arXiv subjects

Virginie Ehrlacher

Publications and source records attributed to Virginie Ehrlacher.

At least 19 recordsLinked to original sources

arXiv2026

Two-layers neural networks for Schr{ö}dinger eigenvalue problems

The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of high-dimensional Schr{ö}dinger eigenvalue problems with smooth interaction potentials and Neumann boundary condition on the unit cube in any dimension. More precisely, any eigenfunction associated to the lowest eigenvalue of the Schr{ö}dinger operator is a unit L 2 norm minimizer of the associated energy. Using Barron's representation of the solution with a probability measure defined on the set of parameter values and following the approach initially suggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curve dynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. We prove the existence of solutions to this constrained gradient curve. Furthermore, we prove that, if it converges, the represented function is then an eigenfunction of the considered Schr{ö}dinger operator. At least up to our knowledge, this is the first work where this type of analysis is carried out to deal with the minimization of non-convex functionals.

math.AP↗

arXiv2026

A Hyper-Reduced Neural Network-Augmented Semi-smooth Newton Method for Nonlinear Parametric Variational Inequalities

We propose a model order reduction framework for nonlinear parametrized variational inequalities arising in computational mechanics. The high-dimensional model is written in mixed primal-dual form with projection-based complementarity conditions, leading to nonlinear nonsmooth algebraic systems solved by a semi-smooth Newton method in primal-dual form. On this basis, reduced models are constructed by proper orthogonal decomposition (POD) of both primal and dual solution snapshots, and the resulting reduced systems are solved by semi-smooth Newton iterations in the reduced space. To address cases where low-dimensional linear spaces provide limited approximation efficiency, we introduce a neural-network-augmented reduced model. Two feedforward networks learn corrections in the truncated POD coordinates of the primal and dual variables, defining a nonlinear manifold approximation that is embedded directly in the semi-smooth Newton iterations. The online cost associated with high-dimensional residual evaluations is reduced through hyper-reduction, using a sparse cubature approach based on greedy nonnegative least squares. Particular attention is paid to the interaction between hyper-reduction and the learned nonlinear manifold. The proposed methodology is assessed on two nonlinear variational inequalities with distinct sources of nonlinearity: a two-dimensional obstacle problem with a cubic nonlinearity in the state equation, and a three-dimensional frictional contact problem in which the Coulomb law induces a nonlinear projection in the constraint equation. Numerical results compare the high-dimensional model, the linear reduced model, the neural-network-augmented reduced model, and their hyper-reduced variants, demonstrating accurate approximations with substantial reductions in online computational cost.

math.NA↗

arXiv2026

Debiasing optimal transport: classical and entropic

We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge \tfrac{1}{2}c(x,x)+\tfrac{1}{2}c(y,y)$ for all $x,y\in\mathscr{X}$. Building on an equivalent characterization by an inf-representation $c(x,y)=\inf_{z\in\mathscr{Z}}ψ(x,z)+ψ(y,z)$ for some set $\mathscr{Z}$ and some function $ψ: \mathscr{X}\times \mathscr{Z} \to \mathbb{R} \cup \{+\infty\}$, interpreted as a generalization of the midpoint identity for squared geodesic distances, we investigate the debiasability of costs defined on spaces of probability measures. Our primary focus is the entropic regularization of optimal transport across different regimes of the regularization parameter $\varepsilon \in [0,+\infty]$, encompassing classical optimal transport ($\varepsilon=0$), entropic optimal transport ($\varepsilon>0$), and the Maximum Mean Discrepancy ($\varepsilon=+\infty$). For $\varepsilon \in (0,+\infty]$, we investigate sufficient conditions, such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, handled then via a convex-nonconcave minimax argument. All our results extend naturally to unbalanced optimal transport settings. As a byproduct, we derive novel barycentric formulas for entropic optimal transport. Extending our factorizations to include a time parameter $t\in(0,1)$, we construct two canonical entropic interpolation curves sharing the same variational origin: an intermediate measure $η_t\in\mathscr{P}(\mathscr{Z})$, which coincides with the marginals of the Schrödinger bridge between $μ$ and $ν$, and an intermediate potential $z_t^*\in\mathscr{H}_k$, which follows a segment in a flat reproducing kernel Hilbert space.

