Solving Quantum Mechanical Many-body Problems with Machine Learning Algorithms
Contents
What is this talk about?
More material
Why? Basic motivation
Overview
What is Machine Learning?
What are the Machine Learning calculations here based on?
A new world
Lots of room for creativity
Knowledge of Statistical analysis and optimization of data
Some members of the ML family
What are the basic ingredients?
Types of Machine Learning
References
Fitting Nuclear Masses
Organizing our data
Artificial neurons
A simple perceptron model
Neural network types
The system: two electrons in a harmonic oscillator trap in two dimensions
Quantum Monte Carlo Motivation
Quantum Monte Carlo Motivation
Quantum Monte Carlo Motivation
Quantum Monte Carlo
The trial wave function
The correlation part of the wave function
Resulting ansatz
The VMC code
Technical aspect, improvements and how to define the cost function
Energy derivatives
Derivatives of the local energy
How do we define our cost function?
Meet the variance and its derivatives
The variance defines the cost function
The code for two electrons in two dims with no Coulomb interaction
Why Boltzmann machines?
Boltzmann Machines
Some similarities and differences from DNNs
Boltzmann machines (BM)
A standard BM setup
The structure of the RBM network
The network
Goals
Joint distribution
Network Elements, the energy function
Defining different types of RBMs
Sampling: Metropolis sampling
RBMs for the quantum many body problem
Choose the right RBM
Representing the wave function
Choose the cost/loss function
Running the codes
Energy as function of iterations, \( N=2 \) electrons
Energy as function of iterations, no Physics info \( N=2 \) electrons
Energy as function of iterations, \( N=6 \) electrons
Wave function analysis, onebody densities \( N=2 \)
Wave function analysis, onebody densities \( N=6 \)
Wave function analysis, onebody densities \( N=12 \)
Conclusions and where do we stand
Appendix: Mathematical details
Marginal Probability Density Functions
Conditional Probability Density Functions
Python version for the two non-interacting particles
Sampling: Metropolis sampling
In order to sample from the RBM probability distribution it is common to use Markov Chain Monte Carlo (MCMC) algorithms such as Metropolis-Hastings or Gibbs sampling.
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