arXiv subjects

Johann Guilleminot

Publications and source records attributed to Johann Guilleminot.

9 recordsLinked to original sources

arXiv2026

Stable by Construction: Variational Latent Markov Operators for Long-Horizon PDE Prediction

Neural PDE solvers provide efficient surrogates for time-dependent physical systems, but autoregressive prediction over long horizons remains challenging because local errors can induce distribution shift and accumulate under recursive deployment. We develop a variational approach to this problem by introducing latent Markov dynamics in which physical states are represented by latent distributions and evolved through probabilistic transitions. The framework is formulated directly on function spaces and specialized to functional Gaussian models, where structured latent perturbations induce a spectral geometry and variational transition alignment regularizes the learned dynamics. We further analyze how these mechanisms affect autoregressive error propagation, providing a theoretical connection between variational training and long-horizon prediction. We instantiate the framework as the Variational Autoencoding Markov Operator (VAMO), which combines spatially resolved latent fields, structured Gaussian perturbations, and a neural-operator transition. Empirically, we demonstrate the effectiveness of VAMO on several fluid-dynamics benchmarks with prediction horizons extending substantially beyond those represented during training, where it consistently reduces error accumulation and improves rollout stability over several deterministic and noise-injection baselines. Overall, these results highlight variational modeling as a complementary approach to robust long-horizon neural PDE dynamics.

cs.LG↗

arXiv2026

Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators

Machine learning surrogates based on neural operators have shown broad applicability in solving forward PDE problems. However, eigenvalue problems, in which an eigenparameter and one of several valid eigenmodes must be simultaneously solved, remain difficult because standard operator learning formulations assume a unique input-output map. This work demonstrates that Fourier Neural Operators (FNOs), combined with wavelet-based encodings of PDE inputs, can learn and predict multiple eigenmodes of the elastic wave equation, corresponding to deformation modes of acoustic waves propagating through arbitrary metamaterial geometries. We provide a mechanistic explanation and experimental evidence for why wavelet encodings are well matched to the dual spatial-spectral structure of the FNO, enabling deterministic mode selection on both continuous-valued and binary-valued geometries within a single model, and for why prediction accuracy varies with geometric discontinuities. For metamaterial design, the resulting surrogate accelerates the simulation stage of the design cycle by three orders of magnitude relative to finite element analysis on a consumer-grade CPU, while preserving high fidelity. These results also carry broader implications for designing input encodings in other multi-mode PDE solvers based on spectral neural operators.

cs.LG↗

arXiv2026

A Dynamic Subspace Approach for Low-rank Approximation of Large-scale Nonlinear Systems

We present a dynamic subspace approach for efficiently approximating large-scale systems by learning time-continuous trajectories on the Grassmannian manifold. By parameterizing a low-dimensional basis as a geodesic path, the method allows for adaptive tracking of evolving physics. Our approach decouples the geometric drift of the subspace from the intrinsic state evolution. This avoids the typical rank inflation required by static low-dimensional approximation methods to maintain accuracy, effectively breaking the Kolmogorov barrier in transport-dominated phenomena. To ensure scalability for high-dimensional data, the optimization is performed in a reduced feature space, rendering the computational cost independent of the large original state dimension. Numerical results for a 1D transport equation and a large-scale turbulent airfoil wake demonstrate that this dynamic subspace approach achieves higher accuracy than static linear approximations at equivalent ranks, positioning it as a robust and scalable method for the low-rank modeling of complex, non-stationary dynamical systems.

math.NA↗

arXiv2026

Stochastic Modeling of Anisotropic Strength Surfaces from Atomistic Simulations

This work develops a unified framework for inferring, representing, and statistically characterizing an anisotropic strength surface directly from molecular dynamics data. Large-scale tensile loading simulations are used to generate failure data across all principal stress ratios and loading orientations, facilitated by a data-driven mapping between imposed strain-rate tensors and resulting stresses. The orientation-dependent strength surface is then represented using a constrained parametric formulation in which the surface parameters vary smoothly with loading angle through a low-dimensional functional encoding. To deploy the framework, we specifically consider the case of monocrystalline graphene, which is a prototypical two-dimensional material that has been extensively characterized, both experimentally and computationally, in the literature. For defective graphene, multiple random realizations of vacancy defect distributions are used to construct a stochastic ensemble of angular strength surfaces. Because each anisotropic strength surface requires substantial atomistic sampling to construct, the resulting ensemble is inherently limited in size, motivating the use of compact encoding, dimensionality reduction, and probabilistic modeling to characterize strength variability. Dimensionality reduction via Principal Component Analysis reveals a condensed latent representation of the fitted, encoded surfaces, where a Gaussian mixture model is employed to capture defect-induced variability, including rare outlier behaviors arising from clustered vacancy defects. Sampling from this probabilistic model enables the generation of new, physically admissible strength surfaces and the construction of confidence intervals in both parameter space and stress space. (Abstract shortened to meet arXiv limits.)

