arXiv subjects

Geneviève Dusson

Publications and source records attributed to Geneviève Dusson.

At least 19 recordsLinked to original sources

arXiv2026

Benchmarking machine-learning interatomic potentials using classical force fields

We propose a rational comparison between several well-known machine-learning interatomic potentials (linear atomic cluster expansion, the Behler-Parrinello high-dimensional neural network potential, and the graph neural network-based MACE potential) by testing their performance for fitting classical potentials such as the Lennard-Jones and Stillinger-Weber potential. With this we assess their advantages and limitations regarding several aspects: varying training dataset size, robustness to noise, carbon footprint, as well as the ability to reproduce the radial distribution function as an example observable. Overall the MACE models are shown to most accurately reproduce the potential energy surface, with little sign of overfitting, but at a significant computational cost, highlighting important questions regarding the choice of models in computational materials science.

cond-mat.mtrl-sci↗

arXiv2026

Machine learning of electronic structure and atomistic properties from the external potential

Electronic structure calculations remain a major bottleneck in atomistic simulations and, not surprisingly, have attracted significant attention in machine learning (ML). Most existing approaches learn a direct map from molecular geometries, typically represented as graphs or encoded local environments, to molecular properties or use ML as a surrogate for electronic structure theory by targeting quantities such as Fock or density matrices expressed in an atomic orbital (AO) basis. Inspired by the Hohenberg-Kohn theorem, in this work, we propose an operator-centered framework in which the external (nuclear) potential, expressed in an AO basis, serves as the model input. From this operator, we construct hierarchical, body-ordered representations of atomic configurations that closely mirror the principles underlying several popular atom-centered descriptors. At the same time, the matrix-valued nature of the external potential provides a natural connection to equivariant message-passing neural networks. In particular, we show that successive products of the external potential provide a scalable route to equivariant message passing and enable an efficient description of long-range effects. We demonstrate that this approach can be used to model molecular properties, such as energies and dipole moments, from the external potential, or learn effective operator-to-operator maps, including mappings to the Fock matrix and the reduced density matrix from which multiple molecular observables can be simultaneously derived.

physics.chem-ph↗

arXiv2026

Computing resonances of perturbed Schrödinger equations: Application to Reissner-Norsdtröm-de Sitter black holes

We present a numerical method for computing resonances of one-dimensional Schrödinger equations perturbed by a compactly supported potential, via finding zeros of the Wronskian associated with Jost solutions of the reference equation, computed through the resolution of Cauchy problems. All resonances located in a given domain are found efficiently using a defeated Newton algorithm. A key ingredient of the method is the choice of reference potential for which Jost solutions are known, which removes spurious resonances often encountered numerically. We test this method on three types of reference potentials and perturbations thereof: Pöschl-Teller potentials, exponentially decaying potentials, and potentials associated with Reissner-Nordströmde Sitter black holes. In particular we study the impact of perturbations on the resonances, and the stability of small resonances under perturbation. As an illustration, we use the method to numerically study the strong cosmic censorship hypothesis.

math.NA↗

arXiv2026

Efficient construction of Lie group-equivariant and permutation-invariant spaces

We introduce a practical construction of group-equivariant and permutation-invariant functions of $N$ variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix $M$ whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-invariance. For the groups $SO(3)$ and $SU(2)$, we further exploit the structure of the Lie algebra to demonstrate the sparsity pattern and rank of the matrix $M$, which yields the exact dimension of the group-equivariant and permutation-invariant space, as well as the dimension of the group-equivariant space alone. We demonstrate analytically and verify numerically that the proposed method scales linearly with respect to the dimensionality of the basis, offering a high computational gain compared to existing methods in the literature which typically scale exponentially. We finally perform a dimensionality comparison, showing that for large values of~$N$, the dimension of group-equivariant and permutation-invariant spaces is of comparable order as the dimension of permutation-invariant spaces, while pre-asymptotically, the first dimensionality is orders of magnitude lower than the second. Hence a substantial computational gain can be achieved by explicitly enforcing group-equivariance on top of permutation-invariance when approximating such functions.

