{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "6c97c1ad",
   "metadata": {},
   "source": [
    "# A physics-informed neural network for the Fokker-Planck equation\n",
    "\n",
    "We solve the Fokker-Planck equation associated with the overdamped Langevin SDE\n",
    "\n",
    "$$ \\mathrm{d}X_t = -\\nabla V(X_t)\\,\\mathrm{d}t + \\sqrt{2D}\\,\\mathrm{d}W_t, $$\n",
    "\n",
    "which governs the time-dependent probability density $p(t, x)$ of $X_t$:\n",
    "\n",
    "$$ \\partial_t p = \\nabla \\cdot \\big( \\nabla V\\, p \\big) + D\\, \\Delta p, \\qquad p(0, \\cdot) = p_0 . $$\n",
    "\n",
    "As potential we take a **banana** in the first two coordinates and a harmonic well in the remaining ones,\n",
    "\n",
    "$$ V(x) = \\frac{x_1^2}{2\\sigma_1^2} + \\frac{(x_2 - b\\,x_1^2)^2}{2\\sigma_2^2} + \\sum_{k \\geq 3} \\frac{x_k^2}{2}, $$\n",
    "\n",
    "so the stationary density $\\propto e^{-V/D}$ is the classic banana shape. Starting from a Gaussian $p_0$ at the vertex, the mass spreads along both arms until the banana is filled.\n",
    "\n",
    "The density is represented with a `torchtt.nn.TTDensityLayer`: a squared functional Tensor-Train over a Gaussian basis, composed with a **nonlinear diffeomorphism** (a rank-1 polynomial shear that can bend the density, followed by an affine map). A small network takes the time $t$ and outputs the flat parameter vector of the layer — TT cores and transform parameters — so that\n",
    "\n",
    "$$ p_\\theta(t, x) = \\texttt{TTDensityLayer}\\big(\\mathrm{net}_\\theta(t),\\, x\\big). $$\n",
    "\n",
    "Two properties make this ansatz attractive for Fokker-Planck PINNs: $p_\\theta \\geq 0$ and $\\int p_\\theta(t, x)\\,\\mathrm{d}x = 1$ hold **by construction for every $t$**, so no normalization penalty is needed — only the PDE residual and the initial condition enter the loss. The code is written for general dimension $d$; here we fix $d = 2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "95eacf96",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/yonnss/repos/torchTT/torchtt/_dmrg.py:19: UserWarning: \u001b[33m\n",
      "C++ implementation not available. Using pure Python.\n",
      "\u001b[0m\n",
      "  warnings.warn(\"\\x1B[33m\\nC++ implementation not available. Using pure Python.\\n\\033[0m\")\n",
      "/home/yonnss/repos/torchTT/torchtt/_amen.py:21: UserWarning: \u001b[33m\n",
      "C++ implementation not available. Using pure Python.\n",
      "\u001b[0m\n",
      "  warnings.warn(\n",
      "/home/yonnss/repos/torchTT/torchtt/solvers.py:21: UserWarning: \u001b[33m\n",
      "C++ implementation not available. Using pure Python.\n",
      "\u001b[0m\n",
      "  warnings.warn(\n",
      "/home/yonnss/repos/torchTT/torchtt/cpp.py:12: UserWarning: \u001b[33m\n",
      "C++ implementation not available. Using pure Python.\n",
      "\u001b[0m\n",
      "  warnings.warn(\"\\x1B[33m\\nC++ implementation not available. Using pure Python.\\n\\033[0m\")\n",
      "/home/yonnss/repos/torchTT/torchtt/__init__.py:34: UserWarning: \u001b[33m\n",
      "C++ implementation not available. Using pure Python.\n",
      "\u001b[0m\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<torch._C.Generator at 0x77e512cb11b0>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import torchtt.functional\n",
