Christian Soize

In this paper, the problem of helium gas transport in graphene nanochannels is investigated by a multilevel approach coupling surrogate modeling and Monte Carlo simulations. Under the rarefaction condition, the Knudsen diffusion is the dominant gas flow regime, characterized by gas-wall collisions. Furthermore, at low temperatures, helium gas atoms can be adsorbed on the surface and move like Brownian particles. To address the problem, the collision data between gas atoms and wall surfaces in equilibrium state are first generated by molecular dynamics simulations. A general statistical surrogate model based on polynomial chaos expansion accounting for adsorption and surface diffusion is constructed. The model exploits the Langevin dynamics analogy to satisfy physical constraints on the velocity distribution, the decaying time-dependent velocity correlation, and time-dependent displacement behavior. The trained model is then used in Monte Carlo simulations to compute various transport quantities in long nanochannels. Numerical results from the Monte Carlo study show that the approach can capture the size effect on the diffusion coefficient and transmission probability. Without being limited to helium and graphene, the developed method can be applied to any gas and solid couple and the problem of construction of kinetic boundary conditions.

Numerical simulations are crucial for modeling complex systems, but calibrating them becomes challenging when data are noisy or incomplete and likelihood evaluations are computationally expensive. Bayesian calibration offers an interesting way to handle uncertainty, yet computing the posterior distribution remains a major challenge under such conditions. To address this, we propose a sequential surrogate-based approach that incrementally improves the approximation of the log-likelihood using Gaussian Process Regression. Starting from limited evaluations, the surrogate and its gradient are refined step by step. At each iteration, new evaluations of the expensive likelihood are added only at informative locations, that is to say where the surrogate is most uncertain and where the potential impact on the posterior is greatest. The surrogate is then coupled with the Metropolis-Adjusted Langevin Algorithm, which uses gradient information to efficiently explore the posterior. This approach accelerates convergence, handles relatively high-dimensional settings, and keeps computational costs low. We demonstrate its effectiveness on both a synthetic benchmark and an industrial application involving the calibration of high-speed train parameters from incomplete sensor data.

This paper presents a probabilistic learning method on random manifolds for building and updating statistical surrogate models using small datasets. The approach accounts for various types of variables, including random, controlled, and latent, forming a random manifold that links quantities of interest to controlled variables. This paper consolidates the Probabilistic Learning on Manifolds (PLoM) methodology, previously dispersed across multiple publications, and demonstrates its effectiveness through three diverse applications: Molecular dynamics, nonlinear elasticity, and updating under-observed nonlinear dynamic system with incomplete data. These examples illustrate the robustness of the method in managing complex and uncertain problems.