math.OC↗

arXiv2026

Cut-off Jastrow Factors and Spectral Barron Regularity of Coulombic Electronic Wave Functions

We study the spectral Barron regularity of Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. Let $H=-Δ+V$ be an $N$-electron Coulomb Hamiltonian with clamped nuclei, and let $ψ$ be an eigenfunction associated with a discrete eigenvalue below the bottom of the essential spectrum. For the cut-off Jastrow factor $F_{\rm cut}$ of Fournais--Hoffmann-Ostenhof--Hoffmann-Ostenhof--Sørensen, we set \[ ϕ=e^{-F_{\rm cut}}ψ. \] Whereas the original wave function satisfies the sharp global threshold $ψ\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})$ for every $0\leq s<1$, we prove that the Jastrow quotient gains one full order: \[ ϕ\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{for every } 0\le s<2 . \] The endpoint $s=2$ is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument. After conjugation by the cut-off Jastrow factor, the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures. Low frequencies are controlled by the a priori $H^1$-bound, while high frequencies are recovered by a Neumann fixed-point argument using the resolvent multiplier and annular estimates for the coefficient measures.

math.AP↗

arXiv2026

An algorithm for dynamical quantum optimal transport with applications to quantum chemistry

Quantum optimal transport (QOT) is a rapidly developing field. Among the many formulations of this adaptation of classical optimal transport (OT) to spaces of density matrices, we numerically study a family of distances based on a dynamical formulation inspired by the Benamou-Brenier OT formulation. We introduce an interior-point regularized method to compute geodesics between positive semidefinite matrices and visualize the results in terms of integral kernels and densities, inspired by quantum chemistry applications. We show that dynamical QOT may provide a good approximation to certain problems in quantum chemistry with appropriate parameter tuning. We also study the numerical properties of the distances at hand, and the convergence of the objects when the size of the matrices increases.

math.OC↗

arXiv2026

A land of monotone plenty, bis repetita: from classical to weak optimal transport

The celebrated c-cyclical monotonicity property is shown to boil down to the zeroth-order optimality condition for the optimal transport problem. More precisely, we show that optimality is equivalent to the non-negativity of the linear transport cost functional on the radial cone of admissible perturbations. We then utilise this point of view to extend the c-cyclical monotonicity property to the weak optimal transport problem, for which it corresponds to the first-order optimality condition, namely to the non-negativity of the linearisation of the weak transport cost functional near the optimiser. Altogether, this sheds new light on this monotonicity concept. For both classical and weak optimal transport, we show that this property characterises (under suitable assumptions) optimal transport plans. In the classical case, we recover known results of the literature but with revisited proofs.

math.OC↗

arXiv2026

Model order reduction for parametrized variational inequalities: application to crowd motion

This work investigates model order reduction for time-dependent parametrized variational inequalities, with a focus on discrete contact problems. As a prototypical example, we consider an agent-based crowd model [Maury et al., 2011] in which agent velocities are obtained at each time step from a constrained least-squares problem. Geometric parameter variations induce significant variability in both agent positions and contact forces, leading to a slowly decaying Kolmogorov $n$-width of the solution manifold. We propose a nonlinear approach that combines a linear reduced-order model with a deep-learning-based correction. The method utilizes a greedy index selection (gIS) algorithm for compressing Lagrange multipliers and Proper Orthogonal Decomposition (POD) applied to velocity snapshots. Additionally, we explore hyper-reduction techniques, comparing the Empirical Interpolation Method (EIM) and the Empirical Quadrature (EQ) procedure from both computational complexity and accuracy perspectives. Finally, we demonstrate the applicability of the methodology in a complex scenario involving many agents in a highly congested geometric configuration. This work represents the first attempt to apply model order reduction to a discrete contact problem of the type introduced in [Maury et al., 2011] and paves the way for future advancements in nonlinear MOR specifically for this class of problems.