    "from torchtt.nn import TTDensityLayer, AffineTransform, Rank1Shear, ComposedTransform\n",
    "\n",
    "torch.manual_seed(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d8bc9fe5",
   "metadata": {},
   "source": [
    "## Problem setup\n",
    "\n",
    "Dimension, potential parameters, diffusion coefficient, time horizon and the box on which collocation points will be placed. The initial condition is an isotropic Gaussian at the banana vertex,\n",
    "\n",
    "$$ p_0(x) = (2\\pi s_0^2)^{-d/2} \\exp\\!\\Big( -\\tfrac{\\|x - m_0\\|^2}{2 s_0^2} \\Big), $$\n",
    "\n",
    "and the drift is $f = -\\nabla V$, which for the banana potential reads\n",
    "\n",
    "$$ f_1 = -\\frac{x_1}{\\sigma_1^2} + \\frac{2 b\\, x_1 (x_2 - b x_1^2)}{\\sigma_2^2}, \\qquad f_2 = -\\frac{x_2 - b x_1^2}{\\sigma_2^2}, \\qquad f_k = -x_k \\ \\ (k \\geq 3). $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2a6122f8",
   "metadata": {},
   "outputs": [],
   "source": [
    "d = 2                 # spatial dimension (the script is generic in d)\n",
    "sig1, sig2, bend = 1.0, 0.4, 0.5   # banana parameters\n",
    "D = 1.0               # diffusion coefficient\n",
    "T = 2.5               # time horizon\n",
    "\n",
    "m0 = torch.zeros(d)   # initial Gaussian at the banana vertex\n",
    "s0 = 0.5\n",
    "\n",
    "lo = torch.tensor([-3.5, -1.5] + [-3.0] * (d - 2))   # collocation box\n",
    "hi = torch.tensor([3.5, 5.0] + [3.0] * (d - 2))\n",
    "\n",
    "\n",
    "def drift(x):\n",
    "    \"\"\"f(x) = -grad V(x) for the banana potential, any d >= 2.\"\"\"\n",
    "    w = (x[..., 1] - bend * x[..., 0] ** 2) / sig2 ** 2\n",
    "    f0 = -x[..., 0] / sig1 ** 2 + 2.0 * bend * x[..., 0] * w\n",
    "    f1 = -w\n",
    "    return torch.cat([torch.stack([f0, f1], dim=-1), -x[..., 2:]], dim=-1)\n",
    "\n",
    "\n",
    "def pdf0(x):\n",
    "    \"\"\"Initial density: isotropic Gaussian N(m0, s0^2 I).\"\"\"\n",
    "    q = ((x - m0) ** 2).sum(-1) / (2 * s0 ** 2)\n",
    "    return torch.exp(-q) / (2 * torch.pi * s0 ** 2) ** (d / 2)\n",
    "\n",
    "\n",
    "def sample_box(m):\n",
    "    return lo + (hi - lo) * torch.rand(m, d)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63bf0f65",
   "metadata": {},
   "source": [
    "## The model\n",
    "\n",
    "The reference density lives on the unit cube: each dimension gets a Gaussian basis of $N$ functions on $[0,1]$, and the TT cores use a bond rank of $6$. On top of it sits a diffeomorphism $T_t$, so the model density is the push-forward\n",
    "\n",
    "$$ p_\\theta(t, x) = p_\\text{ref}\\big(T_t(x);\\, G(t)\\big)\\, \\big|\\det J_{T_t}(x)\\big|, $$\n",
    "\n",
    "where both the TT cores $G(t)$ and the parameters of $T_t$ are emitted by the network `net` as one flat vector, as a function of $t$. The transform is a `Rank1Shear` of degree 2 — exactly the map that can unbend a parabola, $z_2 = x_2 + \\alpha_2 x_1^2$ — followed by an `AffineTransform` that carries the physical box onto the unit cube.\n",
    "\n",
    "The last layer of `net` is initialized with tiny weights and a bias `theta0` chosen so that at start the transform is the plain box-to-cube map — a sane density to begin training from."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0bd51170",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "parameters per density: 156  (TT cores: 144, transform: 12)\n"
