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feat(mean_field): Implement Mean Field Games module with PDE solvers

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- 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
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Melvin Alvarez
2026-01-04 13:30:57 +01:00
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{
"cells": [
{
"cell_type": "markdown",
"id": "d18b6d0c",
"metadata": {},
"source": [
"## Setup\n",
"\n",
"First, let's import the necessary libraries and set up the environment."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d33ec4c7",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib import cm\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"import seaborn as sns\n",
"\n",
"# Set plotting style\n",
"sns.set_style('whitegrid')\n",
"plt.rcParams['figure.figsize'] = (12, 8)\n",
"plt.rcParams['font.size'] = 11\n",
"\n",
"print(\"✓ Libraries loaded\")"
]
},
{
"cell_type": "markdown",
"id": "fb55613b",
"metadata": {},
"source": [
"## Example 1: Congestion Game\n",
"\n",
"Consider a simple congestion game where agents want to move from an initial distribution to a target distribution, but experience congestion costs.\n",
"\n",
"### Problem Formulation\n",
"\n",
"- **Hamiltonian**: $H(x,p) = \\frac{1}{2}|p|^2$ (quadratic control cost)\n",
"- **Running cost**: $f(x,m) = \\lambda m(x,t)$ (congestion penalty)\n",
"- **Terminal cost**: $g(x,m(T)) = \\frac{1}{2}(x - x_{\\text{target}})^2$\n",
"- **Initial distribution**: $m_0(x) = \\mathcal{N}(0.3, 0.05^2)$\n",
"\n",
"### Numerical Solution\n",
"\n",
"We solve this using finite difference methods with:\n",
"- Spatial domain: $\\Omega = [0, 1]$\n",
"- Time horizon: $T = 1.0$\n",
"- Grid: $n_x = 100$, $n_t = 100$\n",
"- Viscosity: $\\nu = 0.01$"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "4e31056d",
"metadata": {},
"outputs": [],
"source": [
"# Problem parameters\n",
"nx = 100 # Spatial grid points\n",
"nt = 100 # Time steps\n",
"T = 1.0 # Time horizon\n",
"nu = 0.01 # Viscosity\n",
"lambda_congestion = 0.5 # Congestion penalty\n",
"x_target = 0.7 # Target location\n",
"\n",
"# Spatial and temporal grids\n",
"x = np.linspace(0, 1, nx)\n",
"t = np.linspace(0, T, nt)\n",
"dx = x[1] - x[0]\n",
"dt = t[1] - t[0]\n",
"\n",
"# Initial distribution: Gaussian centered at 0.3\n",
"m0 = np.exp(-((x - 0.3)**2) / (2 * 0.05**2))\n",
"m0 /= np.sum(m0) * dx # Normalize\n",
"\n",
"# Plot initial distribution\n",
"plt.figure(figsize=(10, 4))\n",
"plt.plot(x, m0, 'b-', linewidth=2, label='Initial distribution $m_0(x)$')\n",
"plt.axvline(x_target, color='r', linestyle='--', label=f'Target: $x={x_target}$')\n",
"plt.xlabel('Space $x$')\n",
"plt.ylabel('Density')\n",
"plt.title('Initial Agent Distribution')\n",
"plt.legend()\n",
"plt.grid(True, alpha=0.3)\n",
"plt.tight_layout()\n",
"plt.show()\n",
"\n",
"print(f\"Grid: {nx} × {nt}\")\n",
"print(f\"dx = {dx:.4f}, dt = {dt:.4f}\")"
]
},
{
"cell_type": "markdown",
"id": "dd49d343",
"metadata": {},
"source": [
"### Fixed-Point Iteration Algorithm\n",
"\n",
"The algorithm proceeds as follows:\n",
"\n",
"1. **Initialize**: Start with uniform distribution $m^{(0)}(x,t) = 1$\n",
"2. **Iterate** until convergence:\n",
" - Solve HJB backward: $-\\partial_t u^{(k)} - \\nu \\Delta u^{(k)} + \\frac{1}{2}|\\nabla u^{(k)}|^2 = \\lambda m^{(k-1)}$\n",
" - Solve FP forward: $\\partial_t m^{(k)} - \\nu \\Delta m^{(k)} - \\text{div}(m^{(k)} \\nabla u^{(k)}) = 0$\n",
" - Update: $m^{(k)} \\leftarrow \\alpha m^{(k)} + (1-\\alpha) m^{(k-1)}$ (relaxation)\n",
