arXiv subjects

Moaad Khamlich

Publications and source records attributed to Moaad Khamlich.

9 recordsLinked to original sources

arXiv2026

A Multi-Fidelity Parametric Framework for Reduced-Order Modeling using Optimal Transport-based Interpolation: Applications to Diffused-Interface Two-Phase Flows

This work introduces a data-driven, non-intrusive reduced-order modeling (ROM) framework that leverages Optimal Transport (OT) for multi-fidelity and parametric problems in two-phase flows modelling. Building upon the success of displacement interpolation for data augmentation in handling nonlinear dynamics, we extend its application to more complex and practical scenarios. The framework is designed to correct a computationally inexpensive low-fidelity (LF) model to match an accurate high-fidelity (HF) one by capturing its temporal evolution via displacement interpolation while preserving the problem's physical consistency. The framework is further extended to address systems dependent on a physical parameter, for which we construct a surrogate model using a hierarchical, two-level interpolation strategy. First, it creates synthetic HF checkpoints via displacement interpolation in the parameter space. Second, the residual between these synthetic HF checkpoints and a true LF solution is interpolated in the time domain using the multi-fidelity OT-based methodology. This strategy provides a robust and efficient way to explore the parameter space and to obtain a refined description of the dynamical system. The potential of the method is discussed in the context of complex and computationally expensive diffuse-interface methods for two-phase flow simulations, which are characterized by moving interfaces and nonlinear evolution, and challenging to be dealt with traditional ROM techniques.

math.NA↗

arXiv2025

Projection-based model order reduction for residence time distribution analysis of an industrial-scale continuous casting tundish

The flow behavior in the continuous casting tundish plays a critical role in steel quality and is typically characterized via residence time distribution (RTD) curves. This study investigates the fluid flow behaviour in a single-strand tundish using numerical and experimental approaches. Full-order model (FOM) steady-state simulations were conducted under both isothermal and non-isothermal conditions to assess the influence of thermal buoyancy on the flow characteristics. The results show that buoyancy effects under non-isothermal conditions have a negligible impact on the overall velocity field. The converged flow fields serve as initial conditions for transient tracer transport simulations, enabling evaluation of RTD curves and volume partitioning. A Galerkin projection-based reduced-order model (ROM) is developed to efficiently derive RTD curves. Comparison of RTD curves from experiments, FOM simulations, and ROM predictions demonstrates strong agreement, with both computational approaches closely matching experimental data. A parameter-time dependent ROM is subsequently developed using Galerkin projection and operator interpolation. This enables efficient evaluation of RTD curves across varying parameter values with significantly reduced computational cost compared to full-order simulations. The ROM framework is well-suited for real-time analysis, design processes, optimization, and digital twin applications in metallurgical processes.

physics.flu-dyn↗

arXiv2025

Efficient Numerical Strategies for Entropy-Regularized Semi-Discrete Optimal Transport

Semi-discrete optimal transport (SOT), which maps a continuous probability measure to a discrete one, is a fundamental problem with wide-ranging applications. Entropic regularization is often employed to solve the SOT problem, leading to a regularized (RSOT) formulation that can be solved efficiently via its convex dual. However, a significant computational challenge emerges when the continuous source measure is discretized via the finite element (FE) method to handle complex geometries or densities, such as those arising from solutions to Partial Differential Equations (PDEs). The evaluation of the dual objective function requires dense interactions between the numerous source quadrature points and all target points, creating a severe bottleneck for large-scale problems. This paper presents a cohesive framework of numerical strategies to overcome this challenge. We accelerate the dual objective and gradient evaluations by combining distance-based truncation with fast spatial queries using R-trees. For overall convergence, we integrate multilevel techniques based on hierarchies of both the FE source mesh and the discrete target measure, alongside a robust scheduling strategy for the regularization parameter. When unified, these methods drastically reduce the computational cost of RSOT, enabling its practical application to complex, large-scale scenarios. We provide an open-source C++ implementation of this framework, built upon the deal.II finite element library, available at https://github.com/SemiDiscreteOT/SemiDiscreteOT.

math.NA↗

arXiv2025

Sparse Identification for bifurcating phenomena in Computational Fluid Dynamics

This work investigates model reduction techniques for nonlinear parameterized and time-dependent PDEs, specifically focusing on bifurcating phenomena in Computational Fluid Dynamics (CFD). We develop interpretable and non-intrusive Reduced Order Models (ROMs) capable of capturing dynamics associated with bifurcations by identifying a minimal set of coordinates. Our methodology combines the Sparse Identification of Nonlinear Dynamics (SINDy) method with a deep learning framework based on Autoencoder (AE) architectures. To enhance dimensionality reduction, we integrate a nested Proper Orthogonal Decomposition (POD) with the SINDy-AE architecture, enabling a sparse discovery of system dynamics while maintaining efficiency of the reduced model. We demonstrate our approach via two challenging test cases defined on sudden-expansion channel geometries: a symmetry-breaking bifurcation and a Hopf bifurcation. Starting from a comprehensive analysis of their high-fidelity behavior, i.e. symmetry-breaking phenomena and the rise of unsteady periodic solutions, we validate the accuracy and computational efficiency of our ROMs. The results show successful reconstruction of the bifurcations, accurate prediction of system evolution for unseen parameter values, and significant speed-up compared to full-order methods.

