Artificial Intelligence and Machine Learning in Physics

Morten Hjorth-Jensen [1, 2]
[1] Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, USA
[2] Department of Physics and Center for Computing in Science Education, University of Oslo, Oslo, Norway

Universidad Autonoma de Madrid, November 28, 2022


What is this talk about?

The main emphasis is to give you a short and pedestrian introduction to the whys and hows we can use (with several examples) machine learning methods in physics. And why this could (or should) be of interest. I will also try to link to educational activities if time.

Additional info

These slides are at https://mhjensenseminars.github.io/MachineLearningTalk/doc/pub/UAutonomaMadrid/html/UAutonomaMadrid-reveal.html

Many to thank

Thanks to

  • MSU: Ben Hall, Jane Kim, Julie Butler, Danny Jammoa, Nicholas Cariello, Johannes Pollanen (Expt), Niyaz Beysengulov (Expt), Dean Lee, Scott Bogner, Heiko Hergert, Matt Hirn, Huey-Wen Lin, Alexei Bazavov, Andrea Shindler and Angela Wilson

  • UiO: Stian Bilek, Robert Solli, Heine Åbø (OsloMet), Håkon Emil Kristiansen, Øyvind Schøyen Sigmundsson, Jonas Boym Flaten, Linus Ekstrøm, Oskar Leinoen, Kristian Wold (OsloMet), Lasse Vines (Expt) Andrej Kuznetsov (Expt), David Rivas Gongora (Expt) and Marianne Bathen (Expt, now PD at ETH)

  • ANL: Bryce Fore, Alessandro Lovato

  • Trento: Mauro Rigo and Francesco Pederiva
Sponsors

This work is supported by the U.S. Department of Energy, Office of Science, office of Nuclear Physics under grant No. DE-SC0021152 and U.S. National Science Foundation Grants No. PHY-1404159 and PHY-2013047 and the Norwegian ministry of Education and Research for PhD fellowships.

A simple perspective on the interface between ML and Physics





ML in Nuclear Physics

Figure 1: u

AI/ML and some statements you may have heard

  1. Fei-Fei Li on ImageNet: map out the entire world of objects (The data that transformed AI research)

  2. Russell and Norvig in their popular textbook: t_relevant to any intellectual task; it is truly a universal field_ (Artificial Intelligence, A modern approach)

  3. Woody Bledsoe puts it more bluntly: in the long run, AI is the only science (quoted in Pamilla McCorduck, Machines who think)

If you wish to have a critical read on AI/ML from a societal point of view, see Kate Crawford's recent text Atlas of AI

Here: with AI/ML we intend a collection of machine learning methods with an emphasis on statistical learning and data analysis

Scientific Machine Learning

An important and emerging field is what has been dubbed as scientific ML, see the article by Deiana et al Applications and Techniques for Fast Machine Learning in Science, arXiv:2110.13041

The authors discuss applications and techniques for fast machine learning (ML) in science – the concept of integrating power ML methods into the real-time experimental data processing loop to accelerate scientific discovery. The report covers three main areas

  1. applications for fast ML across a number of scientific domains;

  2. techniques for training and implementing performant and resource-efficient ML algorithms;

  3. and computing architectures, platforms, and technologies for deploying these algorithms.

Physics driven Machine Learning

Another hot topic is what has loosely been dubbed Physics-driven deep learning. See the recent work on Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Machine Learning, vol 3, 218 (2021).

From their abstract

A less known but powerful result is that an NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations.

Machine Learning and Physics

Machine learning is an extremely rich field, in spite of its young age. The increases we have seen during the last three decades in computational capabilities have been followed by developments of methods and techniques for analyzing and handling large date sets, relying heavily on statistics, computer science and mathematics. The field is rather new and developing rapidly.

Popular software packages written in Python for ML are

and more. 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.

Lots of room for creativity

Not all the algorithms and methods can be given a rigorous mathematical justification, opening up thereby for experimenting and trial and error and thereby exciting new developments.

A solid command of linear algebra, multivariate theory, probability theory, statistical data analysis, optimization algorithms, understanding errors and Monte Carlo methods is important in order to understand many of the various algorithms and methods.

Job market, a personal statement: A familiarity with ML is almost becoming a prerequisite for many of the most exciting employment opportunities. And add quantum computing and there you are!

