Introduction to the artificial neural network-based variational Monte Carlo method

Abstract

In this self-contained tutorial, the variational Monte Carlo method with trial wave functions based on artificial neural networks is detailed. Unfolding the historical background we illustrate how machine learning interacts with physics research. The mathematical tools for both, artificial neural networks and variational Monte Carlo, are introduced. We demonstrate, how the algorithm operates with generic examples from chemical physics, including the harmonic, the Morse, the Poschl-Teller and the Yukawa potential, as well as the simplest molecules, the hydrogen molecular ion and the hydrogen molecule.

Figures (8)

• Figure 1: Schematic representations of a artificial neuron (left) and a FFNN (right).
• Figure 2: Analogies between the variational method and machine learning regression showing the equivalent components in each method.
• Figure 3: Minimization process for the harmonic oscillator potential showing the energy evolution as a function of the iterations. The inset presents the analytical ground state and optimized trial state.
• Figure 4: Minimization process for the Morse oscillator. The main graph shows the energy progression as a function of the number of iterations. The inset presents the analytical ground state and optimized trial state.
• Figure 5: Minimization process for the Poschl-Teller potential. The main figure shows energy values as a function of optimization iterations. The inset presents the analytical ground state and optimized trial state.