Progress in nuclear and many-body physics today is predicated on the ability to solve large-scale, strongly correlated quantum many-body problems. As the theoretical models become more sophisticated, they also become more computationally complex. Simultaneously, quantifying uncertainty in model predictions and fitting free parameters to experimental observations requires repeated evaluation of these expensive models. Surrogate models---known as emulators---provide the means of accomplishing these goals. This thesis provides an introduction into the current state of emulation in nuclear and many-body physics. The motivations, goals, and origins of currently popular emulation methods are discussed along with selected examples. We see how many methods are closely mathematically related and how trade-offs are made to optimize specific properties or applications. The central work in this thesis is the method of parametric matrix models (PMMs), an emulation and general machine learning framework which combines aspects of traditional reduced basis method with modern parametric machine learning. PMMs are able to retain as much or as little physical information about the underlying system as desired, yielding not only excellent performance but also nearly unparalleled adaptability, interpretability, and trustworthiness as an emulation method. A formal mathematical framework for PMMs is developed and accompanied by practical step-by-step procedures for the application of the method. As part of this thesis, the open-source pyPMM package was developed. This package enables any researcher to construct, train, share, and deploy PMM-based emulators with modular, extendable, and graphics processing unit (GPU)-optimized code. All PMM examples in this thesis were created using this package.
Progress in nuclear and many-body physics today is predicated on the ability to solve large-scale, strongly correlated quantum many-body problems. As the theoretical models become more sophisticated, they also become more computationally complex. Simultaneously, quantifying uncertainty in model predictions and fitting free parameters to experimental observations requires repeated evaluation of these expensive models. Surrogate models---known as emulators---provide the means of accomplishing these goals. This thesis provides an introduction into the current state of emulation in nuclear and many-body physics. The motivations, goals, and origins of currently popular emulation methods are discussed along with selected examples. We see how many methods are closely mathematically related and how trade-offs are made to optimize specific properties or applications. The central work in this thesis is the method of parametric matrix models (PMMs), an emulation and general machine learning framework which combines aspects of traditional reduced basis method with modern parametric machine learning. PMMs are able to retain as much or as little physical information about the underlying system as desired, yielding not only excellent performance but also nearly unparalleled adaptability, interpretability, and trustworthiness as an emulation method. A formal mathematical framework for PMMs is developed and accompanied by practical step-by-step procedures for the application of the method. As part of this thesis, the open-source pyPMM package was developed. This package enables any researcher to construct, train, share, and deploy PMM-based emulators with modular, extendable, and graphics processing unit (GPU)-optimized code. All PMM examples in this thesis were created using this package.
The validation, verification, and uncertainty quantification of computationally expensive theoretical models of quantum many-body systems require the construction of fast and accurate emulators. In this work, we develop emulators for auxiliary field diffusion Monte Carlo (AFDMC), a powerful many-body method for nuclear systems. We introduce a reduced-basis method (RBM) emulator for AFDMC and study it in the simple case of the deuteron. Furthermore, we compare our RBM emulator with the recently proposed parametric matrix model (PMM) that combines elements of RBMs with machine learning. We contrast these two approaches with a traditional Gaussian Process emulator. All three emulators constructed here are based on a very limited set of 5 training points, as expected for realistic AFDMC calculations, but validated against O ( 10 3 ) exact solutions. We find that the PMM, with emulator errors of only ≈ 0.1 % and speed-up factors of ≈ 10 7 , outperforms our implementation of the other two emulators when applied to AFDMC.
Recent advances in both theoretical and computational methods have enabled large-scale, precision calculations of the properties of atomic nuclei. With the growing complexity of modern nuclear theory, however, also comes the need for novel methods to perform systematic studies and quantify the uncertainties of models when confronted with experimental data. This study presents an application of such an approach, the reduced basis method, to substantially lower computational costs by constructing a significantly smaller Hamiltonian subspace informed by previous solutions. Our method shows comparable efficiency and accuracy to other dimensionality reduction techniques on an artificial three-body bound system while providing a richer representation of physical information in its projection and training subspace. This methodological advancement can be applied in other contexts and has the potential to greatly improve our ability to systematically explore theoretical models and thus enhance our understanding of the fundamental properties of nuclear systems.