Types of machine learning

The approaches to machine learning are many, but are often split into two main categories. In supervised learning we know the answer to a problem, and let the computer deduce the logic behind it. On the other hand, unsupervised learning is a method for finding patterns and relationship in data sets without any prior knowledge of the system. Some authours also operate with a third category, namely reinforcement learning. This is a paradigm of learning inspired by behavioural psychology, where learning is achieved by trial-and-error, solely from rewards and punishment.

Another way to categorize machine learning tasks is to consider the desired output of a system. Some of the most common tasks are:

  • Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.

  • Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.

  • Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.

The plethora of machine learning algorithms/methods

  1. Deep learning: Neural Networks (NN), Convolutional NN, Recurrent NN, Boltzmann machines, autoencoders and variational autoencoders and generative adversarial networks

  2. Bayesian statistics and Bayesian Machine Learning, Bayesian experimental design, Bayesian Regression models, Bayesian neural networks, Gaussian processes and much more

  3. Dimensionality reduction (Principal component analysis), Clustering Methods and more

  4. Ensemble Methods, Random forests, bagging and voting methods, gradient boosting approaches

  5. Linear and logistic regression, Kernel methods, support vector machines and more

  6. Reinforcement Learning

  7. and more

Examples of applications of ML in physics

The large amount of degrees of freedom pertain to both theory and experiment in physics. With increasingly complicated experiments that produce large amounts data, automated classification of events becomes increasingly important. Here, deep learning methods offer a plethora of interesting research avenues.

  • Reconstruction of particle trajectories or classification of events are typical examples where ML methods are being used. However, since these data can often be extremely noisy, the precision necessary for discovery in physics requires algorithmic improvements. Research along such directions, interfacing nuclear physics with AI/ML is expected to play a significant role in physics discoveries related to new facilities. The treatment of corrupted data in imaging and image processing is also a relevant topic.

  • Design of detectors represents an important area of applications for ML/AI methods in subatomic physics.

And more

  • An important application of AI/ML methods is to improve the estimation of bias or uncertainty due to the introduction of or lack of physical constraints in various theoretical models.

  • In theory, we expect to use AI/ML algorithms and methods to improve our knowledge about correlations of physical model parameters in data for quantum many-body systems. Deep learning methods show great promise in circumventing the exploding dimensionalities encountered in quantum mechanical many-body studies.

  • Merging a frequentist approach (the standard path in ML theory) with a Bayesian approach, has the potential to infer better probabilitity distributions and error estimates. As an example, methods for fast Monte-Carlo- based Bayesian computation of nuclear density functionals show great promise in providing a better understanding of nuclear data and constraining density functionals.

  • Machine Learning and Quantum Computing is a very interesting avenue to explore. See for example a recent talk by Sofia Vallecorsa.

Selected references

What are the basic ingredients?

Almost every problem in ML and data science starts with the same ingredients:

  • The dataset \( \mathbf{x} \) (could be some observable quantity of the system we are studying)

  • A model which is a function of a set of parameters \( \mathbf{\alpha} \) that relates to the dataset, say a likelihood function \( p(\mathbf{x}\vert \mathbf{\alpha}) \) or just a simple model \( f(\mathbf{\alpha}) \)

  • A so-called loss/cost/risk function \( C (\mathbf{x}, f(\mathbf{\alpha})) \) which allows us to decide how well our model represents the dataset.

We seek to minimize the function \( C (\mathbf{x}, f(\mathbf{\alpha})) \) by finding the parameter values which minimize \( \mathcal{C} \). This leads to various minimization algorithms. It may surprise many, but at the heart of all machine learning algortihms there is an optimization problem.

Unsupervised learning in nuclear physics, Argon-46 by Solli, Bazin, Kuchera, MHJ, Strauss, NIMA 1010, 165461 (2020)

Two- and three-dimensional representations of two events from the Argon-46 experiment. Each row is one event in two projections, where the color intensity of each point indicates higher charge values recorded by the detector. The bottom row illustrates a carbon event with a large fraction of noise, while the top row shows a proton event almost free of noise.