     ]
    }
   ],
   "source": [
    "N = [12] * d                       # basis functions per dimension\n",
    "R = [1] + [6] * (d - 1) + [1]      # TT ranks\n",
    "basis = [torchtt.functional.GaussianBasis(torch.linspace(0, 1, n), delta_overlap=1) for n in N]\n",
    "\n",
    "transform = ComposedTransform([Rank1Shear(d, degree=2), AffineTransform(d)])\n",
    "layer = TTDensityLayer(N, R, basis, transform=transform)\n",
    "n_params = layer.input_requirement()\n",
    "n_core = sum(n * r0 * r1 for n, r0, r1 in zip(N, R, R[1:]))\n",
    "print(f\"parameters per density: {n_params}  (TT cores: {n_core}, transform: {n_params - n_core})\")\n",
    "\n",
    "# initial parameter vector: random cores, identity shear, box -> unit cube affine\n",
    "shear0 = torch.zeros(2 * d + 1 + 2)\n",
    "shear0[d] = 1.0                                   # u = e_2 (shear direction)\n",
    "shear0[1] = 1.0                                   # v = e_1 (shear \"driven by\" x_1)\n",
    "affine0 = torch.cat([torch.zeros(d * (d - 1) // 2),          # no rotation\n",
    "                     torch.log(1.0 / (hi - lo)),             # scales\n",
    "                     -lo / (hi - lo)])                       # offsets\n",
    "theta0 = torch.cat([torch.rand(n_core), shear0, affine0])\n",
    "\n",
    "net = torch.nn.Sequential(\n",
    "    torch.nn.Linear(1, 64), torch.nn.Tanh(),\n",
    "    torch.nn.Linear(64, 64), torch.nn.Tanh(),\n",
    "    torch.nn.Linear(64, n_params),\n",
    ")\n",
    "with torch.no_grad():\n",
    "    net[-1].weight *= 0.01\n",
    "    net[-1].bias.copy_(theta0)\n",
    "\n",
    "\n",
    "def pdf_model(t, x):\n",
    "    \"\"\"p(t, x) for a batch: t is (M, 1), x is (M, d).\"\"\"\n",
    "    return layer(net(t / T), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd131551",
   "metadata": {},
   "source": [
    "## The PDE residual\n",
    "\n",
    "The Fokker-Planck equation in divergence form uses the probability flux $J = f\\,p - D \\nabla p$, and the residual of the ansatz is\n",
    "\n",
    "$$ r_\\theta(t, x) = \\partial_t p_\\theta(t, x) + \\nabla \\cdot \\big( f(x)\\, p_\\theta(t, x) - D\\, \\nabla p_\\theta(t, x) \\big). $$\n",
    "\n",
    "All derivatives come from automatic differentiation; the divergence costs one backward pass per dimension."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ae5ad21a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def residual(t, x):\n",
    "    t = t.detach().requires_grad_(True)\n",
    "    x = x.detach().requires_grad_(True)\n",
    "    p = pdf_model(t, x)\n",
    "    p_t = torch.autograd.grad(p.sum(), t, create_graph=True)[0][:, 0]\n",
    "    p_x = torch.autograd.grad(p.sum(), x, create_graph=True)[0]\n",
    "    flux = drift(x) * p.unsqueeze(-1) - D * p_x\n",
    "    div = 0.0\n",
    "    for i in range(d):\n",
    "        div = div + torch.autograd.grad(flux[:, i].sum(), x, create_graph=True)[0][:, i]\n",
    "    return p_t + div"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dce4e541",
   "metadata": {},
   "source": [
    "## Training\n",
    "\n",
