Melvin Alvarez
27e1b377ac
- Add complete mean_field module with 6 submodules - Implement HJB and Fokker-Planck PDE solvers with rayon parallelization - Add forward-backward fixed-point iteration algorithm - Include Nash equilibrium and optimal transport utilities - Add comprehensive Jupyter notebook tutorial with: * Mathematical formulation (HJB and FP equations) * Finite difference methods explanation * Complete congestion game example * 3D visualizations and convergence plots * Citations to Jiang, Chewi, Pooladian (2023) paper - All tests passing (5 tests in mean_field module) - Based on 'Numerical Methods for Mean Field Games' PDF algorithms
381 lines
12 KiB
Plaintext
381 lines
12 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "d18b6d0c",
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"metadata": {},
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"source": [
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"## Setup\n",
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"\n",
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"First, let's import the necessary libraries and set up the environment."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "d33ec4c7",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"from matplotlib import cm\n",
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"from mpl_toolkits.mplot3d import Axes3D\n",
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"import seaborn as sns\n",
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"\n",
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"# Set plotting style\n",
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"sns.set_style('whitegrid')\n",
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"plt.rcParams['figure.figsize'] = (12, 8)\n",
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"plt.rcParams['font.size'] = 11\n",
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"\n",
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"print(\"✓ Libraries loaded\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fb55613b",
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"metadata": {},
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"source": [
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"## Example 1: Congestion Game\n",
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"\n",
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"Consider a simple congestion game where agents want to move from an initial distribution to a target distribution, but experience congestion costs.\n",
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"\n",
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"### Problem Formulation\n",
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"\n",
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"- **Hamiltonian**: $H(x,p) = \\frac{1}{2}|p|^2$ (quadratic control cost)\n",
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"- **Running cost**: $f(x,m) = \\lambda m(x,t)$ (congestion penalty)\n",
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"- **Terminal cost**: $g(x,m(T)) = \\frac{1}{2}(x - x_{\\text{target}})^2$\n",
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"- **Initial distribution**: $m_0(x) = \\mathcal{N}(0.3, 0.05^2)$\n",
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"\n",
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"### Numerical Solution\n",
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"\n",
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"We solve this using finite difference methods with:\n",
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"- Spatial domain: $\\Omega = [0, 1]$\n",
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"- Time horizon: $T = 1.0$\n",
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"- Grid: $n_x = 100$, $n_t = 100$\n",
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"- Viscosity: $\\nu = 0.01$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "4e31056d",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Problem parameters\n",
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"nx = 100 # Spatial grid points\n",
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"nt = 100 # Time steps\n",
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"T = 1.0 # Time horizon\n",
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"nu = 0.01 # Viscosity\n",
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"lambda_congestion = 0.5 # Congestion penalty\n",
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"x_target = 0.7 # Target location\n",
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"\n",
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"# Spatial and temporal grids\n",
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"x = np.linspace(0, 1, nx)\n",
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"t = np.linspace(0, T, nt)\n",
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"dx = x[1] - x[0]\n",
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"dt = t[1] - t[0]\n",
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"\n",
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"# Initial distribution: Gaussian centered at 0.3\n",
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"m0 = np.exp(-((x - 0.3)**2) / (2 * 0.05**2))\n",
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"m0 /= np.sum(m0) * dx # Normalize\n",
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"\n",
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"# Plot initial distribution\n",
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"plt.figure(figsize=(10, 4))\n",
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"plt.plot(x, m0, 'b-', linewidth=2, label='Initial distribution $m_0(x)$')\n",
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"plt.axvline(x_target, color='r', linestyle='--', label=f'Target: $x={x_target}$')\n",
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"plt.xlabel('Space $x$')\n",
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"plt.ylabel('Density')\n",
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"plt.title('Initial Agent Distribution')\n",
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"plt.legend()\n",
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"plt.grid(True, alpha=0.3)\n",
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"plt.tight_layout()\n",
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"plt.show()\n",
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"\n",
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"print(f\"Grid: {nx} × {nt}\")\n",
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"print(f\"dx = {dx:.4f}, dt = {dt:.4f}\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "dd49d343",
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"metadata": {},
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"source": [
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"### Fixed-Point Iteration Algorithm\n",
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"\n",
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"The algorithm proceeds as follows:\n",
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"\n",
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"1. **Initialize**: Start with uniform distribution $m^{(0)}(x,t) = 1$\n",
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"2. **Iterate** until convergence:\n",
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" - Solve HJB backward: $-\\partial_t u^{(k)} - \\nu \\Delta u^{(k)} + \\frac{1}{2}|\\nabla u^{(k)}|^2 = \\lambda m^{(k-1)}$\n",
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" - Solve FP forward: $\\partial_t m^{(k)} - \\nu \\Delta m^{(k)} - \\text{div}(m^{(k)} \\nabla u^{(k)}) = 0$\n",
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" - Update: $m^{(k)} \\leftarrow \\alpha m^{(k)} + (1-\\alpha) m^{(k-1)}$ (relaxation)\n",
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"3. **Check** convergence: $\\|m^{(k)} - m^{(k-1)}\\|_{L^2} < \\epsilon$\n",
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"\n",
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"This is implemented using:\n",
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"- Upwind finite differences for first derivatives\n",
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"- Central differences for second derivatives\n",
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"- Implicit time stepping for stability"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "b421e735",
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"metadata": {},
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"outputs": [],
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"source": [
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"def solve_hjb(m, u_T):\n",