Quantum Monte Carlo and deep learning

Given a hamiltonian \( H \) and a trial wave function \( \Psi_T \), the variational principle states that the expectation value of \( \langle H \rangle \), defined through

 
$$ \langle E \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})H(\boldsymbol{R})\Psi_T(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})\Psi_T(\boldsymbol{R})}, $$

 

is an upper bound to the ground state energy \( E_0 \) of the hamiltonian \( H \), that is

 
$$ E_0 \le \langle E \rangle. $$

 

In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be adequate for say the computation of the energy of a many-body system. Basic philosophy: Let a neural network find the optimal wave function

Monte Carlo methods and Neural Networks

Machine Learning and the Deuteron by Kebble and Rios and Variational Monte Carlo calculations of \( A\le 4 \) nuclei with an artificial neural-network correlator ansatz by Adams et al. PRL 127, 022502

Adams et al:

 
$$ \begin{align} H_{LO} &=-\sum_i \frac{{\vec{\nabla}_i^2}}{2m_N} +\sum_{i < j} {\left(C_1 + C_2\, \vec{\sigma_i}\cdot\vec{\sigma_j}\right) e^{-r_{ij}^2\Lambda^2 / 4 }} \nonumber\\ &+D_0 \sum_{i < j < k} \sum_{\text{cyc}} {e^{-\left(r_{ik}^2+r_{ij}^2\right)\Lambda^2/4}}\,, \tag{1} \end{align} $$

 

where \( m_N \) is the mass of the nucleon, \( \vec{\sigma_i} \) is the Pauli matrix acting on nucleon \( i \), and \( \sum_{\text{cyc}} \) stands for the cyclic permutation of \( i \), \( j \), and \( k \). The low-energy constants \( C_1 \) and \( C_2 \) are fit to the deuteron binding energy and to the neutron-neutron scattering length

Deep learning neural networks, Variational Monte Carlo calculations of \( A\le 4 \) nuclei with an artificial neural-network correlator ansatz by Adams et al.

An appealing feature of the neural network ansatz is that it is more general than the more conventional product of two- and three-body spin-independent Jastrow functions

 
$$ \begin{align} |\Psi_V^J \rangle = \prod_{i < j < k} \Big( 1-\sum_{\text{cyc}} u(r_{ij}) u(r_{jk})\Big) \prod_{i < j} f(r_{ij}) | \Phi\rangle\,, \tag{2} \end{align} $$

 

which is commonly used for nuclear Hamiltonians that do not contain tensor and spin-orbit terms. The above function is replaced by a four-layer Neural Network.





Gnech et al, Variational Monte Carlo calculations of \( A\le 6 \) nuclei Few Body Systems 63, (2022)





Neutron matter with \( N=14 \) neutron

Bryce Fore, Jane Kim, Alessandro Lovato and MHJ, in preparation





The electron gas in three dimensions with \( N=14 \) electrons (Wigner-Seitz radius \( r_s=4 \) Ry)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





Extrapolations and model interpretability

When you hear phrases like predictions and estimations and correlations and causations, what do you think of? May be you think of the difference between classifying new data points and generating new data points. Or perhaps you consider that correlations represent some kind of symmetric statements like if \( A \) is correlated with \( B \), then \( B \) is correlated with \( A \). Causation on the other hand is directional, that is if \( A \) causes \( B \), \( B \) does not necessarily cause \( A \).

Physics based statistical learning and data analysis

The above concepts are in some sense the difference between old-fashioned machine learning and statistics and Bayesian learning. In machine learning and prediction based tasks, we are often interested in developing algorithms that are capable of learning patterns from given data in an automated fashion, and then using these learned patterns to make predictions or assessments of newly given data. In many cases, our primary concern is the quality of the predictions or assessments, and we are less concerned about the underlying patterns that were learned in order to make these predictions.

Physics based statistical learning points however to approaches that give us both predictions and correlations as well as being able to produce error estimates and understand causations. This leads us to the very interesting field of Bayesian statistics.

Bayes' Theorem

Bayes' theorem

 
$$ p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}. $$

 

The quantity \( p(Y\vert X) \) on the right-hand side of the theorem is evaluated for the observed data \( Y \) and can be viewed as a function of the parameter space represented by \( X \). This function is not necessarily normalized and is normally called the likelihood function.

The function \( p(X) \) on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.

Quantified limits of the nuclear landscape

Neufcourt et al., Phys. Rev. C 101, 044307 (2020)

Predictions made with eleven global mass model and Bayesian model averaging. Plot shows the probability of existence that the nucleus is bound with respect to proton and neutron decay.





