V_waves.ipynb

{
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   "attachments": {},
   "cell_type": "markdown",
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   "source": [
    "# Differentiable Physics V: 2D Wave Equation\n",
    "\n",
    "Lets try to solve the wave equation in 2D using the Mac Cormack method. The wave equation is given by:\n",
    "\n",
    "$$\\frac{\\partial ^2 }{\\partial t^2} = c^2 \\left( \\frac{\\partial ^2  u}{\\partial x^2} + \\frac{\\partial ^2  u}{\\partial y^2} \\right)$$\n",
    "\n",
    "Where $u$ is a scalar field, denoting the height or amount of fluid at a given grid point.\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
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   "source": [
    "\n",
    "\n",
    "The Mac Cormack Method in 2D becomes:\n",
    "\n",
    "$$\\bar{u}^{n+1} = u^n - \n",
    "\\frac{\\Delta t}{\\Delta x} \\left( f(u_{i, j}) - f(u_{i-1, j}) \\right) - \n",
    "\\frac{\\Delta t}{\\Delta y} \\left( f(u_{i, j}) - f(u_{i, j-1}) \\right)$$\n",
    "\n",
    "$$u^{n+1} = \\frac{1}{2} \\left( u^n + \\bar{u}^{n+1}\\right) - \n",
    "\\frac{\\Delta t}{2 \\Delta x} \\left( f(\\bar{u}^{n+1}_{i,j}) - f(\\bar{u}^{n+1}_{i-1,j})  \\right) - \n",
    "\\frac{\\Delta t}{2 \\Delta y} \\left( f(\\bar{u}^{n+1}_{i,j}) - f(\\bar{u}^{n+1}_{i,j-1})  \\right) $$\n",
    "\n",
    "where the flux function is defined as:\n",
    "\n",
    "$$f(u) = c \\cdot u$$ \n",
    "\n",
    "where $c$ denotes the wave speed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import jax.numpy as jnp\n",
    "from jax import value_and_grad, jit\n",
    "from functools import partial\n",
    "\n",
    "@partial(jit, static_argnames=(\"c\",))\n",
    "def flux(u : jnp.ndarray, c : jnp.ndarray = -0.2) -> jnp.ndarray:\n",
    "    return c * u\n",
    "\n",
    "@partial(jit, static_argnames=(\"dt\", \"dx\", \"dy\"))\n",
    "def mac_cormack_2d(u : jnp.ndarray, dt : float, dx : float, dy : float) -> jnp.ndarray:\n",
    "    dtdx = dt / dx\n",
    "    dtdy = dt / dy\n",
    "\n",
    "    u = u.at[0, :].set(0.0)\n",
    "    u = u.at[-1, :].set(0.0)\n",
    "    u = u.at[:, 0].set(0.0)\n",
    "    u = u.at[:, -1].set(0.0)\n",
    "\n",
    "    u_pred = u - \\\n",
    "        dtdx * (flux(u) - flux(jnp.roll(u, 1, axis=0))) - \\\n",
    "        dtdy * (flux(u) - flux(jnp.roll(u, 1, axis=1))) \n",
    "\n",
    "    u_fwd = 1/2 *( (u + u_pred) - \\\n",
    "        dtdx * (flux(jnp.roll(u_pred, -1, axis=0)) - flux(u_pred)) - \\\n",
    "        dtdy * (flux(jnp.roll(u_pred, -1, axis=1)) - flux(u_pred)) )\n",
    "\n",
    "\n",
    "    return u_fwd\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets look at a simple forward simulation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n"
     ]
    }
   ],
   "source": [
    "n_center = 50\n",
    "n_paddding = 1\n",
    "n_grid = n_center + 2 * n_paddding\n",
    "u = jnp.zeros((n_grid, n_grid), dtype=jnp.float32)\n",
    "\n",
    "k = 4\n",
    "u = u.at[n_grid//2-k:n_grid//2+k, n_grid//4-k:n_grid//4+k].set(1.0)\n",
    "u = u.at[n_grid//2-k:n_grid//2+k, 3*n_grid//4-k:3*n_grid//4+k].set(-1.0)\n",
    "\n",
    "\n",
    "dt = 1.0\n",
    "dx = 1.0\n",
    "dy = 1.0\n",
    "\n",
    "us = []\n",
    "\n",
    "animation_steps = 50\n",
    "for i in range(animation_steps):\n",
    "    u = mac_cormack_2d(u, dt, dx, dy)\n",
    "    us.append(u.copy())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "BBhtTSKieOqpp4a/5pe//GXxJ3/yJ8WFF15YXHDBBcXv/d7vFUePHq3fpCfIH/7hHxaXXXZZ0dLS\\\n",
       "Ulx88cXFjTfeWPzTP/3T8Hiu61Lmf9bAFEW+63PbbbcVl156adHS0lJ88IMfLG677bbirbfeGh7P\\\n",
       "dV04N5WiKIo6ZU8AAOrAewABADIjAAIAZEYABADIjAAIAJAZARAAIDMCIABAZgRAAIDMCIAAAJkR\\\n",
       "AAEAMiMAAgBkRgAEAMiMAAgAkBkBEAAgMwIgAEBmBEAAgMwIgAAAmREAAQAyIwACAGRGAAQAyIwA\\\n",
       "CACQGQEQACAzAiAAQGYEQACAzAiAAACZEQABADIjAAIAZEYABADIjAAIAJAZARAAIDMCIABAZgRA\\\n",
       "AIDM/P/KjVgB0NaxAQAAAABJRU5ErkJggg==\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim3e14038a79134462baef7ad03a308aca = new Animation(frames, img_id, slider_id, 50.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import animation\n",
    "from IPython.display import HTML\n",
    "\n",
    "fig = plt.figure()\n",
    "\n",
    "imp = plt.imshow(us[0], animated=True)\n",
    "plt.close()\n",
    "\n",
    "def updatefig(i):\n",
    "    imp.set_array(us[i])\n",
    "    return imp,\n",
    "\n",
    "ani = animation.FuncAnimation(fig, updatefig, frames=range(animation_steps), interval=50, blit=True)\n",
    "# js \n",
