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.
Parts of this talk are based on Artificial Intelligence and Machine Learning in Nuclear Physics, Amber Boehnlein et al., Reviews Modern of Physics 94, 031003 (2022)
Jane Kim (MSU), Julie Butler (MSU), Patrick Cook (MSU), Danny Jammooa (MSU), Daniel Bazin (MSU), Dean Lee (MSU), Witek Nazarewicz (MSU), Michelle Kuchera (Davidson College), Even Nordhagen (UiO), Robert Solli (UiO, Expert Analytics), Bryce Fore (ANL), Alessandro Lovato (ANL), Stefano Gandolfi (LANL), Francesco Pederiva (UniTN), and Giuseppe Carleo (EPFL). Niyaz Beysengulov and Johannes Pollanen (experiment, MSU); Zachary Stewart, Jared Weidman, and Angela Wilson (quantum chemistry, MSU) Jonas Flaten, Oskar, Leinonen, Øyvind Sigmundson Schøyen, Stian Dysthe Bilek, and Håkon Emil Kristiansen (UiO). Marianne Bathen and Lasse Vines (experiments (UiO). Excuses to those I have omitted.
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. See also https://www.nationaldefensemagazine.org/articles/2023/3/24/ukraine-a-living-lab-for-ai-warfare
Here: with AI/ML we intend a collection of machine learning methods with an emphasis on statistical learning and data analysis
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.
An emerging third category is 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:
Almost every problem in ML and data science starts with the same ingredients:
We seek to minimize the function \( \mathcal{C} (\mathbf{x}, f(\mathbf{\alpha})) \) 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.
Our data which we want to apply a machine learning method on, consist of a set of inputs \( \boldsymbol{x}^T=[x_0,x_1,x_2,\dots,x_{n-1}] \) and the outputs we want to model \( \boldsymbol{y}^T=[y_0,y_1,y_2,\dots,y_{n-1}] \). We assume that the output data can be represented (for a regression case) by a continuous function \( f \) through
$$ \boldsymbol{y}=f(\boldsymbol{x})+\boldsymbol{\epsilon}. $$In linear regression we approximate the unknown function with another continuous function \( \tilde{\boldsymbol{y}}(\boldsymbol{x}) \) which depends linearly on some unknown parameters \( \boldsymbol{\theta}^T=[\theta_0,\theta_1,\theta_2,\dots,\theta_{p-1}] \).
The input data can be organized in terms of a so-called design matrix with an approximating function \( \boldsymbol{\tilde{y}} \)
$$ \boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\theta}, $$The simplest approach is the mean squared error
$$ C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, $$or using the matrix \( \boldsymbol{X} \) and in a more compact matrix-vector notation as
$$ C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}. $$This function represents one of many possible ways to define the so-called cost function.
Optimizing wrt to th eunknown paramters \( \theta_j \) we get
$$ \boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\theta}, $$and if the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) is invertible we have the optimal values
$$ \hat{\boldsymbol{\theta}} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. $$Our optimization problem is
$$ {\displaystyle \min_{\boldsymbol{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}. $$or we can state it as
$$ {\displaystyle \min_{\boldsymbol{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2, $$where we have used the definition of a norm-2 vector, that is
$$ \vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. $$By minimizing the above equation with respect to the parameters \( \boldsymbol{\theta} \) we could then obtain an analytical expression for the parameters \( \boldsymbol{\theta} \). We can add a regularization parameter \( \lambda \) by defining a new cost function to be optimized, that is
$$ {\displaystyle \min_{\boldsymbol{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\theta}\vert\vert_2^2 $$which leads to the Ridge regression minimization problem where we require that \( \vert\vert \boldsymbol{\theta}\vert\vert_2^2\le t \), where \( t \) is a finite number larger than zero. We do not include such a constraints in the discussions here.
Defining
$$ C(\boldsymbol{X},\boldsymbol{\theta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\theta}\vert\vert_1, $$we have a new optimization equation
$$ {\displaystyle \min_{\boldsymbol{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\theta}\vert\vert_1 $$which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. Here we have defined the norm-1 as
$$ \vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert. $$
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!
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, Big Data 5, 787421 (2022):https://doi.org/10.3389/fdata.2022.787421"
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
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).
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. ... 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.
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.
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
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
$$ \langle E[\boldsymbol{\alpha}] \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})}. $$$$ E_L(\boldsymbol{R},\boldsymbol{\alpha})=\frac{1}{\psi_T(\boldsymbol{R},\boldsymbol{\alpha})}H\psi_T(\boldsymbol{R},\boldsymbol{\alpha}), $$
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 local energy as function of the variational parameters defines now our objective/cost function.
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
$$ \bar{\Psi}_{i}=\frac{d\Psi}{d\alpha_i}. $$
The elements of the gradient of the local energy are
$$ \bar{E}_{i}= 2\left( \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L \rangle\right). $$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). Use methods like stochastic gradient or other minimization methods to find the optimal parameters.
According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function.
Figure 1: In a) we show a single perceptron model while in b) we dispay a network with two hidden layers, an input layer and an output layer.
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.
Adams et al: $$ \begin{align} H_{LO} &=-\sum_i \frac{{\vec{\nabla}_i^2}}{2m_N} +\sum_{i < j} {\left(C_1 + C_2\, \vec{\sigma_i}\cdot\vec{\sigma_j}\right) e^{-r_{ij}^2\Lambda^2 / 4 }} \nonumber\\ &+D_0 \sum_{i < j < k} \sum_{\text{cyc}} {e^{-\left(r_{ik}^2+r_{ij}^2\right)\Lambda^2/4}}\,, \label{_auto1} \end{align} $$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
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
$$ \begin{align} |\Psi_V^J \rangle = \prod_{i < j < k} \Big( 1-\sum_{\text{cyc}} u(r_{ij}) u(r_{jk})\Big) \prod_{i < j} f(r_{ij}) | \Phi\rangle\,, \label{_auto2} \end{align} $$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.
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 \( r_{ij}=\sqrt{\mathbf{r}_i^2 - \mathbf{r}_j^2} \).
Separable Hamiltonian with the relative motion part (\( r_{ij}=r \))
$$ \hat{H}_r=-\nabla^2_r + \frac{1}{4}\omega^2r^2+ \frac{1}{r}, $$Analytical solutions in two and three dimensions (M. Taut 1993 and 1994).
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.
The goal of the hidden layer is to increase the model's expressive power. We encode complex interactions between visible variables by introducing additional, hidden variables that interact with visible degrees of freedom in a simple manner, yet still reproduce the complex correlations between visible degrees in the data once marginalized over (integrated out).
The network parameters, to be optimized/learned:The restricted Boltzmann machine is described by a Boltzmann distribution
$$ \begin{align} P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}, \label{_auto3} \end{align} $$where \( Z \) is the normalization constant or partition function, defined as
$$ \begin{align} Z = \int \int e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})} d\mathbf{x} d\mathbf{h}. \label{_auto4} \end{align} $$It is common to ignore \( T_0 \) by setting it to one.
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 \( A \).
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
$$ p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}. $$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.
Predictions made with eleven global mass model and Bayesian model averaging