Constraining the equation of state for dense nuclear matter

G. Raaijmakers et al., Constraining the Dense Matter Equation of State with Joint Analysis of NICER and LIGO/Virgo Measurements, AJ Letters, 893, L21 (2020)



Observations (or conclusions if you prefer)

  • Need for AI/Machine Learning in physics, lots of ongoing activities

  • To solve many complex problems and facilitate discoveries, multidisciplinary efforts efforts are required involving scientists in physics, statistics, computational science, applied math and other fields.

  • There is a need for focused AI/ML learning efforts that will benefit accelerator science and experimental and theoretical programs

  • How do we develop insights, competences, knowledge in statistical learning that can advance a given field?

    • For example: Can we use ML to find out which correlations are relevant and thereby diminish the dimensionality problem in standard many-body theories?

    • Can we use AI/ML in detector analysis, accelerator design, analysis of experimental data and more?

    • Can we use AL/ML to carry out reliable extrapolations by using current experimental knowledge and current theoretical models?

  • The community needs to invest in relevant educational efforts and training of scientists with knowledge in AI/ML. These are great challenges to the CS and DS communities

  • Quantum computing and quantum machine learning not discussed here

  • Most likely tons of things I have forgotten

Possible start to raise awareness about ML in your own field

Education

  1. Incorporate elements of statistical data analysis and Machine Learning in undergraduate programs

  2. Develop courses on Machine Learning and statistical data analysis

  3. Build up a series of courses in QIS, inspiration QuSTEAM (Quantum Information Science, Technology, Engineering, Arts and Mathematics) initiative from USA

  4. Modifying contents of present Physics programs or new programs on Computational Physics and Quantum Technologies

    1. study direction/option in quantum technologies

    2. study direction/option in Artificial Intelligence and Machine Learning

    3. and more

  5. Master of Science/PhD programs in Computational and Data Science

    1. UiO has already MSc programs in CS and DS

    2. MSU has own graduate programs plus dual degree programs in CS and DS

    3. Many other universities are developing or have similar programs

QuSTEAM Model





Possible courses

Topics in a Bachelor of Science/Master of Science

  1. General university course on quantum mech and quantum technologies

  2. Information Systems

  3. From Classical Information theory to Quantum Information theory

  4. Classical vs. Quantum Logic

  5. Classical and Quantum Laboratory

  6. Discipline-Based Quantum Mechanics

  7. Quantum Software

  8. Quantum Hardware

  9. more

Important Issues to think of

  1. Lots of conceptual learning: superposition, entanglement, QIS applications, etc.

  2. Coding is indispensable. That is why this should be a part of a CS/DS program

  3. Teamwork, project management, and communication are important and highly valued

  4. Engagement with industry: guest lectures, virtual tours, co-ops, and/or internships.

Observations

  1. Students do not really know what QIS is.

  2. ML/AI seen as black boxes/magic!

  3. Students perceive that a graduate degree is necessary to work in QIS. A BSc will help.

Future Needs/Problems

  1. There are already great needs for specialized people (Ph. D. s, postdocs), but also needs of people with a broad overview of what is possible in ML/AI and/or QIS.

  2. There are not enough potential employees in AI/ML and QIS . It is a supply gap, not a skills gap.

  3. A BSc with specialization is a good place to start

  4. It is tremendously important to get everyone speaking the same language. Facility with the vernacular of quantum mechanics is a big plus.

  5. There is a huge list of areas where technical expertise may be important. But employers are often more concerned with attributes like project management, working well in a team, interest in the field, and adaptability than in specific technical skills.

Educational initiatives since 2019 in nuclear physics

Please feel free to propose new schools and/or update the list here.
AI/ML courses

  1. The FRIB-TA Summer School: Machine Learning Applied to Nuclear Physics, FRIB/NSCL (MSU)from May 20 to 23, 2019; organizers and teachers: Matthew Hirn (MSU), Morten Hjorth-Jensen (MSU) and Michelle Kuchera (Davidson)

  2. Nuclear TALENT course Learning from Data: Bayesian Methods and Machine Learning, in York, UK, June 10-28, 2019; Teachers and organizers Christian Forssén, Chalmers University of Technology, Sweden, Dick Furnstahl, Ohio State University, USA, Daniel Phillips, Ohio University, USA

  3. Nuclear TALENT School on Machine learning from 22 June 2020 to 03 July 2020. Teachers and organizers: Daniel Bazin (MSU), Morten Hjorth-Jensen (MSU), Michelle Kuchera (Davidson), Sean Liddick (MSU), Raghuram Ramanujan (Davidson)

