neural-pde.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving differential equations using Neural networks\n",
    "\n",
    "In this project, we attempt to solve ordinary differntial equations using Neural Networks. The Cybenko theorem states that an aritifical neural network with one hidden layer can approximate any continuous function on $\\mathbb{R}^n$ under some mild assumptions on the activate function (for example the Sigmoid activation function). In essence we can discreteize the domain and try and minimize a trial solution involving terms that satsify the boundary condition as well as the output of a neural network. We then need to optimize the parameters such that the LHS of the differential equation written as LHS  = 0 is minimized. The inputs are different discretized points in the domain over which we seek the solution.  \n",
    "\n",
    "See this paper here for a preliminary introduction: http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import torch\n",
    "import torch.nn as nn\n",
    "import torch.optim as optim\n",
    "from torch.utils.data import Dataset, DataLoader\n",
    "device = torch.device('cuda:0' if torch.cuda.is_available() else 'cpu')\n",
    "\n",
    "from tqdm import tqdm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this example, we are going to solve the 2D Laplace equation $\\nabla^2 \\psi(x) = 0$ over the domain $(x_1, x_2) \\in [0,1] \\times [0,1]$ under the boundary conditions $\\psi(x) = 0 $ when $x_1=0, x_1=1$ and $x_2=0$, and $\\psi(x)=\\sin \\pi x$ when $x_2 = 1$ for all $x_1$. First we discretize the domain and create a meshgrid of points. We will convert the meshgrid into a numpy array of tuples so that we can later use this is a DataLoader object to generate this one by one rather than holding them all in memory. It also helps us implement batching nicely."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 250\n",
    "\n",
    "x1 = np.linspace(0, 1, N, endpoint=True)\n",
    "x2 = np.linspace(0, 1, N, endpoint=True)\n",
    "\n",
    "X1, X2 = np.meshgrid(x1, x2)\n",
    "pts = np.vstack([X1.ravel(), X2.ravel()]).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next define a dataset class and create the dataloader object"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "class pt_generator(Dataset):\n",
    "\n",
    "    def __init__(self, pts):\n",
    "        super(pt_generator, self).__init__()\n",
    "        self.pts = pts\n",
    "\n",
    "    def __len__(self):\n",
    "        return self.pts.shape[0]\n",
    "\n",
    "    def __getitem__(self, idx):\n",
    "        p = torch.Tensor([self.pts[idx]])\n",
    "        o = torch.ones((1,1))\n",
    "        t = torch.cat((p, o), dim=-1).view(-1, 3) # Adding a bias terms to capture constants in the solution\n",
    "        t.requires_grad_(True)\n",
    "        return t \n",
    "\n",
    "pt_dataset = pt_generator(pts)\n",
    "pt_dataloader = DataLoader(pt_dataset, batch_size=1, shuffle=True) # WE will start with just a batch size of 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we define the structure of the neural network. We choose one hidden layer with 16 neurons"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "class NET(nn.Module):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(NET, self).__init__()\n",
    "\n",
    "        self.i2h = nn.Linear(3, 16)\n",
    "        self.h2o = nn.Linear(16, 1)\n",
    "        self.sigmoid = nn.Sigmoid()\n",
    "\n",
    "    def forward(self, x):\n",
    "        x = x.view(-1)\n",
    "        x = self.sigmoid(self.i2h(x))\n",
    "        x = self.h2o(x)\n",
    "        return x\n",
    "\n",
    "net = NET()\n",
    "\n",
    "optimizer = optim.SGD(net.parameters(), lr=.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def objective_fn(x):\n",
    "    \"\"\"\n",
    "        We first need to be able to calculate the ojective function,\n",
    "        which involves the derivatives of the equation LHS=0\n",
    "        with respect to inputs. In fact we need the second order derivatives \n",
    "    \"\"\"\n",
    "    x1 = x.view(-1)[0]\n",
    "    x2 = x.view(-1)[1]\n",
    "    # The solution below was constructed based on knowledge of the boundary conditions\n",
    "    sol = x2*torch.sin(np.pi * x1) + x1*(1-x1)*x2*(1-x2)*net(x)\n",
    "    # print(sol)\n",
    "    gradient = torch.autograd.grad(sol, x, create_graph=True, retain_graph=True)[0].view(3, 1)\n",
    "    dx, dy, dc = gradient[0], gradient[1], gradient[2]\n",
    "    second_x_deriv = torch.autograd.grad(dx, x1, create_graph=True, retain_graph=True)[0]\n",
    "    second_y_deriv = torch.autograd.grad(dy, x2, create_graph=True, retain_graph=True)[0]\n",
    "    laplacian = (second_x_deriv + second_y_deriv)**2\n",
    "    return laplacian.view(1, -1)\n",
    "    # print('Does dx require grad?', dx.requires_grad)\n",
    "    # laplacian = torch.autograd.grad(gradient, x, retain_graph=True)\n",
    "    # return laplacian\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def train():\n",
    "    total_loss = 0\n",
    "    # counter = 0\n",
    "    for x in tqdm(pt_dataloader):\n",
    "        optimizer.zero_grad()\n",
    "        loss = objective_fn(x)\n",
    "        total_loss += loss\n",
    "        loss.backward()\n",
    "        optimizer.step()\n",
    "        # if counter%2500 == 0:\n",
    "        #     print('iteration: ', counter, 'loss: ', total_loss/25)\n",
    "        #     total_loss = 0\n",
    "        # counter += 1\n",
    "    return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 62500/62500 [04:10<00:00, 249.07it/s]\n"
     ]
    }
   ],
   "source": [
    "train()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NET(\n",
       "  (i2h): Linear(in_features=3, out_features=16, bias=True)\n",
       "  (h2o): Linear(in_features=16, out_features=1, bias=True)\n",
       "  (sigmoid): Sigmoid()\n",
       ")"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.save(net.state_dict(), './neural-pde-trained.pt')\n",
    "\n",
    "net_loaded = NET()\n",
    "net_loaded.load_state_dict(torch.load('./neural-pde-trained.pt'))\n",
    "net_loaded.eval()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def solution_generator():\n",
    "    with torch.no_grad():\n",
    "        N = 500\n",
    "        X = np.linspace(0, 1, N, endpoint=True)\n",
    "        Y = np.linspace(0, 1, N, endpoint=True)\n",
    "        image = np.empty((500, 500))\n",
    "        for i, x in tqdm(enumerate(X)):\n",
    "            for j, y in enumerate(Y):\n",
    "                image[j, i] = y*np.sin(np.pi * x) + x*(1-x)*y*(1-y)*(net_loaded(torch.Tensor([x, y, 1]))).item() \n",
    "        return image"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we plot this solution of the Laplace equation on a square of side 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "500it [00:25, 19.99it/s]\n"
     ]
    },
    {
     "data": {
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    "im = solution_generator()\n",
    "im_flipped = np.flip(im, axis=0)\n",
    "plt.imshow(im_flipped, cmap='magma')\n",
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    "It can be shown that this solution is very close to the full analytical solution for this problem. The advantage of using this approach is that there is a great speed up in solving the equations this way vs. doing the full blown numerical solution using finite difference or other schemes. We will be applying this technique to more complicated problems in the future and measure how much more efficient this approach can be in specific situations. "
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