math.NA↗

arXiv2025

A posteriori error estimates for Schr{ö}dinger operators discretized with linear combinations of atomic orbitals

We establish guaranteed and practically computable a posteriori error bounds for source problems and eigenvalue problems involving linear Schr{ö}dinger operators with atom-centered potentials discretized with linear combinations of atomic orbitals. We show that the energy norm of the discretization error can be estimated by the dual energy norm of the residual, that further decomposes into atomic contributions, characterizing the error localized on atoms. Moreover, we show that the practical computation of the dual norms of atomic residuals involves diagonalizing radial Schr{ö}dinger operators which can easily be precomputed in practice. We provide numerical illustrations of the performance of such a posteriori analysis on several test cases, showing that the error bounds accurately estimate the error, and that the localized error components allow for optimized adaptive basis sets.

math.NA↗

arXiv2025

A multipoint perturbation formula for eigenvalue problems

Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several nearby operators are known, standard perturbation theory cannot simultaneously use all this knowledge to provide a better approximation. We derive a resolvent formula enabling such an approximation result, and provide numerical examples for which this method is more competitive than standard perturbation theory.

math-ph↗

arXiv2025

Copula methods for modeling pair densities in density functional theory

We propose a new approach towards approximating the density-to-pair-density map based on copula theory from statistics. We extend the copula theory to multi-dimensional marginals, and deduce that one can describe any (exact or approximate) pair density by the single-particle density and a copula. We present analytical formulas for the exact copula in scaling limits, numerically compute the copula for dissociating systems with two to four particles in one dimension, and propose accurate approximations of the copula between equilibrium and dissociation for two-particle systems.

physics.comp-ph↗

arXiv2025

Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals

In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.

math.NA↗

arXiv2024

Comparison between tensor methods and neural networks in electronic structure calculations

This article compares the tensor method density matrix renormalization group (DMRG) with two neural network based methods -namely FermiNet and PauliNet) for determining the ground state wavefunction of the many-body electronic Schr{ö}dinger problem. We provide numerical simulations illustrating the main features of the methods and showing convergence with respect to some parameter, such as the rank for DMRG, and number of pretraining iterations for neural networks. We then compare the obtained energy with the methods for a few atoms and molecules, for some of which the exact value of the energy is known for the sake of comparison. In the last part of the article, we propose a new kind of neural network to solve the Schr{ö}dinger problem based on the training of the wavefunction on a simplex, and an explicit permutation for evaluating the wavefunction on the whole space. We provide numerical results on a toy problem for the sake of illustration.

math.OC↗

arXiv2023

An Application of Reduced Basis Methods to Core Computation in APOLLO3

In the aim of reducing the computational cost of the resolution of parameter-dependent eigenvalue problems, a model order reduction (MOR) procedure is proposed. We focus on the case of non-self-adjoint generalized eigenvalue problems, such as the stationary multigroup neutron diffusion equations. The method lies in an approximation of the manifold of solutions using a Proper Orthogonal Decomposition approach. The numerical method is composed of two stages. In the offline stage, we build a reduced space which approximates the manifold. In the online stage, for any given new set of parameters, we solve a reduced problem on the reduced space within a much smaller computational time than the required time to solve the high-fidelity problem. This method is applied to core computations in the APOLLO3 code.

math.NA↗

arXiv2023

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the eigenvalue and left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

math.NA↗

arXiv2023

A Quasi Time-Reversible scheme based on density matrix extrapolation on the Grassmann manifold for Born-Oppenheimer Molecular Dynamics

This article proposes a so-called Quasi Time-Reversible (QTR G-Ext) scheme based on Grassmann extrapolation of density matrices for an accurate calculation of initial guesses in Born-Oppenheimer Molecular Dynamics simulations. The method shows excellent results on four large molecular systems, ranging from 21 to 94 atoms simulated with Kohn-Sham density functional theory surrounded with a classical environment with 6k to 16k atoms. Namely, it clearly reduces the number of self-consistent field iterations, while keeping a similar energy drift as in the extended Lagrangian Born-Oppenheimer method.