"3. **Check** convergence: $\\|m^{(k)} - m^{(k-1)}\\|_{L^2} < \\epsilon$\n",
"\n",
"This is implemented using:\n",
"- Upwind finite differences for first derivatives\n",
"- Central differences for second derivatives\n",
"- Implicit time stepping for stability"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b421e735",
"metadata": {},
"outputs": [],
"source": [
"def solve_hjb(m, u_T):\n",
" \"\"\"Solve HJB equation backward in time\"\"\"\n",
" u = np.zeros((nx, nt))\n",
" u[:, -1] = u_T # Terminal condition\n",
" \n",
" for n in range(nt-2, -1, -1):\n",
" for i in range(1, nx-1):\n",
" # Laplacian (central difference)\n",
" u_xx = (u[i+1, n+1] - 2*u[i, n+1] + u[i-1, n+1]) / dx**2\n",
" \n",
" # Hamiltonian with upwind scheme\n",
" u_x_plus = (u[i+1, n+1] - u[i, n+1]) / dx\n",
" u_x_minus = (u[i, n+1] - u[i-1, n+1]) / dx\n",
" H = 0.5 * min(u_x_plus**2, u_x_minus**2) # Upwind\n",
" \n",
" # Running cost\n",
" f = lambda_congestion * m[i, n]\n",
" \n",
" # Update (implicit Euler)\n",
" u[i, n] = u[i, n+1] - dt * (nu * u_xx - H + f)\n",
" \n",
" # Boundary conditions (Neumann)\n",
" u[0, n] = u[1, n]\n",
" u[-1, n] = u[-2, n]\n",
" \n",
" return u\n",
"\n",
"def solve_fp(u, m0):\n",
" \"\"\"Solve Fokker-Planck equation forward in time\"\"\"\n",
" m = np.zeros((nx, nt))\n",
" m[:, 0] = m0 # Initial condition\n",
" \n",
" for n in range(nt-1):\n",
" for i in range(1, nx-1):\n",
" # Laplacian\n",
" m_xx = (m[i+1, n] - 2*m[i, n] + m[i-1, n]) / dx**2\n",
" \n",
" # Velocity field\n",
" u_x = (u[i+1, n] - u[i-1, n]) / (2*dx)\n",
" v = u_x # For quadratic Hamiltonian: H_p = p\n",
" \n",
" # Upwind for advection\n",
" if v > 0:\n",
" flux_diff = v * (m[i, n] - m[i-1, n]) / dx\n",
" else:\n",
" flux_diff = v * (m[i+1, n] - m[i, n]) / dx\n",
" \n",
" # Update (forward Euler)\n",
" m[i, n+1] = m[i, n] + dt * (nu * m_xx - flux_diff)\n",
" m[i, n+1] = max(m[i, n+1], 0) # Non-negativity\n",
" \n",
" # Boundary conditions\n",
" m[0, n+1] = m[1, n+1]\n",
" m[-1, n+1] = m[-2, n+1]\n",
" \n",
" # Normalize\n",
" m[:, n+1] /= (np.sum(m[:, n+1]) * dx)\n",
" \n",
" return m\n",
"\n",
"print(\"✓ Solver functions defined\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6ffb4f23",
"metadata": {},
"outputs": [],
"source": [
"# Fixed-point iteration\n",
"max_iter = 50\n",
"tol = 1e-5\n",
"relax = 0.5\n",
"\n",
"# Initialize\n",
"m_old = np.ones((nx, nt)) / nx\n",
"errors = []\n",
"\n",
"print(\"Running fixed-point iteration...\")\n",
"for iter in range(max_iter):\n",
" # Terminal condition for HJB\n",
" u_T = 0.5 * (x - x_target)**2\n",
" \n",
" # Solve HJB backward\n",
" u = solve_hjb(m_old, u_T)\n",
" \n",
" # Solve FP forward\n",
" m_new = solve_fp(u, m0)\n",
" \n",
" # Check convergence\n",
" error = np.sqrt(np.sum((m_new - m_old)**2)) / np.sqrt(np.sum(m_old**2))\n",
" errors.append(error)\n",
" \n",
" if iter % 5 == 0:\n",
" print(f\" Iteration {iter:3d}: error = {error:.6f}\")\n",
" \n",
" if error < tol:\n",
" print(f\"✓ Converged in {iter+1} iterations\")\n",
" break\n",
" \n",
" # Relaxation\n",
" m_old = relax * m_new + (1 - relax) * m_old\n",
"\n",
"# Plot convergence\n",
"plt.figure(figsize=(10, 4))\n",
"plt.semilogy(errors, 'b-', linewidth=2)\n",
"plt.axhline(tol, color='r', linestyle='--', label=f'Tolerance: {tol}')\n",
"plt.xlabel('Iteration')\n",
"plt.ylabel('Relative L² error')\n",
"plt.title('Convergence of Fixed-Point Iteration')\n",
"plt.legend()\n",
"plt.grid(True, alpha=0.3)\n",
"plt.tight_layout()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "be038cc7",
"metadata": {},
"source": [
"### Results Visualization\n",
"\n",
