© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Emulation of PGCM calculations using the Eigenvector continuation method

A. Y. Le Roux

Published Oct 15, 2024• 0 citations

Abstract

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.

DOI: 10.70675/7baeb2eazad6cz4147z9d95z46f51270effd

Related papers

Eigenvector continuation as an efficient and accurate emulator for uncertainty quantification

S. König, A. Ekström, K. Hebeler +2 more

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.

arXiv:1909.08446•Physics Letters B•Sep 30, 2020•108 citations

Multi-reference many-body perturbation theory for nuclei III -- Ab initio calculations at second order in PGCM-PT

Mikaël Frosini, T. Duguet, J.-P. Ebran +6 more

In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small ($e_{\mathrm{max}}=4$) model space demonstrate that exact binding energies of closed- (\nucl{O}{16}) and open-shell (\nucl{O}{18}, \nucl{Ne}{20}) nuclei are reproduced within $0.5-1.5\%$ at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.

arXiv:2111.01461•Nov 2, 2021•4 citations

Emulating the generator coordinate method with extended eigenvector continuation for the Lipkin-Meshkov-Glick model

Q. Y. Luo, Xilin Zhang, Li‐Hui Chen +1 more

We present a benchmark study of the generator coordinate method (GCM) combined with eigenvector continuation (EC) in two different schemes for the low-lying states of the Lipkin-Meshkov-Glick (LMG) model, where the interaction strength is treated as a controlling parameter, simulating quantum many-body systems with the phase transition from noncollective to collective states. We demonstrate that the ${\mathrm{EC}}_{\mathrm{kmax}}$ scheme accurately reproduces the low-lying states of the LMG model. In this scheme, the EC basis consists of the wave functions of low-lying states up to the ${k}_{\mathrm{max}}$-th state of sampling Hamiltonians. Compared to ${\mathrm{EC}}_{1}$, which only includes the wave functions of the $k\mathrm{th}$ state of sampling Hamiltonians for the $k\mathrm{th}$ state of a target Hamiltonian, the ${\mathrm{EC}}_{\mathrm{kmax}}$ scheme exhibits significantly improved efficiency and accuracy. This study suggests the potential utilization of the extended EC scheme as an efficient emulator for GCM calculations.

arXiv:2404.08581•Physical Review C•Jul 2, 2024•5 citations

Eigenvector continuation for the pairing Hamiltonian

M. Companys Franzke, A. Tichai, K. Hebeler +1 more

The development of emulators for the evaluation of many-body observables has gained increasing attention over the last years. In particular the framework of eigenvector continuation (EC) has been identified as a powerful tool when the Hamiltonian admits for a parametric dependence. By training the emulator on a set of training data the many-body solution for arbitrary parameter values can be robustly predicted in many cases. Furthermore, it can be used to resum perturbative expansions that otherwise diverge. In this work, we apply EC to the pairing Hamiltonian and show that EC-resummed perturbation theory is in qualitative agreement with the exact solution and that EC-based emulators robustly predict the ground-state energy once the training data are chosen appropriately. In particular the phase transition from the normal to the superfluid regime is quantitatively predicted using a very low number of training points. Published by the American Physical Society 2024

arXiv:2302.08373•Physical Review C•Feb 13, 2024•10 citations

Fast emulation of quantum three-body scattering

Xilin Zhang, R. J. Furnstahl

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.

arXiv:2110.04269•Physical Review C•Jun 28, 2022•34 citations

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

Emanuel Chew Bonilla, Pablo Giuliani, Kyle Godbey +1 more

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.

arXiv:2203.05284•Physical Review C•Nov 17, 2022•51 citations

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.

In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (

e_{\mathrm{max}}=4

) model space demonstrate that exact binding energies of closed- (\nucl{O}{16}) and open-shell (\nucl{O}{18}, \nucl{Ne}{20}) nuclei are reproduced within

0.5-1.5\%

at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.

We present a benchmark study of the generator coordinate method (GCM) combined with eigenvector continuation (EC) in two different schemes for the low-lying states of the Lipkin-Meshkov-Glick (LMG) model, where the interaction strength is treated as a controlling parameter, simulating quantum many-body systems with the phase transition from noncollective to collective states. We demonstrate that the

{\mathrm{EC}}_{\mathrm{kmax}}

scheme accurately reproduces the low-lying states of the LMG model. In this scheme, the EC basis consists of the wave functions of low-lying states up to the

{k}_{\mathrm{max}}

-th state of sampling Hamiltonians. Compared to

{\mathrm{EC}}_{1}

, which only includes the wave functions of the

k\mathrm{th}

state of sampling Hamiltonians for the

{k}_{\mathrm{max}}

state of a target Hamiltonian, the

{\mathrm{EC}}_{\mathrm{kmax}}

scheme exhibits significantly improved efficiency and accuracy. This study suggests the potential utilization of the extended EC scheme as an efficient emulator for GCM calculations.

The development of emulators for the evaluation of many-body observables has gained increasing attention over the last years. In particular the framework of eigenvector continuation (EC) has been identified as a powerful tool when the Hamiltonian admits for a parametric dependence. By training the emulator on a set of training data the many-body solution for arbitrary parameter values can be robustly predicted in many cases. Furthermore, it can be used to resum perturbative expansions that otherwise diverge. In this work, we apply EC to the pairing Hamiltonian and show that EC-resummed perturbation theory is in qualitative agreement with the exact solution and that EC-based emulators robustly predict the ground-state energy once the training data are chosen appropriately. In particular the phase transition from the normal to the superfluid regime is quantitatively predicted using a very low number of training points. Published by the American Physical Society 2024

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.

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.