For project 3, you can propose own data sets that relate to your research interests or just use existing data sets from say
As an example on applications of the methods we have discussed in this course, see the article from economy that deals with machine learning on bankruptcy data from Spain at URL":"https://link.springer.com/article/10.1007/s10614-020-10078-2". The data are most likely not accessible within short time, but it is a nice demonstration on how to use the various machine learning methods we have discussed during the semester.
The approach to the analysis of these new data sets should follow to a large extent what you did in projects 1 and 2. That is:
For Boosting, feel also free to write your own codes.
All in all, the report should follow the same pattern as the two previous ones, with abstract, introduction, methods, code, results, conclusions etc..
We propose also an alternative to the above. This is a project on using machine learning methods (neural networks mainly) to the solution of ordinary differential equations and partial differential equations, with a final twist on how to diagonalize a symmetric matrix with neural networks.
This is a field with a large interest recently, spanning from studies of turbulence in fluid mechanics and meteorology to the solution of quantum mechanical systems. As reading background you can use the slides from week 42 and/or the textbook by Yadav et al.
Note: Project 3 has an additional exercise which can give you an additional score of 20 (twenty) points. These are added to the total score from all projects. See below for the additional exercise.
Here follows a set up on how to structure your report and analyze the data you have opted for.
The first part deals with structuring and reading the data, much along the same lines as done in projects 1 and 2. Explain how the data are produced and place them in a proper context.
You need to include at least two central algorithms, or as an alternative explore methods from decisions tree to bagging, random forests and boosting. Explain the basics of the methods you have chosen to work with. This would be your theory part.
Then describe your algorithm and its implementation and tests you have performed.
Then presents your results and findings, link with existing literature and more.
Finally, here you should present a critical assessment of the methods you have studied and link your results with the existing literature.
For this variant of project 3, we will assume that you have some background in the solution of partial differential equations using finite difference schemes. We will study the solution of the diffusion equation in one dimension using a standard explicit scheme and neural networks to solve the same equations.
For the explicit scheme, you can study for example chapter 10 of the lecture notes in Computational Physics or alternative sources. For the solution of ordinary and partial differential equations using neural networks, the lectures by Kristine Baluka Hein and included in the lectures of week 42 at this course are highly recommended.
For the machine learning part you can use your own code from project 2 or the functionality of for example Tensorflow/Keras..
The physical problem can be that of the temperature gradient in a rod of length \( L=1 \) at \( x=0 \) and \( x=1 \). We are looking at a one-dimensional problem
$$ \begin{equation*} \frac{\partial^2 u(x,t)}{\partial x^2} =\frac{\partial u(x,t)}{\partial t}, t> 0, x\in [0,L] \end{equation*} $$or
$$ \begin{equation*} u_{xx} = u_t, \end{equation*} $$with initial conditions, i.e., the conditions at \( t=0 \),
$$ \begin{equation*} u(x,0)= \sin{(\pi x)} \hspace{0.5cm} 0 < x < L, \end{equation*} $$with \( L=1 \) the length of the \( x \)-region of interest. The boundary conditions are
$$ \begin{equation*} u(0,t)= 0 \hspace{0.5cm} t \ge 0, \end{equation*} $$and
$$ \begin{equation*} u(L,t)= 0 \hspace{0.5cm} t \ge 0. \end{equation*} $$The function \( u(x,t) \) can be the temperature gradient of a rod. As time increases, the velocity approaches a linear variation with \( x \).
We will limit ourselves to the so-called explicit forward Euler algorithm with discretized versions of time given by a forward formula and a centered difference in space resulting in
$$ \begin{equation*} u_t\approx \frac{u(x,t+\Delta t)-u(x,t)}{\Delta t}=\frac{u(x_i,t_j+\Delta t)-u(x_i,t_j)}{\Delta t} \end{equation*} $$and
$$ \begin{equation*} u_{xx}\approx \frac{u(x+\Delta x,t)-2u(x,t)+u(x-\Delta x,t)}{\Delta x^2}, \end{equation*} $$or
$$ \begin{equation*} u_{xx}\approx \frac{u(x_i+\Delta x,t_j)-2u(x_i,t_j)+u(x_i-\Delta x,t_j)}{\Delta x^2}. \end{equation*} $$Write down the algorithm and the equations you need to implement. Find also the analytical solution to the problem.
Implement the explicit scheme algorithm and perform tests of the solution for \( \Delta x=1/10 \), \( \Delta x=1/100 \) using \( \Delta t \) as dictated by the stability limit of the explicit scheme. The stability criterion for the explicit scheme requires that \( \Delta t/\Delta x^2 \leq 1/2 \).
Study the solutions at two time points \( t_1 \) and \( t_2 \) where \( u(x,t_1) \) is smooth but still significantly curved and \( u(x,t_2) \) is almost linear, close to the stationary state.
Study now the lecture notes on solving ODEs and PDEs with neural network and use either your own code from project 2 or the functionality of tensorflow/keras to solve the same equation as in part b). Discuss your results and compare them with the standard explicit scheme. Include also the analytical solution and compare with that.
Follow the discussion in the work of Yi et al. in the article from Computers and Mathematics with Applications 47, 1155 (2004), and use your differential equation solver with neural networks, set up a simple square, real and symmetric \( 6\times 6 \) matrix and find the eigenvalues. Compare with the solution from numerical diagonalization with standard eigenvalue solvers from linear algebra.
Finally, present a critical assessment of the methods you have studied and discuss the potential for the solving differential equations and eigenvalue problems with machine learning methods.
This exercise can be done independently of the other tasks. Here you can also choose the data set you want to use. Furthermore, you can use your codes from projects 1 and 2 as well as the codes here or simply use libraries like Scikit-Learn, Tensorflow or similar.
Your task is to perform an analysis of the bias-variance tradeoff using at least three of the main sets of algorithms we have discussed in this course. We will limit ourselves to a regression problem (fitting). However, feel free to venture into a classification problem.
You are free to choose between bootstrap for resampling (recommended) or cross-validation in order to get the best possible estimates. The methods you could study are Linear Regression (OLS, Ridge and Lasso), deep learning (feed forward neural networks and/or recurrent neural networks), Ensemble methods (decision trees, bagging, random forests and boosting) and support vector machines.
Study the bias-variance tradeoff for at least three of these sets of algorithms for a data set of your choice as function of the complexity of your model. Comment and discuss the results. Discuss the pros and cons of the various methods. Are there some methods which provide both low variance and low bias?
Hint: when you use different methods, pay attention to how you represent (and understand) the complexity of the model. For example, when using decision trees you may represent the complexity of your model by the depth of the tree.
Here follows a brief recipe and recommendation on how to write a report for each project.
The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:
Finally, we encourage you to collaborate. Optimal working groups consist of 2-3 students. You can then hand in a common report.
If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, we recommend that you install the following Python packages via pip as
For Python3, replace pip with pip3.
See below for a discussion of tensorflow and scikit-learn.
For OSX users we recommend also, after having installed Xcode, to install brew. Brew allows for a seamless installation of additional software via for example
For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution you can use pip as well and simply install Python as
etc etc.
If you don't want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely
Popular software packages written in Python for ML are
These are all freely available at their respective GitHub sites. They encompass communities of developers in the thousands or more. And the number of code developers and contributors keeps increasing.