cond-mat.mtrl-sci↗

arXiv2024

Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach

This paper presents a probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques. Such uncertainties can arise in the selection of an appropriate state-space representation, in the projection step that underlies many reduced-order modeling methods, or as a byproduct of considerations made during training, to name a few. Following previous works in the literature, the proposed method captures these uncertainties by expanding the approximation space through the randomization of the projection matrix. This is achieved by combining Riemannian projection and retraction operators - acting on a subset of the Stiefel manifold - with an information-theoretic formulation. The efficacy of the approach is assessed on canonical problems in fluid mechanics by identifying and quantifying the impact of model-form uncertainties on the inferred operators.

stat.ML↗

arXiv2023

Uncertainty Quantification of Bandgaps in Acoustic Metamaterials with Stochastic Geometric Defects and Material Properties

This paper studies the utility of techniques within uncertainty quantification, namely spectral projection and polynomial chaos expansion, in reducing sampling needs for characterizing acoustic metamaterial dispersion band responses given stochastic material properties and geometric defects. A novel method of encoding geometric defects in an interpretable, resolution independent is showcased in the formation of input space probability distributions. Orders of magnitude sampling reductions down to $\sim10^0$ and $\sim10^1$ are achieved in the 1D and 7D input space scenarios respectively while maintaining accurate output space probability distributions through combining Monte Carlo, quadrature rule, and sparse grid sampling with surrogate model fitting.

cs.SD↗

arXiv2022

A Riemannian Stochastic Representation for Quantifying Model Uncertainties in Molecular Dynamics Simulations

A Riemannian stochastic representation of model uncertainties in molecular dynamics is proposed. The approach relies on a reduced-order model, the projection basis of which is randomized on a subset of the Stiefel manifold characterized by a set of linear constraints defining, e.g., Dirichlet boundary conditions in the physical space. We first show that these constraints are, indeed, preserved through Riemannian pushforward and pullback actions to, and from, the tangent space to the manifold at any admissible point. This fundamental property is subsequently exploited to derive a probabilistic model that leverages the multimodel nature of the atomistic setting. The proposed formulation offers several advantages, including a simple and interpretable low-dimensional parameterization, the ability to constraint the Fréchet mean on the manifold, and ease of implementation and propagation. The relevance of the proposed modeling framework is finally demonstrated on various applications including multiscale simulations on graphene-based systems.

physics.comp-ph↗

arXiv2017

On the construction of multiscale surrogates for design optimization of acoustical materials

This paper is concerned with the use of polynomial metamodels for the design of acoustical materials, considered as equivalent fluids. Polynomial series in microstructural parameters are considered, and allow us to approximate the multiscale solution map in some well-defined sense. The relevance of the framework is illustrated by considering the prediction of the sound absorption coefficient. In accordance with theoretical results provided elsewhere in the literature, it is shown that the surrogate model can accurately approximate the solution map at a reasonable computational cost, depending on the dimension of the input parameter space. Microstructural and process optimization by design are two envisioned applications.

physics.app-ph↗

arXiv2014

Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media

In this paper we develop a multiple scattering model for elastic waves in random anisotropic media. It relies on a kinetic approach of wave propagation phenomena pertaining to the situation whereby the wavelength is comparable to the correlation length of the weak random inhomogeneities--the so-called weak coupling limit. The waves are described in terms of their associated energy densities in the phase space position x wave vector. They satisfy radiative transfer equations in this scaling, characterized by collision operators depending on the correlation structure of the heterogeneities. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators, along the same lines as Bal [Wave Motion 43, 132-157 (2005)]. The model accounts for all possible polarizations of waves in anisotropic elastic media and their interactions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide. Thus it embodies isotropic elasticity which was considered in several previous publications. Some particular anisotropic cases of engineering interest are derived in detail.

math-ph↗