math.NA↗

arXiv2026

A reduced-order model for parametrized Optimal Transport problems

In this work, we aim at efficiently solving a parametrized family of optimal transport problems by using model order reduction methods. We propose a reduced-order model by adding to the primal (respectively dual) version of the high-fidelity model the additional constraint to live in a non negative sub cone (resp. in subspaces) of small dimension. The reduced-order model then reads as a linear program with a small number of degrees of freedom and constraints. We identify explicit conditions under which this reduced-order model has at least one solution. We propose two a posteriori error estimations that bounds the error between the optimal values of the high-fidelity problem and the reduced-order model. As one of these estimations requires the computation of non linear terms (with respect to the reduction of dimension), we use an Empirical Interpolation Method (EIM) (see e.g. \cite{maday2007general} or \cite{barrault2004empirical}) to numerically efficiently compute this estimation. We apply the whole methodology on a simple 1D example and on a problem of color transfer between images, and compare its performances to Sinkhorn algorithm.

math.NA↗

arXiv2026

A space-time variational formulation for the many-body electronic Schr{ö}dinger evolution equation

We prove in this paper that the solution of the time-dependent Schr{ö}dinger equation can be expressed as the solution of a global space-time quadratic minimization problem that is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation. The present analysis can be applied to the electronic many-body time-dependent Schr{ö}dinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities. We motivate the interest of the present approach with two goals: first, the design of Galerkin space-time discretization methods; second, the definition of dynamical low-rank approximations following a variational principle different from the classical Dirac-Frenkel principle, and for which it is possible to prove the global-in-time existence of solutions.

math.NA↗

arXiv2025

Observability inequality for the von Neumann equation in crystals

We provide a quantitative observability inequality for the von Neumann equation on $\mathbb{R}^d$ in the crystal setting, uniform in small $\hbar$. Following the method of Golse and Paul (2022) proving this result in the non-crystal setting, the method relies on a stability argument between the quantum (von Neumann) and classical (Liouville) dynamics and uses an optimal transport-like pseudo-distance between quantum and classical densities. Our contribution yields in the adaptation of all the required tools to the periodic setting, relying on the Bloch decomposition, notions of periodic Schrödinger coherent state, periodic Töplitz operator and periodic Husimi densities.

math.AP↗

arXiv2025

Nonlinear reduced basis using mixture Wasserstein barycenters: application to an eigenvalue problem inspired from quantum chemistry

The aim of this article is to propose a new reduced-order modelling approach for parametric eigenvalue problems arising in electronic structure calculations. Namely, we develop nonlinear reduced basis techniques for the approximation of parametric eigenvalue problems inspired from quantum chemistry applications. More precisely, we consider here a one-dimensional model which is a toy model for the computation of the electronic ground state wavefunction of a system of electrons within a molecule, solution to the many-body electronic Schrödinger equation, where the varying parameters are the positions of the nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of the set of solutions for this parametric problem in several settings, including the standard L2-norm as well as with distances based on optimal transport. The fact that the latter decays much faster than in the traditional L2-norm setting motivates us to propose a practical nonlinear reduced basis method, which is based on an offline greedy algorithm, and an efficient stochastic energy minimization in the online phase. We finally provide numerical results illustrating the capabilities of the method and good approximation properties, both in the offline and the online phase.

math.NA↗

arXiv2025

Optimal morphings for model-order reduction for poorly reducible problems with geometric variability

We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov $N$-width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first $r$ modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries.

math.NA↗

arXiv2025

Low-complexity approximations with least-squares formulation of the time-dependent Schr{ö}dinger equation

We propose new methods designed to numerically approximate the solution to the time dependent Schr{ö}dinger equation, based on two types of ansatz: tensors, and approximation by a linear combination of gaussian wave packets. In both cases, the method can be seen as a restricted optimization problem, which can be solved by adapting either the Alternating Least Square algorithm in the tensor case, or some greedy algorithm in the gaussian wavepacket case. We also discuss the efficiency of both approaches.