In this digital age, machine learning (a prominent element within the context of artificial intelligence) has become an integral part of modern life, influencing various aspects of our daily activities.The predictive power of these tools is increasing with the advancements in algorithms, increased computational power, and the availability of large data sets: one basic example of this is language-based models, which have revolutionized our capabilities of transferring and processing information.Data is the foundational element of machine learning, and the quality and relevance of the data sets have a profound impact on the performance and accuracy of machine learning models.Thus "data is the new gold", captures the immense value that data holds in our modern world.With the exponential growth in data, the increasing demand for better machine-learning tools comes as a natural consequence.The field of physics is no exception to this, especially with the rise of big simulations and experiments generating huge amounts of data.Experiments at particle accelerators like the Large Hadron Collider(LHC) and Fermi lab, gravitational waves detector by LIGO and Virgo, climate models and weather simulations, Sloan Digital Sky Survey (SDSS), Cosmic Microwave Background(CMB) radiation providing insights into the early universe; all these studies generate massive data sets.In this context, developing more powerful, accurate, and specialized tools, is essential to meet the evolving needs in physics as well.In this thesis, we aim to study different types of ML tools to characterize the many-body problem in the context of classical and quantum statistical mechanics, with a focus on unbiased and interpretable methods.We use techniques coming from so-called non-parametric learning (NPL), where the analysis is done without specifying a fixed set of parameters in an assumption-free vi manner.NPL is a subset of the broad unsupervised learning category; the algorithm learns from the intrinsic structure of the data set.NPL is particularly valuable in scenarios where data assumptions are uncertain, data distributions are non-standard, or when dealing with complex data sets where relationships are difficult to predict beforehand.This works in our favor as the relation between data sets coming from many-body physics system and their effect on data structures when physical parameters change is not well understood -the simple reason being that the typical physicists mindset is very much focused on performing huge dimensional reduction from microscopic physical information (a canonical example of this being mean field theories, where the behavior of a large number of degrees of freedom is replaced by a single one).The thesis is organized as follows.In Chapter 1, we discuss the theoretical framework behind the NPL methods used.We present basic ideas in our methodologies, which encompass the overarching principles guiding our approach.These include the linkage between statistical physics and data analysis, particularly in relation to intrinsic dimension (Id) and its significance in understanding critical characteristics.Additionally, we delve into the application of Principal Component Analysis (PCA) and introduce the notion of PCA entropy -a witness of information spreading under linear transformation, whose relevance to physical phenomena we start investigating here.In Chapter 2, we describe the data sets we work with; statistical mechanics inspired data sets, such as those generated via sampling classical Ising models through advanced Monte Carlo algorithms, and experimental data of a spinor Bose-Einstein condensate.The following chapters present the applications I worked on.In Chapter 3, we present and thoroughly assess a framework for understanding critical behavior in classical partition functions by employing NPL techniques on data sets containing thermal configurations; provide a robust analysis of phase transitions using different NPL methods.In Chapter 4, we present a theoretical framework for information extraction in synthetic quantum matter for the case of a quantum quench in a spinor Bose-Einstein vii condensate experiment.In Chapter 5, we demonstrate how the field of network science can significantly improve and elevate our assumption-free characterization of various phases of matter.We present compelling evidence that network science offers a transparent and effective set of methods for investigating the properties of distinct phases and phase transitions, enabling us to identify critical points.
