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Lecture

Variational Principle in Quantum Mechanics

In course

PHYS-754: Lecture series on scientific machine learning

This lecture presents ongoing work on how scientific questions can be tackled using machine learning. Machine learning enables extracting knowledge from data computationally and in an automatized way. We will learn on examples how this is influencing the v ...

Description

This lecture introduces the variational principle in quantum mechanics, focusing on optimizing trial wave functions to approximate the ground state energy. The discussion covers the Hamiltonian operator, ground state energy, variational optimization, and Monte Carlo methods. The lecture also explores the application of deep neural networks in solving the Schrodinger equation for complex systems, emphasizing the importance of smart preprocessing and the use of known theory to fix network parameters. The presentation includes a detailed analysis of the electronic cost function, Hartree-Fock method, Slater determinants, and the impact of multi-determinants and backflow functions on computational accuracy. The lecture concludes with a discussion on the efficient computation of large molecules using neural networks and the significance of incorporating physical knowledge into machine learning models.

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Instructors (10)

Anne-Clémence Corminboeuf

David Richard Harvey

Alexander Mathis

Alexander studied pure mathematics with a minor in logic and theory of science at the Ludwig Maximilians University in Munich. For his PhD also at LMU, he worked on optimal coding approaches to elucidate the properties of grid cells. As a postdoctoral fellow with Prof. Venkatesh N. Murthy at Harvard University and Prof. Matthias Bethge at Tuebingen AI, he decided to study olfactory behaviors such as odor-guided navigation, social behaviors and the cocktail party problem in mice. During this time, he increasingly got interested sensorimotor behaviors beyond olfaction and started working on proprioception, motor adaption, as well as computer vision tools for measuring animal behavior. In his group, he is interested in elucidating how the brain gives rise to adaptive behavior. One of the major goals is to synthesize large datasets into computationally useful information. For those purposes, he develops algorithms and systems to analyze animal behavior (e.g. DeepLabCut), neural data, as well as creates experimentally testable computational models.

Philippe Schwaller

Matthieu Wyart

Giuseppe Carleo

Giuseppe Carleo is a computational quantum physicist, whose main focus is the development of advanced numerical algorithms tostudy challenging problems involving strongly interacting quantum systems.He is best known for the introduction of machine learning techniques to study both equilibrium and dynamical properties,based on a neural-network representations of quantum states, as well for the time-dependent variational Monte Carlo method.He earned a Ph.D. in Condensed Matter Theory from the International School for Advanced Studies (SISSA) in Italy in 2011.He held postdoctoral positions at the Institut d’Optique in France and ETH Zurich in Switzerland, where he alsoserved as a lecturer in computational quantum physics.In 2018, he joined the Flatiron Institute in New York City in 2018 at the Center for Computational Quantum Physics (CCQ), working as a Research Scientist and project leader, and also leading the development of the open-source project NetKet.Since September 2020 he is an assistant professor at EPFL, in Switzerland, leading the Computational Quantum Science Laboratory (CQSL).

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Related concepts (32)

Recurrent neural network

A recurrent neural network (RNN) is one of the two broad types of artificial neural network, characterized by direction of the flow of information between its layers. In contrast to uni-directional feedforward neural network, it is a bi-directional artificial neural network, meaning that it allows the output from some nodes to affect subsequent input to the same nodes. Their ability to use internal state (memory) to process arbitrary sequences of inputs makes them applicable to tasks such as unsegmented, connected handwriting recognition or speech recognition.

Artificial neural network

Artificial neural networks (ANNs, also shortened to neural networks (NNs) or neural nets) are a branch of machine learning models that are built using principles of neuronal organization discovered by connectionism in the biological neural networks constituting animal brains. An ANN is based on a collection of connected units or nodes called artificial neurons, which loosely model the neurons in a biological brain. Each connection, like the synapses in a biological brain, can transmit a signal to other neurons.

Slater determinant

In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons (or other fermions). Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.

Types of artificial neural networks

There are many types of artificial neural networks (ANN). Artificial neural networks are computational models inspired by biological neural networks, and are used to approximate functions that are generally unknown. Particularly, they are inspired by the behaviour of neurons and the electrical signals they convey between input (such as from the eyes or nerve endings in the hand), processing, and output from the brain (such as reacting to light, touch, or heat). The way neurons semantically communicate is an area of ongoing research.

Configuration interaction

Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination of Slater determinants used for the wave function. In terms of a specification of orbital occupation (for instance, (1s)2(2s)2(2p)1...), interaction means the mixing (interaction) of different electronic configurations (states).

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