{"id":16915,"date":"2018-06-07T04:00:21","date_gmt":"2018-06-07T04:00:21","guid":{"rendered":"https:\/\/data-flair.training\/blogs\/?p=16915"},"modified":"2021-05-14T11:00:13","modified_gmt":"2021-05-14T05:30:13","slug":"tensorflow-pde","status":"publish","type":"post","link":"https:\/\/data-flair.training\/blogs\/tensorflow-pde\/","title":{"rendered":"Setup – TensorFlow PDE (Partial Differentiation Equation)"},"content":{"rendered":"

Today is another tutorial of applied mathematics with TensorFlow<\/strong>, where you\u2019ll be learning how to solve partial differential equations (PDE) using the machine learning<\/strong> library. <\/span><\/p>\n

Moreover, in this TensorFlow PDE tutorial, we will be going to learn the setup and convenience\u00a0function for Partial Differentiation Equation. Also, we will see TensorFlow PDE simulation with codes and examples.<\/span><\/p>\n

So, let’s start TensorFlow PDE (Partial Differentiation Equation) tutorial.<\/p>\n

What is PDE (Partial Differential Equation)?<\/span><\/h2>\n

This TensorFlow tutorial is for those who already have an understanding of partial differential equations. But don\u2019t worry if you\u2019re not familiar with the topic. Here\u2019s a great PDE tutorial link that you can go through to know about PDEs:<\/span><\/p>\n

As you have already seen in the previous tutorials, TensorFlow isn’t just for machine learning. Here, in this tutorial, you are presented with an example of using TensorFlow for simulating the behaviour of a\u00a0<\/span>partial differential equation<\/strong>. <\/span><\/p>\n

You will be simulating the surface of the square pond as a few raindrops land on it.<\/span><\/p>\n

