Data Analysis and Machine Learning: Using Neural networks to solve ODEs and PDEs
Contents
Differential equations
Description of the equation to solve for
Ordinary Differential Equations
The trial solution
Minimizing the cost function using gradient descent and automatic differentiation
Example: Exponential decay and setting up the network using Autograd
The function to solve for
The trial solution
Reformulating the problem
A possible implementation of a neural network using Autograd
Backpropagation using Autograd
Gradient descent
The network with one input, hidden, and output layer
The network with one input layer, specified number of hidden layers, and one output layer output layer
Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
Setting up the problem
The trial solution
The program using Autograd
Using forward Euler to solve the ODE
Using TensorFlow to model logistic population growth
The general program flow in TensorFlow
Program flow in TensorFlow - Construction phase
Program flow in TensorFlow - Execution phase
The full program modeling logistic population growth using TensorFlow
Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
The specific equation to solve for
Solving the equation using Autograd
Comparing with a numerical scheme
Using gradient descent in TensorFlow to solve Poisson equation
Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
Partial Differential Equations
Example: The diffusion equation
Defining the problem
Setting up the network using Autograd
Setting up the network using Autograd; The trial solution
Setting up the network using Autograd; The full program
Example: Solving the wave equation using Autograd and TensorFlow
The problem to solve for
The trial solution
The analytical solution
Solving the wave equation - the full program using Autograd
Solving the wave equation - the full program using TensorFlow
Resources
Data Analysis and Machine Learning: Using Neural networks to solve ODEs and PDEs
Kristine Baluka Hein
Department of Informatics, University of Oslo, Norway
Nov 11, 2018
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