physics.chem-ph↗

arXiv2023

An overview of a posteriori error estimation and post-processingmethods for nonlinear eigenvalue problems

In this article, we present an overview of different a posteriori error analysis and postprocessing methods proposed in the context of nonlinear eigenvalue problems, e.g. arising inelectronic structure calculations for the calculation of the ground state and compare them. Weprovide two equivalent error reconstructions based either on a second-order Taylor expansionof the minimized energy, or a first-order expansion of the nonlinear eigenvalue equation. Wethen show how several a posteriori error estimations as well as post-processing methods can beformulated as specific applications of the derived reconstructed errors, and we compare theirrange of applicability as well as numerical cost and precision.

math.NA↗

arXiv2023

Polynomial Approximation of Symmetric Functions

We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$ arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function $f$, and in particular study the dependence of that ratio on $d, N$ and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where $N$ becomes a parameter of the input.

math.NA↗

arXiv2023

A Wasserstein-type metric for generic mixture models, including location-scatter and group invariant measures

In this article, we study Wasserstein-type metrics and corresponding barycenters for mixtures of a chosen subset of probability measures called atoms hereafter. In particular, this works extends what was proposed by Delon and Desolneux [A Wasserstein-Type Distance in the Space of Gaussian Mixture Models. SIAM J. Imaging Sci. 13, 936-970 (2020)] for mixtures of gaussian measures to other mixtures. We first prove in a general setting that for a set of atoms equipped with a metric that defines a geodesic space, the set of mixtures based on this set of atoms is also geodesic space for the defined modified Wasserstein metric. We then focus on two particular cases of sets of atoms: (i) the set of location-scatter atoms and (ii) the set of measures that are invariant with respect to some symmetry group. Both cases are particularly relevant for various applications among which electronic structure calculations. Along the way, we also prove some sparsity and symmetry properties of optimal transport plans between measures that are invariant under some well-chosen symmetries.

math.OC↗

arXiv2022

Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models

We propose a scheme to construct predictive models for Hamiltonian matrices in atomic orbital representation from ab initio data as a function of atomic and bond environments. The scheme goes beyond conventional tight binding descriptions as it represents the ab initio model to full order, rather than in two-centre or three-centre approximations. We achieve this by introducing an extension to the Atomic Cluster Expansion (ACE) descriptor that represents Hamiltonian matrix blocks that transform equivariantly with respect to the full rotation group. The approach produces analytical linear models for the Hamiltonian and overlap matrices. Through an application to aluminium, we demonstrate that it is possible to train models from a handful of structures computed with density functional theory, and apply them to produce accurate predictions for the electronic structure. The model generalises well and is able to predict defects accurately from only bulk training data.

cond-mat.mtrl-sci↗

arXiv2022

On basis set optimisation in quantum chemistry

In this article, we propose general criteria to construct optimal atomic centered basis sets in quantum chemistry. We focus in particular on two criteria, one based on the ground-state one-body density matrix of the system and the other based on the ground-state energy. The performance of these two criteria are then numerically tested and compared on a parametrized eigenvalue problem, which corresponds to a one-dimensional toy version of the ground-state dissociation of a diatomic molecule.

math.NA↗

arXiv2022

Practical error bounds for properties in plane-wave electronic structure calculations

We propose accurate computable error bounds for quantities of interest in plane-wave electronic structure calculations, in particular ground-state density matrices and energies, and interatomic forces. These bounds are based on an estimation of the error in terms of the residual of the solved equations, which is then efficiently approximated with computable terms. After providing coarse bounds based on an analysis of the inverse Jacobian, we improve on these bounds by solving a linear problem in a small dimension that involves a Schur complement. We numerically show how accurate these bounds are on a few representative materials, namely silicon, gallium arsenide and titanium dioxide.

math.NA↗