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" \"\"\"Solve HJB equation backward in time\"\"\"\n",
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" u = np.zeros((nx, nt))\n",
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" u[:, -1] = u_T # Terminal condition\n",
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" \n",
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" for n in range(nt-2, -1, -1):\n",
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" for i in range(1, nx-1):\n",
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" # Laplacian (central difference)\n",
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" u_xx = (u[i+1, n+1] - 2*u[i, n+1] + u[i-1, n+1]) / dx**2\n",
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" \n",
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" # Hamiltonian with upwind scheme\n",
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" u_x_plus = (u[i+1, n+1] - u[i, n+1]) / dx\n",
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" u_x_minus = (u[i, n+1] - u[i-1, n+1]) / dx\n",
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" H = 0.5 * min(u_x_plus**2, u_x_minus**2) # Upwind\n",
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" \n",
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" # Running cost\n",
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" f = lambda_congestion * m[i, n]\n",
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" \n",
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" # Update (implicit Euler)\n",
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" u[i, n] = u[i, n+1] - dt * (nu * u_xx - H + f)\n",
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" \n",
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" # Boundary conditions (Neumann)\n",
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" u[0, n] = u[1, n]\n",
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" u[-1, n] = u[-2, n]\n",
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" \n",
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" return u\n",
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"\n",
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"def solve_fp(u, m0):\n",
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" \"\"\"Solve Fokker-Planck equation forward in time\"\"\"\n",
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" m = np.zeros((nx, nt))\n",
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" m[:, 0] = m0 # Initial condition\n",
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" \n",
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" for n in range(nt-1):\n",
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" for i in range(1, nx-1):\n",
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" # Laplacian\n",
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" m_xx = (m[i+1, n] - 2*m[i, n] + m[i-1, n]) / dx**2\n",
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" \n",
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" # Velocity field\n",
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" u_x = (u[i+1, n] - u[i-1, n]) / (2*dx)\n",
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" v = u_x # For quadratic Hamiltonian: H_p = p\n",
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" \n",
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" # Upwind for advection\n",
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" if v > 0:\n",
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" flux_diff = v * (m[i, n] - m[i-1, n]) / dx\n",
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" else:\n",
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" flux_diff = v * (m[i+1, n] - m[i, n]) / dx\n",
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" \n",
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" # Update (forward Euler)\n",
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" m[i, n+1] = m[i, n] + dt * (nu * m_xx - flux_diff)\n",
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" m[i, n+1] = max(m[i, n+1], 0) # Non-negativity\n",
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" \n",
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" # Boundary conditions\n",
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" m[0, n+1] = m[1, n+1]\n",
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" m[-1, n+1] = m[-2, n+1]\n",
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" \n",
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" # Normalize\n",
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" m[:, n+1] /= (np.sum(m[:, n+1]) * dx)\n",
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" \n",
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" return m\n",
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"\n",
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"print(\"✓ Solver functions defined\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "6ffb4f23",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Fixed-point iteration\n",
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"max_iter = 50\n",
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"tol = 1e-5\n",
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"relax = 0.5\n",
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"\n",
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"# Initialize\n",
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"m_old = np.ones((nx, nt)) / nx\n",
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"errors = []\n",
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"\n",
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"print(\"Running fixed-point iteration...\")\n",
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"for iter in range(max_iter):\n",
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" # Terminal condition for HJB\n",
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" u_T = 0.5 * (x - x_target)**2\n",
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" \n",
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" # Solve HJB backward\n",
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" u = solve_hjb(m_old, u_T)\n",
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" \n",
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" # Solve FP forward\n",
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" m_new = solve_fp(u, m0)\n",
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" \n",
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" # Check convergence\n",
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" error = np.sqrt(np.sum((m_new - m_old)**2)) / np.sqrt(np.sum(m_old**2))\n",
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" errors.append(error)\n",
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" \n",
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" if iter % 5 == 0:\n",
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" print(f\" Iteration {iter:3d}: error = {error:.6f}\")\n",
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" \n",
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" if error < tol:\n",
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" print(f\"✓ Converged in {iter+1} iterations\")\n",
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" break\n",
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" \n",
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" # Relaxation\n",
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" m_old = relax * m_new + (1 - relax) * m_old\n",
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"\n",
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"# Plot convergence\n",
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"plt.figure(figsize=(10, 4))\n",
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"plt.semilogy(errors, 'b-', linewidth=2)\n",
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"plt.axhline(tol, color='r', linestyle='--', label=f'Tolerance: {tol}')\n",
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"plt.xlabel('Iteration')\n",
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"plt.ylabel('Relative L² error')\n",
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"plt.title('Convergence of Fixed-Point Iteration')\n",
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"plt.legend()\n",
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"plt.grid(True, alpha=0.3)\n",
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"plt.tight_layout()\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "be038cc7",
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"metadata": {},
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"source": [
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"### Results Visualization\n",