In recent years, reduced-basis methods (RBMs) have been adapted to the many-body eigenvalue problem and they have been used, largely in nuclear physics, as fast emulators able to bypass expensive direct computations while still providing highly accurate results. This work is meant to show that the RBM is an efficient and accurate emulator for the strong correlations induced by the pairing interaction in a variety of finite systems like ultrasmall superconducting grains, interacting topological superfluids, and mesoscopic hybrid superconductor-semiconductor devices, all of which require an expensive, beyond-mean-field, particle-number conserving description. These systems are modeled by the number-conserving Richardson pairing Hamiltonian and its appropriate generalizations. Their ground state is solved for exactly using the density matrix renormalization group. The reduced basis is assembled iteratively from a small number of exact ground-state vectors, well-chosen from across the relevant parameter space using a fast estimate of the emulation error and a greedy local optimization algorithm. The RBM emulation is found to accurately describe the weak-to-strong pairing crossover in small grains, the third-order topological phase transition of the interacting Richardson-Kitaev chain, and the complex charge stability diagram of a hybrid quantum dot--superconductor device. For all considered systems the emulation error decreases exponentially with the reduced-basis dimensionality. The number of basis vectors necessary to reach a fixed emulation accuracy only shows a modest linear growth with increasing system size. RBMs are confirmed to be cheap and accurate emulators for the widely encountered superconducting phenomena. Capable of providing orders of magnitude computational speedup with respect to approaches based only on traditional many-body solvers, they open new possibilities in building and solving models of interacting many-body systems and in better interfacing them with experimental design and data analysis.
This thesis focuses on the development, application, and high-performance implementation of numerical methods for simulating quantum many-body systems. In the first chapter, I provide an overview of several most successful conventional approaches for simulating quantum many-body systems, along with their current limitations. Then, the basic properties of the quantum spin model, which will be the subject of study in the following chapters, and its ground state are briefly summarized. Additionally, I discuss the background and the motivations of the research projects carried out during my PhD research. Chapter 2 introduces two numerical algorithms that are extensively used in this thesis, which are exact diagonalization (ED) and variational Monte Carlo. I outline the process of constructing a Hamiltonian matrix and employing iterative methods for exact diagonalization. Subsequently, I introduce the mechanism of the four important components in the ``standard" variational Monte Carlo that are widely used in the most recent research. In Chapter 3, we propose a symmetry-projected variational ansatz in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics,and achieve state-of-the-art accuracy in the representation of the ground state. In Chapter 4, we present a variational approach based on a Chebyshev expansion of the spectral function, and each Chebyshev polynomial is represented by a neural network quantum state. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave functions." We present results for the one-dimensional and two-dimensional Heisenberg models on the square lattice and compare them with those obtained by other methods in the literature. In Chapter 5, we describe a method to carry out thermodynamic simulations of quantum many-body models using typicality and variational representations of quantum states. Minimally entangled typical thermal states (METTS) are a construction that allows one to solve for the imaginary time evolution of quantum many-body systems. By using wave functions that are weakly entangled, one can take advantage of efficient representations in the form of matrixproduct states. We generalize these ideas to arbitrary variational wave functions and we focus, as an illustration, on the particular case of restricted Boltzmann machines. We demonstrate the application of the method to quantum spin systems in one and two spatial dimensions. In Chapter 6, we study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band. In the final chapter, Chapter 7, we discuss the implementation of architecture-aware optimizations for spin dynamics simulations on x86 CPUs and Nvidia GPUs. The time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. We propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel to reduce memory latency. On Nvidia GPU, we perform the on-the-fly calculation in the epilogue of the GEMM subroutine to avoid saving intermediate results on global memory, which significantly reduces memory usage and achieves better performance than cuBLAS.--Author's abstract
We develop a class of emulators for solving quantum three-body scattering problems. They are based on combining the variational method for scattering observables and the recently proposed eigenvector continuation concept. The emulators are first trained by the exact scattering solutions of the governing Hamiltonian at a small number of points in its parameter space, and then employed to make interpolations and extrapolations in that space. Through a schematic nuclear-physics model with finite-range two and three-body interactions, we demonstrate the emulators to be extremely accurate and efficient. The computing time for emulation is on the scale of milliseconds (on a laptop), with relative errors ranging from ${10}^{\ensuremath{-}13}$ to ${10}^{\ensuremath{-}4}$ depending on the case. The emulators also require little memory. We argue that these emulators can be generalized to even more challenging scattering problems. Furthermore, this general strategy may be applicable for building the same type of emulators in other fields, wherever variational methods can be developed for evaluating physical models.