viscous_burgers.ipynb

Solving Viscous Burger's Equation using TensorFlow.jl.ipynb

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    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## Burger's Equation\n\n$$u_t + uu_{x} = \\frac{0.01}{\\pi}u_{xx}$$\n\n$$u(x,0) = \\sin(\\pi x)$$\n\nReference: \nRaissi, Maziar, Paris Perdikaris, and George Em Karniadakis. \"Physics Informed Deep Learning (Part I): Data-driven solutions of nonlinear partial differential equations.\" arXiv preprint arXiv:1711.10561 (2017)."
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "using TensorFlow\nusing PyPlot",
      "execution_count": 1,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "num_layers = 10\n\nx = placeholder(Float32,shape=[-1,1])\nt = placeholder(Float32,shape=[-1,1])\npde = placeholder(Float32,shape=[-1,1])\nx0 = placeholder(Float32,shape=[-1,1])\nt0 = placeholder(Float32,shape=[-1,1])\nu0 = placeholder(Float32,shape=[-1,1])",
      "execution_count": 105,
      "outputs": [
        {
          "data": {
            "text/plain": "<Tensor placeholder_80:1 shape=(?, 1) dtype=Float32>"
          },
          "execution_count": 105,
          "metadata": {},
          "output_type": "execute_result"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "function neural(x, reuse=false)\n    variable_scope(\"U1\", reuse=reuse) do\n        W1 = get_variable(\"W0\", shape=[2,20], dtype=Float32) \n        b1 = get_variable(\"b0\", shape=[20], dtype=Float32)\n        net = x*W1 + b1\n        net = nn.tanh(net)\n        for i = 1:num_layers-2\n            W1 = get_variable(\"W$i\", shape=[20,20], dtype=Float32) \n            b1 = get_variable(\"b$i\", shape=[20], dtype=Float32)\n            net = net*W1 + b1\n            net = nn.tanh(net)\n        end\n        W1 = get_variable(\"Wend\", shape=[20,1], dtype=Float32) \n        b1 = get_variable(\"bend\", shape=[1], dtype=Float32) \n        z = net*W1 + b1\n        return z\n    end\nend",
      "execution_count": 106,
      "outputs": [
        {
          "data": {
            "text/plain": "neural (generic function with 3 methods)"
          },
          "execution_count": 106,
          "metadata": {},
          "output_type": "execute_result"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "u = neural(concat([x0,t0],2), false)\nuf = neural(concat([x,t],2), true)\nf = gradients(uf,t) + uf .* gradients(uf, x) -\n             0.01f0/3.141592654f0*gradients(gradients(uf,x),x)\nloss = reduce_sum((f-pde).^2) + reduce_sum((u0-u).^2)",
      "execution_count": 100,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# create a new session\nsess = Session()\ntrain_step = train.minimize(train.AdamOptimizer(), loss)\nrun(sess, global_variables_initializer())\n",
      "execution_count": null,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "function create_training_data(N1, N2)\n    # boundary\n    xx1 = rand(Float32,N1)\n    tt1 = zeros(Float32,N1)\n    uu1 = sin.(2*pi*xx1)\n    xx2 = zeros(Float32,N1)\n    tt2 = rand(Float32,N1)\n    uu2 = zeros(Float32,N1)\n    xx3 = ones(Float32,N1)\n    tt3 = rand(Float32,N1)\n    uu3 = zeros(Float32,N1)\n    xx = [xx1;xx2;xx3]\n    tt = [tt1;tt2;tt3]\n    uu = [uu1;uu2;uu3]\n    \n    xxx = rand(Float32,N2)\n    ttt = rand(Float32,N2)\n    fff = zeros(Float32,N2)\n    xx, tt, uu, xxx, ttt, fff = map(x->Array{Float32}(reshape(x, length(x),1)), [xx, tt, uu, xxx, ttt, fff])\n    return xx, tt, uu, xxx, ttt, fff\nend\n\nfunction testing_data(N)\n    a = LinRange{Float32}(0.0,1.0,N)\n    X = zeros(Float32,N,N)\n    Y = zeros(Float32,N,N)\n    for i = 1:N\n        for j = 1:N\n            X[i,j] = a[i]\n            Y[i,j] = a[j]\n        end\n    end\n    return X, Y\nend",
      "execution_count": null,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "xx, tt, uu, xxx, ttt, fff = create_training_data(1000,10000)\nX,Y = testing_data(200)\nLos = Float64[]\nfor i in 1:50000\n    run(sess, train_step, Dict(x=>xxx, t=>ttt, pde=>fff, x0=>xx, t0=>tt, u0=>uu))\n    los = sqrt(run(sess, loss, Dict(x=>xxx, t=>ttt, pde=>fff, x0=>xx, t0=>tt, u0=>uu)))\n    if mod(i,1000)==0\n        close(\"all\")\n        U = run(sess, u, Dict(x0=>reshape(X, length(X),1), t0=>reshape(Y, length(Y),1)));\n        pcolor(X,Y,reshape(U,200,200))\n        colorbar()\n        xlabel(\"x\")\n        ylabel(\"t\")\n        savefig(\"xt.png\")\n    end\n    push!(Los, los)\n    println(\"$i: $los\")\nend",
      "execution_count": null,
      "outputs": []
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "![title](xt.png)"
    }
  ],
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