Setup for Partial Differentiation Equation<\/span><\/h2>\n

Some necessary imports in TensorFlow PDE:<\/span><\/p>\n

#Import libraries for simulation\r\nimport tensorflow as tf\r\nimport numpy as np\r\n#Imports for visualization\r\nimport PIL.Image\r\nfrom io import BytesIO\r\nfrom IPython.display import clear_output, Image, display\r\n<\/pre>\n
Function for displaying the pond’s surface as an image:<\/p>\n
def DisplayArray(a, fmt='jpeg', rng=[0,1]):\r\n  \"\"\"Display an array as a picture.\"\"\"\r\n  a = (a - rng[0])\/float(rng[1] - rng[0])*255\r\n  a = np.uint8(np.clip(a, 0, 255))\r\n  f = BytesIO()\r\n  PIL.Image.fromarray(a).save(f, fmt)\r\n  clear_output(wait = True)\r\n  display(Image(data=f.getvalue()))\r\n<\/pre>\n
Again, you will be making use of an interactive session, but it is not compulsory.
\nsess = tf.InteractiveSession()<\/strong><\/p>\n
Convenience Functions in TensorFlow PDE<\/span><\/h2>\n
def<\/span> make_kernel(a):<\/span><\/p>\n
\"\"\"Transform a 2D array into a convolution kernel\"\"\"\r\n  a = np.asarray(a)\r\n  a = a.reshape(list(a.shape) + [1,1])\r\n  return tf.constant(a, dtype=1)<\/pre>\n
def<\/span> simple_conv(x, k):<\/span><\/p>\n
\"\"\"A simplified 2D convolution operation\"\"\"\r\n x = tf.expand_dims(tf.expand_dims(x, 0), -1)\r\n y = tf.nn.depthwise_conv2d(x, k, [1, 1, 1, 1], padding='SAME')\r\n return y[0, :, :, 0]\r\n<\/pre>\n
def<\/span> laplace(x):<\/span><\/p>\n
\"\"\"Compute the 2D laplacian of an array\"\"\"\r\n  laplace_k = make_kernel([[0.5, 1.0, 0.5],\r\n                           [1.0, -6., 1.0],\r\n                           [0.5, 1.0, 0.5]])\r\n  return simple_conv(x, laplace_k)<\/pre>\n
Defining the Partial Differential Equations<\/h2>\n
The pond is a perfect 500 x 500 square.<\/span>
\nN = <\/span>500<\/span>
\nHitting the pond with some raindrops:<\/span><\/p>\n
# Initial Conditions -- some rain drops hit a pond\r\n# Set everything to zero\r\nu_init = np.zeros([N, N], dtype=np.float32)\r\nut_init = np.zeros([N, N], dtype=np.float32)\r\n# Some rain drops hit a pond at random points\r\nfor n in range(40):\r\n  a,b = np.random.randint(0, N, 2)\r\n  u_init[a,b] = np.random.uniform()\r\nDisplayArray(u_init, rng=[-0.1, 0.1])<\/pre>\n
\"TensorFlow<\/a>
TensorFlow PDE- Defining the Partial Differentiation Equation<\/p><\/div>\n
Specifying the details of the PDE:<\/span><\/p>\n
# Parameters:\r\n# eps -- time resolution\r\n# damping -- wave damping\r\neps = tf.placeholder(tf.float32, shape=())\r\ndamping = tf.placeholder(tf.float32, shape=())\r\n# Create variables for simulation state\r\nU  = tf.Variable(u_init)\r\nUt = tf.Variable(ut_init)\r\n# Discretized PDE update rules\r\nU_ = U + eps * Ut\r\nUt_ = Ut + eps * (laplace(U) - damping * Ut)\r\n# Operation to update the state\r\nstep = tf.group(\r\n  U.assign(U_),\r\n  Ut.assign(Ut_))<\/pre>\n
 <\/p>\n
Simulation in PDE<\/span><\/h2>\n
Using a simple for loop, running the time forward:<\/span><\/p>\n
# Initialize state to initial conditions\r\ntf.global_variables_initializer().run()\r\n# Run 1000 steps of PDE\r\nfor i in range(1000):\r\n  # Step simulation\r\n  step.run({eps: 0.03, damping: 0.04})\r\n  DisplayArray(U.eval(), rng=[-0.1, 0.1])\r\n<\/pre>\n
\"TensorFlow<\/a>
TensorFlow PDE- Simulation in PDE<\/p><\/div>\n
You\u2019ll get an image like this representing the ripples.<\/span>
\nSo, this was all about PDE (Partial Differentiation Equation) using machine learning in TensorFlow. Hope you like our explanation.<\/p>\n
Conclusion<\/span><\/h2>\n
Hence, in this TensorFlow PDE tutorial, we saw Partial Differential Equations can be implemented using other libraries as well including Theano and Numpy and as shown here, using TensorFlow of course. <\/span><\/p>\n
There are many other mathematical simulations that can be represented using this machine learning library such as complex calculus and different kinds of probability distributions. Next up, is embedding in TensorFlow<\/strong>. <\/span><\/p>\n
Furthermore, if you have any query regarding PDE in TensorFlow, feel free to ask through the comment section.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"
Today is another tutorial of applied mathematics with TensorFlow, where you\u2019ll be learning how to solve partial differential equations (PDE) using the machine learning library. Moreover, in this TensorFlow PDE tutorial, we will be...<\/p>\n","protected":false},"author":6,"featured_media":16936,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[73],"tags":[9419,9451,12771,12899,14592,14608],"class_list":["post-16915","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tensorflow","tag-partial-differentiation-in-tensorflow","tag-pde-in-tensorflow","tag-set-up-for-pde","tag-simulating-in-tensorflow","tag-tensorflow-pde","tag-tensorflow-tutorial"],"yoast_head":"\nSetup - TensorFlow PDE (Partial Differentiation Equation) - DataFlair<\/title>\n<meta name=\"description\" content=\"TensorFlow PDE,PDE set up &Convenience Functions,Defining Partial Differential Equation,Simulation in PDE,solve differential equation by machine learning\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/data-flair.training\/blogs\/tensorflow-pde\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Setup - 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