    "Since normalization and positivity are built into the layer, the loss has only two terms — the PDE residual on uniformly drawn collocation points $(t, x)$ and the initial condition at $t = 0$:\n",
    "\n",
    "$$ \\mathcal{L} = \\frac{1}{M} \\sum_{m=1}^{M} r_\\theta\\big(t_m, x_m\\big)^2 \\;+\\; \\frac{\\lambda}{M'} \\sum_{m=1}^{M'} \\Big( p_\\theta\\big(0, x'_m\\big) - p_0\\big(x'_m\\big) \\Big)^2 . $$\n",
    "\n",
    "The IC points $x'_m$ are drawn half from $p_0$ itself and half uniformly from the box, so the density is also pushed to zero away from the initial blob. All points are redrawn every step."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "9a799be1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter     0   PDE loss 5.357e-01   IC loss 6.282e-02\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter   500   PDE loss 5.310e-03   IC loss 3.175e-06\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter  1000   PDE loss 1.958e-03   IC loss 1.072e-06\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter  1500   PDE loss 7.172e-04   IC loss 5.989e-06\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter  2000   PDE loss 5.551e-04   IC loss 1.174e-06\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter  2500   PDE loss 3.507e-04   IC loss 2.938e-06\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter  3000   PDE loss 3.877e-04   IC loss 4.858e-06\n"
     ]
    }
   ],
   "source": [
    "n_iters = 3000\n",
    "M_pde, M_ic = 2048, 512\n",
    "opt = torch.optim.Adam(net.parameters(), lr=1e-3)\n",
    "\n",
    "for it in range(n_iters + 1):\n",
    "    t_c = T * torch.rand(M_pde, 1)\n",
    "    x_c = sample_box(M_pde)\n",
    "    loss_pde = residual(t_c, x_c).pow(2).mean()\n",
    "\n",
    "    x_ic = torch.cat([m0 + s0 * torch.randn(M_ic // 2, d), sample_box(M_ic // 2)])\n",
    "    p_ic = pdf_model(torch.zeros(M_ic, 1), x_ic)\n",
    "    loss_ic = (p_ic - pdf0(x_ic)).pow(2).mean()\n",
    "\n",
    "    loss = loss_pde + 100.0 * loss_ic\n",
    "    opt.zero_grad()\n",
    "    loss.backward()\n",
    "    opt.step()\n",
    "    if it % 500 == 0:\n",
    "        print(f\"iter {it:5d}   PDE loss {loss_pde.item():.3e}   IC loss {loss_ic.item():.3e}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd0de837",
   "metadata": {},
   "source": [
    "## Reference solution by sampling\n",
    "\n",
    "The same SDE is simulated for a large ensemble of particles with the Euler-Maruyama scheme\n",
    "\n",
    "$$ X_{n+1} = X_n + f(X_n)\\, \\Delta t + \\sqrt{2 D \\Delta t}\\; \\xi_n, \\qquad \\xi_n \\sim \\mathcal{N}(0, I), $$\n",
    "\n",
    "and the empirical histograms serve as ground truth. Snapshots are kept at four times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8a5b580d",
   "metadata": {},
   "outputs": [],
   "source": [
    "t_snap = [0.0, 0.5, 1.25, 2.5]\n",
    "n_particles = 200_000\n",
    "dt = 2e-3\n",
    "\n",
    "x_p = m0 + s0 * torch.randn(n_particles, d)\n",
    "snapshots, t_cur = [x_p.clone()], 0.0\n",
    "with torch.no_grad():\n",
    "    for ts in t_snap[1:]:\n",
    "        while t_cur < ts - 1e-9:\n",
    "            x_p = x_p + drift(x_p) * dt + (2 * D * dt) ** 0.5 * torch.randn_like(x_p)\n",
    "            t_cur += dt\n",
    "        snapshots.append(x_p.clone())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22ec9ca0",