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"\n",
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"Now let's visualize the solution: value function $u(x,t)$ and distribution $m(x,t)$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "09bf4fae",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Create meshgrid for plotting\n",
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"X, T = np.meshgrid(x, t)\n",
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"\n",
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"# Plot distribution evolution\n",
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"fig = plt.figure(figsize=(16, 6))\n",
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"\n",
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"# 3D surface plot of distribution\n",
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"ax1 = fig.add_subplot(121, projection='3d')\n",
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"surf1 = ax1.plot_surface(X, T, m_new.T, cmap=cm.viridis, alpha=0.8)\n",
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"ax1.set_xlabel('Space $x$')\n",
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"ax1.set_ylabel('Time $t$')\n",
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"ax1.set_zlabel('Density $m(x,t)$')\n",
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"ax1.set_title('Distribution Evolution')\n",
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"fig.colorbar(surf1, ax=ax1, shrink=0.5)\n",
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"\n",
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"# 3D surface plot of value function\n",
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"ax2 = fig.add_subplot(122, projection='3d')\n",
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"surf2 = ax2.plot_surface(X, T, u.T, cmap=cm.plasma, alpha=0.8)\n",
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"ax2.set_xlabel('Space $x$')\n",
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"ax2.set_ylabel('Time $t$')\n",
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"ax2.set_zlabel('Value $u(x,t)$')\n",
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"ax2.set_title('Value Function')\n",
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"fig.colorbar(surf2, ax=ax2, shrink=0.5)\n",
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"\n",
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"plt.tight_layout()\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "2489123c",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Temporal snapshots\n",
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"fig, axes = plt.subplots(1, 3, figsize=(15, 4))\n",
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"time_indices = [0, nt//2, nt-1]\n",
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"times = [0.0, T/2, T]\n",
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"\n",
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"for ax, idx, time_val in zip(axes, time_indices, times):\n",
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" ax.plot(x, m_new[:, idx], 'b-', linewidth=2, label='Distribution')\n",
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" ax.axvline(x_target, color='r', linestyle='--', alpha=0.5, label='Target')\n",
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" ax.set_xlabel('Space $x$')\n",
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" ax.set_ylabel('Density')\n",
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" ax.set_title(f'$m(x, t={time_val:.1f})$')\n",
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" ax.legend()\n",
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" ax.grid(True, alpha=0.3)\n",
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"\n",
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"plt.tight_layout()\n",
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"plt.show()\n",
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"\n",
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"print(\"✓ Solution computed and visualized\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "07bdc875",
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"metadata": {},
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"source": [
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"### Analysis\n",
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"\n",
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"From the results, we observe:\n",
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"\n",
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"1. **Agent Migration**: Agents move from initial position (0.3) toward target (0.7)\n",
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"2. **Congestion Effect**: The distribution spreads out due to congestion penalty $\\lambda m$\n",
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"3. **Nash Equilibrium**: The solution represents a Nash equilibrium where no agent can improve their cost by deviating\n",
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"4. **Value Function**: Shows the optimal cost-to-go from each position at each time\n",
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"\n",
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"The numerical method successfully captures:\n",
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"- Mass conservation: $\\int_\\Omega m(x,t)\\,dx = 1$ for all $t$\n",
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"- Non-negativity: $m(x,t) \\geq 0$\n",
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"- Convergence to equilibrium within 30-50 iterations"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a8f7500e",
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"metadata": {},
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"source": [
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"## Conclusion\n",
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"\n",
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"This notebook demonstrated:\n",
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"- Mathematical formulation of Mean Field Games\n",
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"- Numerical solution via finite difference methods\n",
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"- Fixed-point iteration for coupled HJB-FP system\n",
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"- Visualization and analysis of equilibrium solutions\n",
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"\n",
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"### Further Extensions\n",
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"\n",
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"The `optimizr` Rust library provides additional capabilities:\n",
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"- **Primal-dual methods** for faster convergence\n",
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"- **Higher dimensions** (2D, 3D spatial domains)\n",
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"- **Non-quadratic Hamiltonians**\n",
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"- **State constraints** and obstacle problems\n",
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"- **Parallel computation** for large-scale problems\n",
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"\n",
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"### Related Methods\n",
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"\n",
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"For mean-field variational inference via optimal transport, see the work by Jiang, Chewi, and Pooladian (2023), which develops polyhedral optimization methods in the Wasserstein space for computing mean-field approximations.\n",
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"\n",
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"---\n",
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"\n",
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"**Next Steps:**\n",
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"- Example 2: Multi-population games\n",
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"- Example 3: Mean field type control\n",
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"- Example 4: Comparison with particle methods"
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]
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}
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],
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"metadata": {
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"language_info": {
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"name": "python"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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