    "HTML(ani.to_jshtml())"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inverse Case"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7fc8bc6e1ab0>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u_target = jnp.array(plt.imread(\"img/frog.jpg\"))\n",
    "u_target = u_target[0:400 , 150:550, :]\n",
    "u_target = jnp.mean(u_target, axis=-1)\n",
    "\n",
    "# downsample to 40 x 40\n",
    "u_target = u_target[::10, ::10]\n",
    "n_grid = 40\n",
    "\n",
    "plt.imshow(u_target)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def animation_loss(\n",
    "    u : jnp.ndarray, \n",
    "    target : jnp.ndarray, \n",
    "    dt : float, \n",
    "    dx : float, \n",
    "    dy : float, \n",
    "    iterations : int) -> float:\n",
    "\n",
    "    for i in range(iterations):\n",
    "        u = mac_cormack_2d(u, dt, dx, dy)\n",
    "\n",
    "    return jnp.mean((u - target)**2)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "loss_and_grad = value_and_grad(animation_loss, argnums=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Loss: 15461.716796875\n",
      "Loss: 11654.669921875\n",
      "Loss: 9771.9375\n",
      "Loss: 8510.9755859375\n",
      "Loss: 7604.91650390625\n",
      "Loss: 6932.95556640625\n",
      "Loss: 6424.728515625\n",
      "Loss: 6034.466796875\n",
      "Loss: 5730.72607421875\n",
      "Loss: 5491.232421875\n",
      "Loss: 5299.9013671875\n",
      "Loss: 5144.9658203125\n",
      "Loss: 5017.73876953125\n",
      "Loss: 4911.7568359375\n",
      "Loss: 4822.1826171875\n",
      "Loss: 4745.376953125\n",
      "Loss: 4678.587890625\n",
      "Loss: 4619.724609375\n",
      "Loss: 4567.1923828125\n",
      "Loss: 4519.76611328125\n",
      "Loss: 4476.50341796875\n",
      "Loss: 4436.6748046875\n",
      "Loss: 4399.70703125\n",
      "Loss: 4365.15673828125\n",
      "Loss: 4332.669921875\n",
      "Loss: 4301.9658203125\n",
      "Loss: 4272.8193359375\n",
      "Loss: 4245.0478515625\n",
      "Loss: 4218.5029296875\n",
      "Loss: 4193.06103515625\n",
      "Loss: 4168.62060546875\n",
      "Loss: 4145.095703125\n",
      "Loss: 4122.41357421875\n",
      "Loss: 4100.50927734375\n",
      "Loss: 4079.3310546875\n",
      "Loss: 4058.829833984375\n",
      "Loss: 4038.96435546875\n",
      "Loss: 4019.69580078125\n",
      "Loss: 4000.99169921875\n",
      "Loss: 3982.822509765625\n",
      "Loss: 3965.157470703125\n",
      "Loss: 3947.974853515625\n",
      "Loss: 3931.25\n",
      "Loss: 3914.961181640625\n",
      "Loss: 3899.089599609375\n",
      "Loss: 3883.61669921875\n",
      "Loss: 3868.52392578125\n",
      "Loss: 3853.79736328125\n",
      "Loss: 3839.420654296875\n",
      "Loss: 3825.381103515625\n",
      "Loss: 3811.66357421875\n",
      "Loss: 3798.257080078125\n",
      "Loss: 3785.149169921875\n",
      "Loss: 3772.329833984375\n",
      "Loss: 3759.786865234375\n",
      "Loss: 3747.51123046875\n",
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      "Loss: 2859.820556640625\n",
      "Loss: 2858.63232421875\n"
     ]
    }
   ],
   "source": [
    "dt = 1.0\n",
    "dx = 1.0\n",
    "dy = 1.0\n",
    "\n",
    "u0 = jnp.zeros((n_grid, n_grid), dtype=jnp.float32)\n",
    "u0s = []\n",
    "\n",
    "optimization_steps = 300\n",
    "animation_steps = 50\n",
    "\n",
    "learning_rate = 1500\n",
    "\n",
    "for i in range(optimization_steps):\n",
    "    loss, grad = loss_and_grad(u0, u_target, dt, dx, dy, animation_steps)\n",
    "    u0s.append(u0.copy())\n",
    "    \n",
    "    u0 -= learning_rate * grad\n",
    "\n",
    "    print(f\"Loss: {loss}\")\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "MovieWriter imagemagick unavailable; using Pillow instead.\n"
     ]
    }
   ],
   "source": [
    "us = []\n",
    "\n",
    "animation_steps = 50\n",
    "u = u0s[-1]\n",
    "for i in range(animation_steps):\n",
    "    u = mac_cormack_2d(u, dt, dx, dy)\n",
    "    us.append(u.copy())\n",
    "\n",
    "fig = plt.figure()\n",
    "\n",
    "imp = plt.imshow(us[0], animated=True)\n",
    "plt.close()\n",
    "\n",
    "def updatefig(i):\n",
    "    imp.set_array(us[i] * 10)\n",
    "    return imp,\n",
    "\n",
    "ani = animation.FuncAnimation(fig, updatefig, frames=range(animation_steps), interval=50, blit=True)\n",
    "# video\n",
    "# HTML(ani.to_jshtml())\n",
    "# export to gif\n",
    "ani.save(\"img/frog.gif\", writer=\"imagemagick\", fps=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "pbdl",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.9"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}