  4. Nuclear TALENT School on Machine learning from 19 July 2021 to 30 July 2021. Teachers and organizers: Daniel Bazin (MSU), Morten Hjorth-Jensen (MSU), Michelle Kuchera (Davidson), Sean Liddick (MSU), Raghuram Ramanujan (Davidson)

  5. Intensive course on Machine Learning at FRIB/MSU, summer 2019; teacher Morten Hjorth-Jensen, MSU

  6. Four two-week intensive course on Machine Learning for Nuclear Physics held at Ganil, France, 2019, 2020, 2021 and 2022. Teacher and organizer Morten Hjorth-Jensen (MSU)

  7. AI4NP Winter School, 11-15 Jan 2021, (Virtual). Organizers Amber Boehnlein (JLAB), Paulo Bedaque (University of Maryland), Tanja Horn (Catholic University of America)

  8. 2022?

Additional material

Examples of Machine Learning methods and applications in nuclear physics

  • Machine learning for data mining: Oftentimes, it is necessary to be able to accurately calculate observables that have not been measured, to supplement the existing databases.

  • Nuclear density functional theory: Energy density functional calibration involving Bayesian optimization and NN ML. A promising avenue for ML applications is the emulation of DFT results.

  • Nuclear properties with ML: Improving predictive power of nuclear models by emulating model residuals.

  • Effective field theory and A-body systems: Truncation errors and low-energy coupling constant calibration, nucleon-nucleon scattering calculations, variational calculations with ANN for light nuclei, NN extrapolation of nuclear structure observables

  • Nuclear shell model UQ: ML methods have been used to provide UQ of configuration interaction calculations.

Machine learning and nuclear theory: Why?

  1. ML tools can help us to speed up the scientific process cycle and hence facilitate discoveries

  2. Enabling fast emulation for big simulations

  3. Revealing the information content of measured observables w.r.t. theory

  4. Identifying crucial experimental data for better constraining theory

  5. Providing meaningful input to applications and planned measurements

  6. ML tools can help us to reveal the structure of our models

  7. Parameter estimation with heterogeneous/multi-scale datasets

  8. Model reduction

  9. ML tools can help us to provide predictive capability

  10. Theoretical results often involve ultraviolet and infrared extrapolations due to Hilbert-space truncations

  11. Uncertainty quantification essential

  12. Theoretical models are often applied to entirely new nuclear systems and conditions that are not accessible to experiment

Examples of Machine Learning methods and applications in nuclear physics, continues

  • Low-energy nuclear reactions UQ: Bayesian optimization studies of the nucleon-nucleus optical potential, R-matrix analyses, and statistical spatial networks to study patterns in nuclear reaction networks.

  • Neutron star properties and nuclear matter equation of state: constraining the equation of state by properties on neutron stars and selected properties of finite nuclei

  • Experimental design: Bayesian ML provides a framework to maximize the success of on experiment based on the best information available on existing data, experimental conditions, and theoretical models.

Addendum: Quantum Monte Carlo Motivation

Given a hamiltonian \( H \) and a trial wave function \( \Psi_T \), the variational principle states that the expectation value of \( \langle H \rangle \), defined through

 
$$ \langle E \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})H(\boldsymbol{R})\Psi_T(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})\Psi_T(\boldsymbol{R})}, $$

 

is an upper bound to the ground state energy \( E_0 \) of the hamiltonian \( H \), that is

 
$$ E_0 \le \langle E \rangle. $$

 

In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be adequate for say the computation of the energy of a many-body system.

Quantum Monte Carlo Motivation

Basic steps

Choose a trial wave function \( \psi_T(\boldsymbol{R}) \).

 
$$ P(\boldsymbol{R},\boldsymbol{\alpha})= \frac{\left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2}{\int \left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2d\boldsymbol{R}}. $$

 

This is our model, or likelihood/probability distribution function (PDF). It depends on some variational parameters \( \boldsymbol{\alpha} \). The approximation to the expectation value of the Hamiltonian is now

 
$$ \langle E[\boldsymbol{\alpha}] \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})}. $$

 

Quantum Monte Carlo Motivation

Define a new quantity

 
$$ E_L(\boldsymbol{R},\boldsymbol{\alpha})=\frac{1}{\psi_T(\boldsymbol{R},\boldsymbol{\alpha})}H\psi_T(\boldsymbol{R},\boldsymbol{\alpha}), $$

 

called the local energy, which, together with our trial PDF yields

 
$$ \langle E[\boldsymbol{\alpha}] \rangle=\int P(\boldsymbol{R})E_L(\boldsymbol{R},\boldsymbol{\alpha}) d\boldsymbol{R}\approx \frac{1}{N}\sum_{i=1}^NE_L(\boldsymbol{R_i},\boldsymbol{\alpha}) $$

 

with \( N \) being the number of Monte Carlo samples.

