Many-body perturbation theory provides a powerful framework to study the ground state and thermodynamic properties of nuclear matter as well as associated single-particle potentials and response functions within a systematic order-by-order expansion. However, computational challenges can emerge beyond the lowest orders of perturbation theory, especially when computing both single-particle potentials and response functions, which in general are complex-valued and require Cauchy principal value calculations of high-dimensional integrals. We demonstrate that normalizing flows are suitable for Monte Carlo importance sampling of both regular and irregular functions appearing in nuclear many-body calculations. Normalizing flows are a class of machine learning models that can be used to build and sample from complicated distributions through a bijective mapping from a simple base distribution. Furthermore, a well-trained model for a certain target integrand can be efficiently transferred to calculate related integrals with varying physical conditions. These features can enable more efficient tabulations of nuclear physics inputs to numerical simulations of supernovae and neutron star mergers across varying physical conditions and nuclear force models.
Many-body perturbation theory provides a powerful framework to study the ground state and thermodynamic properties of nuclear matter as well as associated single-particle potentials and response functions within a systematic order-by-order expansion. However, computational challenges can emerge beyond the lowest orders of perturbation theory, especially when computing both single-particle potentials and response functions, which in general are complex-valued and require Cauchy principal value calculations of high-dimensional integrals. We demonstrate that normalizing flows are suitable for Monte Carlo importance sampling of both regular and irregular functions appearing in nuclear many-body calculations. Normalizing flows are a class of machine learning models that can be used to build and sample from complicated distributions through a bijective mapping from a simple base distribution. Furthermore, a well-trained model for a certain target integrand can be efficiently transferred to calculate related integrals with varying physical conditions. These features can enable more efficient tabulations of nuclear physics inputs to numerical simulations of supernovae and neutron star mergers across varying physical conditions and nuclear force models.
Normalizing flows are a class of machine learning models used to construct a complex distribution through a bijective mapping of a simple base distribution. We demonstrate that normalizing flows are particularly well suited as a Monte Carlo integration framework for quantum many-body calculations that require the repeated evaluation of high-dimensional integrals across smoothly varying integrands and integration regions. As an example, we consider the finite-temperature nuclear equation of state. An important advantage of normalizing flows is the ability to build highly expressive models of the target integrand, which we demonstrate enables precise evaluations of the nuclear free energy and its derivatives. Furthermore, we show that a normalizing flow model trained on one target integrand can be used to efficiently calculate related integrals when the temperature, density, or nuclear force is varied. This work will support future efforts to build microscopic equations of state for numerical simulations of supernovae and neutron star mergers that employ state-of-the-art nuclear forces and many-body methods.
An efficient and expressive wave function Ansatz is key to scalable solutions for complex many-body electronic structures. While Slater determinants are predominantly used for constructing antisymmetric electronic wave function Ansätze, this construction can result in limited expressiveness when the targeted wave function is highly complex. In this work, we introduce Waveflow, an innovative framework for learning many-body fermionic wave functions using boundary-conditioned normalizing flows. Instead of relying on Slater determinants, Waveflow imposes antisymmetry by defining the fundamental domain of the wave function and applying necessary boundary conditions. A key challenge in using normalizing flows for this purpose is addressing the topological mismatch between the prior and target distributions. We propose using O-spline priors and I-spline bijections to handle this mismatch, which allows for flexibility in the node number of the distribution while automatically maintaining its square-normalization property. We apply Waveflow to a one-dimensional many-electron system, where we variationally minimize the system’s energy using variational quantum Monte Carlo (VQMC). Our experiments demonstrate that Waveflow can effectively resolve topological mismatches and faithfully learn the ground-state wave function.
A long-standing goal of nuclear theory is to explain how the structure and dynamics of atomic nuclei and neutron-star matter emerge from the underlying interactions among protons and neutrons. Achieving this goal requires solving the nuclear quantum many-body problem with high accuracy across a wide range of length scales and density regimes. In this review, we discuss how artificial neural network representations of the nuclear many-body wave function have significantly extended the capabilities of continuum quantum Monte Carlo methods. In particular, neural network quantum states enable calculations of larger systems than were previously accessible and provide a flexible framework for capturing phenomena that challenge conventional approaches, including the emergence of nuclear clusters and superfluid phases in dense matter. We highlight recent applications to finite nuclei, infinite nuclear and neutron matter, and dynamical processes relevant to lepton-nucleus and nucleus-nucleus scattering. We also discuss conceptual and methodological connections with condensed matter physics, emphasizing developments in neural network quantum states that bridge strongly correlated systems across disciplines. Together, these developments demonstrate how neural-network methods open new avenues toward unified and accurate descriptions of nuclear structure, matter, and reactions.
Learning many-body quantum systems is of fundamental importance in quantum information processing, however, it is a challenging task which typically requires estimating quantum distributions of dimensionality exponentially scaling to the system size. As generative models have shown a great scalability to learn high-dimensional distributions and found wide applications in the domain of image and text, they can be a powerful tool to facilitate us to accomplish the challenging quantum tasks. In this work, we propose using normalizing flow (NF) models with fast sampling to learn discrete quantum distributions for quantum state tomography. Particularly, three NF models, including denoising flow, argmax flow, and tree flow, are first adapted to the task of explicit quantum probability density estimation. We then perform extensive experiments on a large scale of quantum systems, and our numerical results demonstrate that these discrete NFs admit an excellent sampling efficiency in the sense that they are insensitive to the system size to learn the high-dimensional quantum distributions, without compromising the learning performance. Finally, in comparison to the other generative models, such as autoregressive, the NFs avoid the problem of slow sequential sampling.
Diagrammatic Monte Carlo provides a systematically improvable framework for stochastically resumming many-body expansions to high orders through direct sampling of diagram topologies. We advance our earlier work by introducing a novel time-ordered Diagrammatic Monte Carlo algorithm for the single-particle Green's function. The algorithm is tailored to finite nuclei, formulated in discrete model spaces and applicable to arbitrary two-body interactions. The new time-ordered diagrammatic Monte Carlo algorithm is based on the on-the-fly evaluation of time-ordered Goldstone diagrams, avoiding explicit diagram enumeration and expensive frequency integration. We show the algorithm by computing 16O up to fifth order in a reduced model space using optimized reference state orbitals and including effective three-body forces. Benchmarking against established truncation schemes in ab initio nuclear theory demonstrates its potential to overcome the limitations of current many-body approaches.
We investigate the applicability of machine learning techniques in studying the finite-size effects associated with many-body physics. These techniques have an emerging presence in many-body theory as they have been used for interpolations, extrapolations, and in modeling wave functions. We will resolve several issues associated with machine learning and many-body calculations such as small datasets, outliers, and discontinuities, for the purpose of extrapolating finite calculations to macroscopic scales. We carry out a systematic investigation of two related systems by developing metrics that aim to avoid spurious effects and capture desired features. This work uses neural networks to extrapolate the unitary gas to the thermodynamic limit at zero range, which is otherwise difficult to reach. The effective mass of strongly interacting neutron matter is also studied and makes use of the noninteracting problem to resolve discontinuous predictions. For this investigation, we also carried out new auxiliary field diffusion Monte Carlo (AFDMC) calculations for a variety of densities and particle numbers. Ultimately, we demonstrate an effective utility for neural networks in this context.