   "metadata": {},
   "source": [
    "## Comparison\n",
    "\n",
    "Top row: 2-D histograms of the particles. Bottom row: the learned $p(t, x)$ on a grid (for $d > 2$ one would plot a slice or marginal instead). The PINN starts as the round Gaussian and bends into the banana, matching the sampled evolution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9ea3d38a",
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 1600x750 with 8 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ng = 150\n",
    "gx = torch.linspace(lo[0], hi[0], ng)\n",
    "gy = torch.linspace(lo[1], hi[1], ng)\n",
    "Xg, Yg = torch.meshgrid(gx, gy, indexing=\"ij\")\n",
    "grid = torch.stack([Xg.flatten(), Yg.flatten()], dim=1)\n",
    "if d > 2:\n",
    "    grid = torch.cat([grid, torch.zeros(grid.shape[0], d - 2)], dim=1)  # slice at x_k = 0\n",
    "\n",
    "fig, axes = plt.subplots(2, len(t_snap), figsize=(4 * len(t_snap), 7.5), sharex=True, sharey=True)\n",
    "for j, (ts, xs) in enumerate(zip(t_snap, snapshots)):\n",
    "    axes[0, j].hist2d(xs[:, 0].numpy(), xs[:, 1].numpy(), bins=100,\n",
    "                      range=[[lo[0], hi[0]], [lo[1], hi[1]]], density=True, cmap=\"viridis\")\n",
    "    axes[0, j].set_title(f\"$t = {ts}$\")\n",
    "    with torch.no_grad():\n",
    "        p_grid = pdf_model(ts * torch.ones(grid.shape[0], 1), grid).reshape(ng, ng)\n",
    "    axes[1, j].contourf(Xg.numpy(), Yg.numpy(), p_grid.numpy(), levels=50, cmap=\"viridis\")\n",
    "axes[0, 0].set_ylabel(\"SDE samples\\n$x_2$\")\n",
    "axes[1, 0].set_ylabel(\"PINN density\\n$x_2$\")\n",
    "for ax in axes[1]:\n",
    "    ax.set_xlabel(\"$x_1$\")\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b93d172",
   "metadata": {},
   "source": [
    "## Moments and normalization\n",
    "\n",
    "A quantitative check: the mean $\\int x\\, p_\\theta(t, x)\\,\\mathrm{d}x$ of the learned density (by quadrature on the grid) against the sample mean, plus the integral $\\int_\\text{box} p_\\theta(t, x)\\,\\mathrm{d}x$ — which should be $\\approx 1$ without ever having been enforced during training."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "fc64e5e7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t = 0.00   integral 0.986   mean PINN (+0.003, +0.000)   mean MC (-0.002, -0.001)\n",
      "t = 0.50   integral 0.986   mean PINN (+0.004, +0.217)   mean MC (+0.004, +0.222)\n",
      "t = 1.25   integral 0.986   mean PINN (-0.001, +0.305)   mean MC (+0.002, +0.345)\n",
      "t = 2.50   integral 0.986   mean PINN (+0.015, +0.389)   mean MC (+0.004, +0.427)\n"
     ]
    }
   ],
   "source": [
    "cell = ((hi[0] - lo[0]) / ng * (hi[1] - lo[1]) / ng).item()\n",
    "for ts, xs in zip(t_snap, snapshots):\n",
    "    with torch.no_grad():\n",
    "        p_grid = pdf_model(ts * torch.ones(grid.shape[0], 1), grid)\n",
    "    w = p_grid * cell\n",
    "    mean_pinn = (grid[:, :2] * w.unsqueeze(-1)).sum(0)\n",
    "    mean_mc = xs[:, :2].mean(0)\n",
    "    print(f\"t = {ts:4.2f}   integral {w.sum():.3f}   \"\n",
    "          f\"mean PINN ({mean_pinn[0]:+.3f}, {mean_pinn[1]:+.3f})   \"\n",
    "          f\"mean MC ({mean_mc[0]:+.3f}, {mean_mc[1]:+.3f})\")"
   ]
  }
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