The trial wave function

We want to perform a Variational Monte Carlo calculation of the ground state of two electrons in a quantum dot well with different oscillator energies, assuming total spin \( S=0 \). Our trial wave function has the following form

 
$$ \begin{equation} \psi_{T}(\boldsymbol{r}_1,\boldsymbol{r}_2) = C\exp{\left(-\alpha_1\omega(r_1^2+r_2^2)/2\right)} \exp{\left(\frac{r_{12}}{(1+\alpha_2 r_{12})}\right)}, \tag{3} \end{equation} $$

 

where the variables \( \alpha_1 \) and \( \alpha_2 \) represent our variational parameters.

Why does the trial function look like this? How did we get there? This is one of our main motivations for switching to Machine Learning.

The correlation part of the wave function

To find an ansatz for the correlated part of the wave function, it is useful to rewrite the two-particle local energy in terms of the relative and center-of-mass motion. Let us denote the distance between the two electrons as \( r_{12} \). We omit the center-of-mass motion since we are only interested in the case when \( r_{12} \rightarrow 0 \). The contribution from the center-of-mass (CoM) variable \( \boldsymbol{R}_{\mathrm{CoM}} \) gives only a finite contribution. We focus only on the terms that are relevant for \( r_{12} \) and for three dimensions. The relevant local energy operator becomes then (with \( l=0 \))

 
$$ \lim_{r_{12} \rightarrow 0}E_L(R)= \frac{1}{R_T(r_{12})}\left(-2\frac{d^2}{dr_{ij}^2}-\frac{4}{r_{ij}}\frac{d}{dr_{ij}}+ \frac{2}{r_{ij}}\right)R_T(r_{12}). $$

 

In order to avoid divergencies when \( r_{12}\rightarrow 0 \) we obtain the so-called cusp condition

 
$$ \frac{d R_T(r_{12})}{dr_{12}} = \frac{1}{2} R_T(r_{12})\qquad r_{12}\to 0 $$

 

Resulting ansatz

The above results in

 
$$ R_T \propto \exp{(r_{ij}/2)}, $$

 

for anti-parallel spins and

 
$$ R_T \propto \exp{(r_{ij}/4)}, $$

 

for anti-parallel spins. This is the so-called cusp condition for the relative motion, resulting in a minimal requirement for the correlation part of the wave fuction. For general systems containing more than say two electrons, we have this condition for each electron pair \( ij \).

Energy derivatives

To find the derivatives of the local energy expectation value as function of the variational parameters, we can use the chain rule and the hermiticity of the Hamiltonian.

Let us define (with the notation \( \langle E[\boldsymbol{\alpha}]\rangle =\langle E_L\rangle \))

 
$$ \bar{E}_{\alpha_i}=\frac{d\langle E_L\rangle}{d\alpha_i}, $$

 

as the derivative of the energy with respect to the variational parameter \( \alpha_i \) We define also the derivative of the trial function (skipping the subindex \( T \)) as

 
$$ \bar{\Psi}_{i}=\frac{d\Psi}{d\alpha_i}. $$

 

Derivatives of the local energy

The elements of the gradient of the local energy are then (using the chain rule and the hermiticity of the Hamiltonian)

 
$$ \bar{E}_{i}= 2\left( \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L \rangle\right). $$

 

From a computational point of view it means that you need to compute the expectation values of

 
$$ \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle, $$

 

and

 
$$ \langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L\rangle $$

 

These integrals are evaluted using MC intergration (with all its possible error sources). We can then use methods like stochastic gradient or other minimization methods to find the optimal variational parameters (I don't discuss this topic here, but these methods are very important in ML).

How do we define our cost function?

We have a model, our likelihood function.

How should we define the cost function?

Meet the variance and its derivatives

Why the variance?

