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arXiv:2203.05284 · Physical Review C

Training and projecting: A reduced basis method emulator for many-body physics

Emanuel Chew Bonilla, Pablo Giuliani, Kyle Godbey, Dean Lee

Published Nov 17, 2022• 51 citations
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Abstract

We present the reduced basis method as a tool for developing emulators for equations with tunable parameters within the context of the nuclear many-body problem. The method uses a basis expansion informed by a set of solutions for a few values of the model parameters and then projects the equations over a well-chosen low-dimensional subspace. We connect some of the results in the eigenvector continuation literature to the formalism of reduced basis methods and show how these methods can be applied to a broad set of problems. As we illustrate, the possible success of the formalism on such problems can be diagnosed beforehand by a principal component analysis. We apply the reduced basis method to the one-dimensional Gross-Pitaevskii equation with a harmonic trapping potential and to nuclear density functional theory for $^{48}\mathrm{Ca}$, achieving speed-ups of more than $\ifmmode\times\else\texttimes\fi{}150$ and $\ifmmode\times\else\texttimes\fi{}250$, respectively, when compared to traditional solvers. The outstanding performance of the approach, together with its straightforward implementation, show promise for its application to the emulation of computationally demanding calculations, including uncertainty quantification.

DOI
10.1103/physrevc.106.054322

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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.
Emulating the Green's function using reduced-order equations is challenging because of the amount of information it contains. This information includes the one-particle spectroscopy over the entire energy spectrum, the one-body density matrix, and the ground-state energy. Nevertheless, the computational speedup offered by emulators is essential for applications requiring repeated model evaluations, which are ubiquitous in many-body physics. These include parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. In this Letter, we introduce a systematically improvable emulator based on eigenvector continuation and moment-matching. The resulting approximation, which we refer to as an ``eigenvector-continued Green's function,'' can be obtained by solving linear equations, enabling an efficient separation of the computational workload into offline and online stages.
The numerical treatment of quantum systems often requires large amounts of computing power and time. As a result, performing calculations repeatedly for different values of the input parameters is often not feasible. One remedy is using eigenvectors describing the system that are analytic functions that vary smoothly for real values of the input parameters. This allows one to replace computationally expensive calculations with emulators that project onto a reduced-basis set. This Colloquium explores a particular class of reduced-basis methods known as eigenvector continuation and its applications, with emphasis on nuclear physics.
An atomic nucleus is a quantum system of interacting nucleons and constitutes a problem difficult to solve exactly. For this reason, a diversity of approximate resolution methods has been designed, and Projected Generator Coordinate Method (PGCM) is one of them. The strong point of PGCM is to construct a physically inspired small dimensional space, in which an approximate solution of the nuclear many-body problem is easily found. However the numerical cost of PGCM space computation make this method inadapted for sensibility analysis of nuclear observables with restect to parametrisation of the interaction model, this analysis requiring an huge number of PGCM computations. In order to make this type of study possible, this thesis explore the concept of PGCM emulator. In this work, a combination of PGCM with Eigenvector Continuation (EC) is constructed and studied. This combination (the PGCM-EC emulator) takes advantage of mathematical similarities between PGCM and EC, and above all of the decomposition of the hamiltonian as a linear combination of parameter-independent terms. The latter property is used to concentrate the heavier numerical cost in the computation of parameter-independent quantities (the elementary kernels), and open the feasability of massive PGCM emulations, the price being having first-handedly computed the costly elementary kernels. Limits of the emulator are also explored, by introducing the concept of over-training, which is exactly a consequence of the aproximativeness of a PGCM computation. Eventually this thesis demonstrates the possibility to emulate millions of PGCM computations with an error on collective spectroscopy less than 3%, and with a low numerical cost fraction of 1% of the million PGCM calculations cost.
First principles calculations of atomic nuclei based on microscopic nuclear forces derived from chiral effective field theory (EFT) have blossomed in the past years. A key element of such ab initio studies is the understanding and quantification of systematic and statistical errors arising from the omission of higher-order terms in the chiral expansion as well as the model calibration. While there has been significant progress in analyzing theoretical uncertainties for nucleon-nucleon scattering observables, the generalization to multi-nucleon systems has not been feasible yet due to the high computational cost of evaluating observables for a large set of low-energy couplings. In this Letter we show that a new method called eigenvector continuation (EC) can be used for constructing an efficient and accurate emulator for nuclear many-body observables, thereby enabling uncertainty quantification in multi-nucleon systems. We demonstrate the power of EC emulation with a proof-of-principle calculation that lays out all correlations between bulk ground-state observables in the few-nucleon sector. On the basis of ab initio calculations for the ground-state energy and radius in 4He, we demonstrate that EC is more accurate and efficient compared to established methods like Gaussian processes.
Effective field theories have established themselves as key pillars of modern nuclear physics. They enable a quantitative understanding of the strong nuclear force, provided low-energy constants that parametrize short-distance physics can be determined from experimental data. This, however, often becomes prohibitively expensive due to a significant computational cost of solving the A-body problem. The computational challenge is particularly severe for three-body forces, which are at the frontier of nuclear and atomic physics and play an important role in the equation of state of neutron stars. Here we prove that for a parametric low-rank update of a Hamiltonian, the A-body problem at a fixed energy exactly reduces to a low-dimensional matrix equation regardless of the size of the Hilbert space. As a proof-of-principle, we present exact and computationally cheap snapshot-based emulators for few-body scattering and bound states. Unlike alternatives, our emulators can be used far away from the snapshot region without loss of precision and yield accurate results for parameter values not accessible using conventional solution techniques. Our approach is not restricted by the interaction type, number of particles, and methods for generating snapshots and can be applied to mitigate the computational burden of the A-body problem to a broad class of problems in nuclear, atomic, and molecular physics.