math.NA↗

arXiv2024

Optimal Transport-Based Displacement Interpolation with Data Augmentation for Reduced Order Modeling of Nonlinear Dynamical Systems

We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses these issues by introducing a data augmentation strategy based on OT principles. The proposed framework generates interpolated solutions tracing geodesic paths in the space of probability distributions, enriching the training dataset for the ROM. A key feature of our approach is its ability to provide a continuous representation of the solution's dynamics by exploiting a virtual-to-real time mapping. This enables the reconstruction of solutions at finer temporal scales than those provided by the original data. To further improve prediction accuracy, we employ Gaussian Process Regression to learn the residual and correct the representation between the interpolated snapshots and the physical solution. We demonstrate the effectiveness of our methodology with atmospheric mesoscale benchmarks characterized by highly nonlinear, advection-dominated dynamics. Our results show improved accuracy and efficiency in predicting complex system behaviors, indicating the potential of this approach for a wide range of applications in computational physics and engineering.

math.NA↗

arXiv2024

A stochastic perturbation approach to nonlinear bifurcating problems

Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems, especially when the solution is not unique or exhibits sudden qualitative changes as parameters vary. However, stochastic models typically require large computational resources to produce meaningful statistics. In this work, we leverage the Polynomial Chaos (PC) expansion to propose a systematic approach for bifurcation detection in parametric systems of equations. We show that the method, exploiting a perturbed version of the deterministic model, avoids repeated costly simulations across multiple parameter values and requires no prior information for initializing numerical solvers, while still providing accurate characterization of the bifurcation branches. We argue that the PC solutions of the perturbed model not only provide access to statistical information about the deterministic branches, but also approximate these branches in a meaningful sense. Finally, we validate our claims by means of numerical tests on the pitchfork bifurcation, examining both its normal form and a classical realization in fluid-dynamics PDEs, namely the Coanda effect.

math.NA↗

arXiv2023

Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Kolmogorov n-width Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel

Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics encompass the underlying structure, relationships, and essential features crucial for accurate modeling. To overcome this limitation, we propose a novel ROM framework that integrates optimal transport (OT) theory and neural network-based methods. Specifically, we investigate the Kernel Proper Orthogonal Decomposition (kPOD) method exploiting the Wasserstein distance as the custom kernel, and we efficiently train the resulting neural network (NN) employing the Sinkhorn algorithm. By leveraging an OT-based nonlinear reduction, the presented framework can capture the geometric structure of the data, which is crucial for accurate learning of the reduced solution manifold. When compared with traditional metrics such as mean squared error or cross-entropy, exploiting the Sinkhorn divergence as the loss function enhances stability during training, robustness against overfitting and noise, and accelerates convergence. To showcase the approach's effectiveness, we conduct experiments on a set of challenging test cases exhibiting a slow decay of the Kolmogorov n-width. The results show that our framework outperforms traditional ROM methods in terms of accuracy and computational efficiency.

math.NA↗

arXiv2023

A physics-based reduced order model for urban air pollution prediction

This article presents an innovative approach for developing an efficient reduced-order model to study the dispersion of urban air pollutants. The need for real-time air quality monitoring has become increasingly important, given the rise in pollutant emissions due to urbanization and its adverse effects on human health. The proposed methodology involves solving the linear advection-diffusion problem, where the solution of the Reynolds-averaged Navier-Stokes equations gives the convective field. At the same time, the source term consists of an empirical time series. However, the computational requirements of this approach, including microscale spatial resolution, repeated evaluation, and low time scale, necessitate the use of high-performance computing facilities, which can be a bottleneck for real-time monitoring. To address this challenge, a problem-specific methodology was developed that leverages a data-driven approach based on Proper Orthogonal Decomposition with regression (POD-R) coupled with Galerkin projection (POD-G) endorsed with the discrete empirical interpolation method (DEIM). The proposed method employs a feedforward neural network to non-intrusively retrieve the reduced-order convective operator required for online evaluation. The numerical framework was validated on synthetic emissions and real wind measurements. The results demonstrate that the proposed approach significantly reduces the computational burden of the traditional approach and is suitable for real-time air quality monitoring. Overall, the study advances the field of reduced order modeling and highlights the potential of data-driven approaches in environmental modeling and large-scale simulations.

math.NA↗

arXiv2021

Model order reduction for bifurcating phenomena in Fluid-Structure Interaction problems

This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand\u{a} effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a Fluid-Structure Interaction problem, we aim at generalizing previous works towards a more reliable description of the physics involved. In particular, we provide several insights on how the introduction of an elastic structure influences the bifurcating behaviour. We have addressed the computational burden by developing a reduced order branch-wise algorithm based on a monolithic Proper Orthogonal Decomposition. We compared different constitutive relations for the solid, and we observed that a nonlinear hyper-elastic law delays the bifurcation w.r.t. the standard model, while the same effect is even magnified when considering linear elastic solid.

math.NA↗