Suppose the trial function (our model) is the exact wave function. The action of the hamiltionan on the wave function

 
$$ H\Psi = \mathrm{constant}\times \Psi, $$

 

The integral which defines various expectation values involving moments of the hamiltonian becomes then

 
$$ \langle E^n \rangle = \langle H^n \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})H^n(\boldsymbol{R})\Psi(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})\Psi(\boldsymbol{R})}= \mathrm{constant}\times\frac{\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})\Psi(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})\Psi(\boldsymbol{R})}=\mathrm{constant}. $$

 
This gives an important information: If I want the variance, the exact wave function leads to zero variance!

The variance is defined as

 
$$ \sigma_E = \langle E^2\rangle - \langle E\rangle^2. $$

 

Variation is then performed by minimizing both the energy and the variance.

The variance defines the cost function

We can then take the derivatives of

 
$$ \sigma_E = \langle E^2\rangle - \langle E\rangle^2, $$

 

with respect to the variational parameters. The derivatives of the variance can then be used to defined the so-called Hessian matrix, which in turn allows us to use minimization methods like Newton's method or standard gradient methods.

This leads to however a more complicated expression, with obvious errors when evaluating integrals by Monte Carlo integration. Less used, see however Filippi and Umrigar. The expression becomes complicated

 
$$ \begin{align} \bar{E}_{ij} &= 2\left[ \langle (\frac{\bar{\Psi}_{ij}}{\Psi}+\frac{\bar{\Psi}_{j}}{\Psi}\frac{\bar{\Psi}_{i}}{\Psi})(E_L-\langle E\rangle)\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\bar{E}_j-\langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\bar{E}_i\right] \tag{4}\\ \nonumber &+\langle \frac{\bar{\Psi}_{i}}{\Psi}E_L{_j}\rangle +\langle \frac{\bar{\Psi}_{j}}{\Psi}E_L{_i}\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L{_j}\rangle \langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\langle E_L{_i}\rangle. \end{align} $$

 

Evaluating the cost function means having to evaluate the above second derivative of the energy.

Why Boltzmann machines?

What is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately. One of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.

The original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images.

Furthermore, they have been used to solve complicated quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models.

A standard BM setup

A standard BM network is divided into a set of observable and visible units \( \hat{x} \) and a set of unknown hidden units/nodes \( \hat{h} \).

Additionally there can be bias nodes for the hidden and visible layers. These biases are normally set to \( 1 \).

BMs are stackable, meaning we can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning

However, they are often hard to train. This leads to the introduction of so-called restricted BMs, or RBMS. Here we take away all lateral connections between nodes in the visible layer as well as connections between nodes in the hidden layer. The network is illustrated in the figure below.

The structure of the RBM network





The network

The network layers:

  1. A function \( \mathbf{x} \) that represents the visible layer, a vector of \( M \) elements (nodes). This layer represents both what the RBM might be given as training input, and what we want it to be able to reconstruct. This might for example be the pixels of an image, the spin values of the Ising model, or coefficients representing speech.

  2. The function \( \mathbf{h} \) represents the hidden, or latent, layer. A vector of \( N \) elements (nodes). Also called "feature detectors".

Joint distribution

The restricted Boltzmann machine is described by a Boltzmann distribution

 
$$ \begin{align} P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}, \tag{5} \end{align} $$

 

where \( Z \) is the normalization constant or partition function, defined as

 
$$ \begin{align} Z = \int \int e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})} d\mathbf{x} d\mathbf{h}. \tag{6} \end{align} $$

 

It is common to ignore \( T_0 \) by setting it to one.

Defining different types of RBMs

There are different variants of RBMs, and the differences lie in the types of visible and hidden units we choose as well as in the implementation of the energy function \( E(\mathbf{x},\mathbf{h}) \).

Binary-Binary RBM:

RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:

 
$$ \begin{align} E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j, \tag{7} \end{align} $$

 

where the binary values taken on by the nodes are most commonly 0 and 1.

