ODEBlackBoxSolver (2).ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "ODEBlackBoxSolver.ipynb",
      "version": "0.3.2",
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "metadata": {
        "id": "MfLXoqSBj7zw",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import matplotlib\n",
        "%matplotlib inline"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "xsKxDb9ykHen",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "```\n",
        "Defining exponential growth equation\n",
        "\n",
        "```\n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "Nj0d5Sw4kNSf",
        "colab_type": "code",
        "outputId": "a773040b-54a5-4a0a-d470-5feb9b0eb606",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 297
        }
      },
      "cell_type": "code",
      "source": [
        "t=np.linspace(0,1,100)\n",
        "λ=5\n",
        "f= lambda t: np.exp(λ*t)\n",
        "plt.plot(t,f(t))\n",
        "plt.xlabel('t')\n",
        "plt.ylabel('f(t)')"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0, 0.5, 'f(t)')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 2
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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W93pyqKksoq6qmLqqYubMKGV2dTGV5QW6zYPIOAoDSSlD/iAdPcN09AzR0T1M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\nQEREUBiIiAgKAxERAf4/Ta1DblyXOq0AAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "O5v6xM7dkXcs",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Defining step functions for methods and gradient function"
      ]
    },
    {
      "metadata": {
        "id": "zp5kdMaCkQb2",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def grad_fn(tk,hk):\n",
        "    return λ*hk\n",
        "\n",
        "def euler_step_fn(dt,tk,hk,fun):\n",
        "    return hk+dt*fun(tk,hk)\n",
        "\n",
        "def mid_pt_step(δt,tk,hk,fun):\n",
        "    k1=fun(tk,hk)\n",
        "    k2=fun(tk+δt,hk+δt*k1)\n",
        "    return hk + δt*(k1+k2)/2\n",
        "  \n",
        "def Runge_kutta_step(δt,tk,hk,fun):\n",
        "    k1=fun(tk,hk)\n",
        "    k2=fun(tk+δt/2,hk+ δt*k1/2)\n",
        "    k3=fun(tk+δt/2,hk+ δt*k2/2)\n",
        "    k4=fun(tk+δt,hk+δt*k3)\n",
        "    return hk+ (1.0/6.0)*δt*(k1+2*k2+2*k3+k4)  "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "tD0EWs8Wksin",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Defining BlackBox ODE solver"
      ]
    },
    {
      "metadata": {
        "id": "wI4ZSZBwkyCv",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def odeint(func,y0,t,solver):\n",
        "    δts=t[1:]-t[:-1]\n",
        "    tk=t[0]\n",
        "    fk=y0\n",
        "    solp=[(tk,fk)]\n",
        "    for δt in δts:\n",
        "        fk=solver(δt,tk,fk,func)\n",
        "        tk=tk+δt\n",
        "        solp.append((tk,fk))\n",
        "    return solp"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "MC3F_VcDk2VO",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Solving using Euler's method,Mid point method and Runge Kutta's Fourth Order method"
      ]
    },
    {
      "metadata": {
        "id": "JseHwThCk_3T",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "tp=np.linspace(0,1,15)\n",
        "sol_euler=odeint(grad_fn,1.0,tp,euler_step_fn)\n",
        "sol_midpt=odeint(grad_fn,1.0,tp,mid_pt_step)\n",
        "sol_rk4=odeint(grad_fn,1.0,tp,Runge_kutta_step)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "EpX1zucIlEgS",
        "colab_type": "code",
        "outputId": "d82f8e00-4970-4909-efe8-fee7480ca734",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 297
        }
      },
      "cell_type": "code",
      "source": [
        "plt.plot(*np.array(sol_euler).T,'-.',label='euler')\n",
        "plt.plot(*np.array(sol_midpt).T,'-.',label='Mid point')\n",
        "plt.plot(*np.array(sol_rk4).T,'-.',label='RungeKutta')\n",
        "plt.plot(t,f(t),label='exact')\n",
        "plt.xlabel('t')\n",
        "plt.ylabel('f(t)')\n",
        "plt.legend()"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7fb3f7bb1390>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 6
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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WsW4LK1cdo80STRStGGe18q53B1G2SO4sWkRqWHBmjGclh5OZFM6t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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "T4oFey_YdQlK",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Integration with neural network"
      ]
    },
    {
      "metadata": {
        "id": "eqIP6nmxdU93",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "import tensorflow as tf\n",
        "from tensorflow.keras.layers import Dense\n",
        "tf.enable_eager_execution()\n",
        "keras=tf.keras\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "8TWL_CZHkqgT",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def grad_fn(tk,hk):\n",
        "    return λ*hk\n",
        "\n",
        "def euler_step_fnk(dt,tk,hk,fun):\n",
        "    return hk+dt*fun([tk,hk])\n",
        "\n",
        "def mid_pt_stepk(δt,tk,hk,fun):\n",
        "    k1=fun([tk,hk])\n",
        "    k2=fun([tk+δt,hk+δt*k1])\n",
        "    return hk + δt*(k1+k2)/2\n",
        "  \n",
        "def Runge_kutta_stepk(δt,tk,hk,fun):\n",
        "    k1=fun([tk,hk])\n",
        "    k2=fun([tk+δt/2,hk+ δt*k1/2])\n",
        "    k3=fun([tk+δt/2,hk+ δt*k2/2])\n",
        "    k4=fun([tk+δt,hk+δt*k3])\n",
        "    return hk+ (1.0/6.0)*δt*(k1+2*k2+2*k3+k4)  "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "9be2FdiFdf7n",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "class Module(keras.Model):\n",
        "    def __init__(self, nf):\n",
        "        super(Module, self).__init__()        \n",
        "        self.dense_1 = Dense(nf, activation='tanh')\n",
        "        self.dense_2 = Dense(nf, activation='tanh')\n",
        "\n",
        "    def call(self, inputs, **kwargs):\n",
        "        t,x = inputs\n",
        "        h = self.dense_1(x)\n",
        "        return self.dense_2(h) - 0.25 * x"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "HLIH-zsed3Y7",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "model = Module(2)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "kdQn1LyQd5ww",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "t_grid = np.linspace(0, 500., 2000)\n",
        "h0 = tf.to_float([[1.0, -1.0]])\n",
        "model = Module(2)\n",
        "hist = odeint(model, h0, t_grid, Runge_kutta_stepk)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "xTMDtn3DefmX",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "\n",
        "h_hist = np.array([h.numpy()[0, :] for t, h in hist])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "VRtM9dRsemjy",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 297
        },
        "outputId": "6d38d838-a09c-485a-8692-63ab6d569a15"
      },
      "cell_type": "code",
      "source": [
        "\n",
        "plt.plot(t_grid, h_hist[:, 0], label=\"h[t, 0]\", lw=2)\n",
        "plt.plot(t_grid, h_hist[:, 1], label=\"h[t, 1]\", lw=2)\n",
        "plt.xlabel(\"time\")\n",
        "plt.legend()"
      ],
      "execution_count": 57,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7fb3e1b43240>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 57
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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FKiIy2jJqwVtr1xhj1htj1uAMeX3GGHMDcNRa+yjwaeDhxOG/tNb+KSulTSGG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\nKAW8iIhHKeBFRDzqfwHILEcx4+suPQAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "UVdo2UsBerod",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}