math.AP↗

arXiv2025

Convergence of two-scale expansions for elastic heterogeneous plates

The aim of this article is to prove strong convergence results on the difference between the solution to highly oscillatory problems posed in thin domains and its two-scale expansion. We first consider the case of the linear diffusion equation and establish such results in arbitrary dimensions, by using a straightforward adaptation of the classical arguments used for the homogenization of highly oscillatory problems posed on fixed (non-thin) domains. We next consider the linear elasticity problem, which raises challenging difficulties in its full generality. Under some classical assumptions on the symmetries of the elasticity tensor, the problem can be split into two independent problems, the membrane problem and the bending problem. Focusing on two-dimensional problems, we show that the membrane case can actually be addressed using a careful adaptation of classical arguments. In the bending case, the scheme of the proof used in the membrane and diffusion cases can however not be straightforwardly adapted. In that bending case, we establish the desired strong convergence results by using a different strategy of proof, which seems, up to our knowledge, to be new.

math.AP↗

arXiv2025

Elasticity-based morphing technique and application to reduced-order modeling

The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. The morphings are obtained by deforming a reference shape, through the resolution of a sequence of linear elasticity equations, onto every target shape. In particular, our approach does not assume any knowledge of a boundary parametrization, and the computation of the boundary deformation is not required beforehand. Furthermore, constraints can be imposed on specific points, lines and surfaces in the reference domain to ensure alignment with their counterparts in the target domain after morphing. Additionally, we show how the proposed methodology can be integrated in an offline and online paradigm, which is useful in reduced-order modeling involving variable shapes. This framework facilitates the efficient computation of the morphings in various geometric configurations, thus improving the versatility and applicability of the approach. The robustness and computational efficiency of the methodology is illustrated on two-dimensional test cases, including the regression problem of the drag and lift coefficients of airfoils of non-parameterized variable shapes.

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↗

arXiv2024

Multi-center decomposition of molecular densities: A numerical perspective

In this study, we analyze various Iterative Stockholder Analysis (ISA) methods for molecular density partitioning, focusing on the numerical performance of the recently proposed Linear approximation of Iterative Stockholder Analysis model (LISA) [J. Chem. Phys. 156, 164107 (2022)]. We first provide a systematic derivation of various iterative solvers to find the unique LISA solution. In a subsequent systematic numerical study, we evaluate their performance on 48 organic and inorganic, neutral and charged molecules and also compare LISA to two other well-known ISA variants: the Gaussian Iterative Stockholder Analysis (GISA) and Minimum Basis Iterative Stockholder analysis (MBIS). The study reveals that LISA-family methods can offer a numerically more efficient approach with better accuracy compared to the two comparative methods. Moreover, the well-known issue with the MBIS method, where atomic charges obtained for negatively charged molecules are anomalously negative, is not observed in LISA-family methods. Despite the fact that LISA occasionally exhibits elevated entropy as a consequence of the absence of more diffuse basis functions, this issue can be readily mitigated by incorporating additional or integrating supplementary basis functions within the LISA framework. This research provides the foundation for future studies on the efficiency and chemical accuracy of molecular density partitioning schemes.

physics.chem-ph↗

arXiv2024

Cross-diffusion systems coupled via a moving interface

We propose and study a one-dimensional model which consists of two cross-diffusion systems coupled via a moving interface. The motivation stems from the modelling of complex diffusion processes in the context of the vapor deposition of thin films. In our model, cross-diffusion of the various chemical species can be respectively modelled by a size-exclusion system for the solid phase and the Stefan-Maxwell system for the gaseous phase. The coupling between the two phases is modelled by linear phase transition laws of Butler-Volmer type, resulting in an interface evolution. The continuous properties of the model are investigated, in particular its entropy variational structure and stationary states. We introduce a two-point flux approximation finite volume scheme. The moving interface is addressed with a moving-mesh approach, where the mesh is locally deformed around the interface. The resulting discrete nonlinear system is shown to admit a solution that preserves the main properties of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints, decay of the free energy and asymptotics. In particular, the moving-mesh approach is compatible with the entropy structure of the continuous model. Numerical results illustrate these properties and the dynamics of the model.

math.AP↗