Gaussian-Binary RBM:

Another variant is the RBM where the visible units are Gaussian while the hidden units remain binary:

 
$$ \begin{align} E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}. \tag{8} \end{align} $$

 

Representing the wave function

The wavefunction should be a probability amplitude depending on \( \boldsymbol{x} \). The RBM model is given by the joint distribution of \( \boldsymbol{x} \) and \( \boldsymbol{h} \)

 
$$ \begin{align} F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. \tag{9} \end{align} $$

 

To find the marginal distribution of \( \boldsymbol{x} \) we set:

 
$$ \begin{align} F_{rbm}(\mathbf{x}) &= \sum_\mathbf{h} F_{rbm}(\mathbf{x}, \mathbf{h}) \tag{10}\\ &= \frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. \tag{11} \end{align} $$

 

Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)

 
$$ \begin{align} \Psi (\mathbf{x}) &= F_{rbm}(\mathbf{x}) \tag{12}\\ &= \frac{1}{Z}\sum_{\boldsymbol{h}} e^{-E(\mathbf{x}, \mathbf{h})} \tag{13}\\ &= \frac{1}{Z} \sum_{\{h_j\}} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2} + \sum_j^N b_j h_j + \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma^2}} \tag{14}\\ &= \frac{1}{Z} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2}} \prod_j^N (1 + e^{b_j + \sum_i^M \frac{x_i w_{ij}}{\sigma^2}}). \tag{15}\\ \tag{16} \end{align} $$

 

Choose the cost/loss function

Now we don't necessarily have training data (unless we generate it by using some other method). However, what we do have is the variational principle which allows us to obtain the ground state wave function by minimizing the expectation value of the energy of a trial wavefunction (corresponding to the untrained NQS). Similarly to the traditional variational Monte Carlo method then, it is the local energy we wish to minimize. The gradient to use for the stochastic gradient descent procedure is

 
$$ \begin{align} \frac{\partial \langle E_L \rangle}{\partial \theta_i} = 2(\langle E_L \frac{1}{\Psi}\frac{\partial \Psi}{\partial \theta_i} \rangle - \langle E_L \rangle \langle \frac{1}{\Psi}\frac{\partial \Psi}{\partial \theta_i} \rangle ), \tag{17} \end{align} $$

 

where the local energy is given by

 
$$ \begin{align} E_L = \frac{1}{\Psi} \hat{\mathbf{H}} \Psi. \tag{18} \end{align} $$

 

Electrons in a harmonic oscillator trap in two dimensions

The Hamiltonian of the quantum dot is given by

 
$$ \hat{H} = \hat{H}_0 + \hat{V}, $$

 

where \( \hat{H}_0 \) is the many-body HO Hamiltonian, and \( \hat{V} \) is the inter-electron Coulomb interactions. In dimensionless units,

 
$$ \hat{V}= \sum_{i < j}^N \frac{1}{r_{ij}}, $$

 

with \( r_{ij}=\sqrt{\mathbf{r}_i^2 - \mathbf{r}_j^2} \).

This leads to the separable Hamiltonian, with the relative motion part given by (\( r_{ij}=r \))

 
$$ \hat{H}_r=-\nabla^2_r + \frac{1}{4}\omega^2r^2+ \frac{1}{r}, $$

 

plus a standard Harmonic Oscillator problem for the center-of-mass motion. This system has analytical solutions in two and three dimensions (M. Taut 1993 and 1994).

Quantum dots and Boltzmann machines, onebody densities \( N=6 \), \( \hbar\omega=0.1 \) a.u.





Onebody densities \( N=30 \), \( \hbar\omega=1.0 \) a.u.





Onebody densities \( N=30 \), \( \hbar\omega=0.1 \) a.u.





Neutron matter with \( N=14 \) neutron

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation





Neutron matter with \( N=14 \) neutron

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation





Neutron matter with \( N=14 \) neutron

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





The electron gas in three dimensions with \( N=14 \) electrons (Various Wigner-Seitz Radii)

Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation and Nordhagen, Kim, Fore, Lovato and MHJ, ArXiv 2210.00365





Neutron matter with \( N=14 \) neutron

Bryce Fore, Jane Kim, Alessandro Lovato and MHJ, in preparation





Folding and unfolding and response functions

A good read on folding/unfolding is An Unfolding Method for High Energy Physics Experiments by Volker Blobel

See notebook details and examples of simple data reconstructed with Gaussian processes.

Experimental design

Beam time and compute cycles are expensive!

  • Bayesian experimental design provides a framework in which experiments can be designed using the best experimental and theoretical information available

  • The utility function is designed to encode the goals of the experiment and the constraints inherent in carrying it out.

  • Once the utility function and the possible designs have been specified, the optimal design is simply the scenario that maximizes the expected utility function over the domain of possible designs.