"Now let's visualize the solution: value function $u(x,t)$ and distribution $m(x,t)$."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "09bf4fae",
"metadata": {},
"outputs": [],
"source": [
"# Create meshgrid for plotting\n",
"X, T = np.meshgrid(x, t)\n",
"\n",
"# Plot distribution evolution\n",
"fig = plt.figure(figsize=(16, 6))\n",
"\n",
"# 3D surface plot of distribution\n",
"ax1 = fig.add_subplot(121, projection='3d')\n",
"surf1 = ax1.plot_surface(X, T, m_new.T, cmap=cm.viridis, alpha=0.8)\n",
"ax1.set_xlabel('Space $x$')\n",
"ax1.set_ylabel('Time $t$')\n",
"ax1.set_zlabel('Density $m(x,t)$')\n",
"ax1.set_title('Distribution Evolution')\n",
"fig.colorbar(surf1, ax=ax1, shrink=0.5)\n",
"\n",
"# 3D surface plot of value function\n",
"ax2 = fig.add_subplot(122, projection='3d')\n",
"surf2 = ax2.plot_surface(X, T, u.T, cmap=cm.plasma, alpha=0.8)\n",
"ax2.set_xlabel('Space $x$')\n",
"ax2.set_ylabel('Time $t$')\n",
"ax2.set_zlabel('Value $u(x,t)$')\n",
"ax2.set_title('Value Function')\n",
"fig.colorbar(surf2, ax=ax2, shrink=0.5)\n",
"\n",
"plt.tight_layout()\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2489123c",
"metadata": {},
"outputs": [],
"source": [
"# Temporal snapshots\n",
"fig, axes = plt.subplots(1, 3, figsize=(15, 4))\n",
"time_indices = [0, nt//2, nt-1]\n",
"times = [0.0, T/2, T]\n",
"\n",
"for ax, idx, time_val in zip(axes, time_indices, times):\n",
" ax.plot(x, m_new[:, idx], 'b-', linewidth=2, label='Distribution')\n",
" ax.axvline(x_target, color='r', linestyle='--', alpha=0.5, label='Target')\n",
" ax.set_xlabel('Space $x$')\n",
" ax.set_ylabel('Density')\n",
" ax.set_title(f'$m(x, t={time_val:.1f})$')\n",
" ax.legend()\n",
" ax.grid(True, alpha=0.3)\n",
"\n",
"plt.tight_layout()\n",
"plt.show()\n",
"\n",
"print(\"✓ Solution computed and visualized\")"
]
},
{
"cell_type": "markdown",
"id": "07bdc875",
"metadata": {},
"source": [
"### Analysis\n",
"\n",
"From the results, we observe:\n",
"\n",
"1. **Agent Migration**: Agents move from initial position (0.3) toward target (0.7)\n",
"2. **Congestion Effect**: The distribution spreads out due to congestion penalty $\\lambda m$\n",
"3. **Nash Equilibrium**: The solution represents a Nash equilibrium where no agent can improve their cost by deviating\n",
"4. **Value Function**: Shows the optimal cost-to-go from each position at each time\n",
"\n",
"The numerical method successfully captures:\n",
"- Mass conservation: $\\int_\\Omega m(x,t)\\,dx = 1$ for all $t$\n",
"- Non-negativity: $m(x,t) \\geq 0$\n",
"- Convergence to equilibrium within 30-50 iterations"
]
},
{
"cell_type": "markdown",
"id": "a8f7500e",
"metadata": {},
"source": [
"## Conclusion\n",
"\n",
"This notebook demonstrated:\n",
"- Mathematical formulation of Mean Field Games\n",
"- Numerical solution via finite difference methods\n",
"- Fixed-point iteration for coupled HJB-FP system\n",
"- Visualization and analysis of equilibrium solutions\n",
"\n",
"### Further Extensions\n",
"\n",
"The `optimizr` Rust library provides additional capabilities:\n",
"- **Primal-dual methods** for faster convergence\n",
"- **Higher dimensions** (2D, 3D spatial domains)\n",
"- **Non-quadratic Hamiltonians**\n",
"- **State constraints** and obstacle problems\n",
"- **Parallel computation** for large-scale problems\n",
"\n",
"### Related Methods\n",
"\n",
"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",
"\n",
"---\n",
"\n",
"**Next Steps:**\n",
"- Example 2: Multi-population games\n",
"- Example 3: Mean field type control\n",
"- Example 4: Comparison with particle methods"
]
}
],
"metadata": {
"language_info": {
"name": "python"
}
},
"nbformat": 4,
"nbformat_minor": 5
}