [0, mid, -1], ['t=0', 't=T/2', 't=T']):\n",
    "    ax.hist(paths[idx], bins=40, density=True,\n",
    "            color='C0', edgecolor='white', alpha=0.85)\n",
    "    ax.set_title(f\"Distribution {label}\")\n",
    "    ax.set_xlabel('opinion'); ax.set_ylabel('densité')\n",
    "    ax.set_xlim(-2, 2)\n",
    "fig.suptitle(\"Convergence d'une population polarisée vers le consensus\",\n",
    "             fontsize=12, y=1.02)\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "001ac849",
   "metadata": {},
   "source": [
    "**Résultat attendu.** L'histogramme bimodal initial fusionne en un\n",
    "pic unique centré à $0$.\n",
    "\n",
    "**Lecture du graphique.** Trois snapshots montrent la fusion\n",
    "progressive des deux modes.\n",
    "\n",
    "**Conclusion.** L'attraction par la moyenne (mean-field) détruit la\n",
    "polarisation initiale en temps fini — illustration du théorème du\n",
    "consensus dans les dynamiques de DeGroot continues.\n"
   ]
  }
 ],
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