project 2 ML added

doc/src/Projects/2024/Project2/Project2ML.do.txt

@@ -48,7 +48,10 @@ two electrons (or bosons) in a quantum dot with a frequency of $\hbar\omega = 1$
 The reason for this is that we have exact closed form expressions 
 for the ground state energy from Taut's work for selected values of $\omega$, 
 see M. Taut, Phys. Rev. A \textbf{48}, 3561 (1993).
-The energy is given by $3$ a.u.  (atomic units) when the interaction between the electrons is included. We can however easily extend our system to say interacting bosons, and in particualr to more than two,  as we did in project 1.
+The energy is given by $3$ a.u.  (atomic units) when the interaction between the electrons is included. We can however easily extend our system to say interacting bosons, and in particular to more than two,  as we did in project 1.
+Thus, _ss an alternative, you can replace the electron system with the same system of Bosons from project 1._
+
+
 
 If only the harmonic oscillator part of the Hamiltonian is included,
 the so-called unperturbed part,
@@ -81,6 +84,10 @@ this specific total spin.
 
 Many of the needed details can be found in the "lecture notes on Boltzmann machines and neural networks":"http://compphysics.github.io/ComputationalPhysics2/doc/LectureNotes/_build/html/boltzmannmachines.html". We recommend also to take a look at the code at the end of these notes.
 
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+
+
+
 === Representing the wave function with a neural network ===
 
 Our neural network of choice is the restricted Boltzmann machine. It
@@ -148,6 +155,7 @@ If $\sigma_i = \sigma$ then
 Here $\mathbf{X}$ are the visible nodes (the position coordinates), $\mathbf{H}$ are the hidden nodes, $\mathbf{a}$ are the visible biases, $\mathbf{b}$ are the hidden biases and $\mathbf{W}$ is a matrix containing the weights characterizing the connection of each visible node to a hidden node.
 
 === The Wave Function ===
+
 To find the marginal probability $F_{rbm}(X)$ we set:
 !bt
 \begin{align}
@@ -230,6 +238,7 @@ Add importance sampling to improve your method. Document the results and compare
 Include a proper statistical analysis by use of the blocking method for your results.
 
 
+
 === Project 2 e): Interaction ===
 
 Include the interaction. Remember that for the interacting case we
@@ -238,7 +247,7 @@ dimensions (the energy shoud be 3 a.u.).
 As before, experiment with the learning rate and
 number of hidden values and document how well the network reproduces
 the analytical value.
-For bosons you can easily extend the code in order to handle more particles. For fermions we would need to include properly the anti-symmetry of the wave function. 
+For bosons you can easily extend the code in order to handle more particles. For fermions we would need to include properly the anti-symmetry of the wave function. As an optional part, you could use the same Hamiltonian for Bosons as you did in project 1.
 
 === Project 2 f): Replacing a Boltzmann machine with a neural network (optional part) ===