arXiv subjects

Shady E. Ahmed

Publications and source records attributed to Shady E. Ahmed.

At least 19 recordsLinked to original sources

arXiv2026

Uncertainty-aware Multi-fidelity Closure via Conditional Normalizing Flows

Reduced-order models (ROMs) provide efficient surrogates for complex multiscale systems, but their predictive accuracy is often compromised by truncation errors and the inadequate representation of interactions between resolved and unresolved scales. The missing effect of truncated (unresolved) scales on ROM (resolved) scales is often denoted as the closure problem. In this work, we formulate ROM closure modeling as a multi-fidelity (MF) learning problem and propose an uncertainty-aware MF framework based on conditional normalizing flows to enhance ROM predictive accuracy. The proposed approach learns a probabilistic mapping from low-fidelity (LF) ROM coefficients to high-fidelity (HF) coefficients, thereby improving predictive fidelity while quantifying the uncertainty associated with the learned closure. Two correction strategies are investigated: direct learning, in which HF coefficients are predicted directly from LF inputs, and residual learning, which learns the discrepancy between LF and HF coefficients. The framework is demonstrated on a double shear layer and vortex merging problems governed by the two-dimensional NavierStokes equations. Results show that both correction strategies improve ROM accuracy over uncorrected ROM, with residual learning achieving consistently better performance than direct learning. Moreover, the two proposed deep generative model-based strategies provide uncertainty quantification for the corrected ROM coefficients, which is critical for assessing prediction confidence and supporting the reliable use of ROMs in practical applications.

cs.LG↗

arXiv2026

From Classification to Regression: Using a Fruitfly to Solve Equations

We present a novel approach to regression tasks using classification which is motivated by the mechanism used by fruitflies to sense their environment. Specifically, we formulate a general framework for learning nonlinear input-output relationships by replacing complex global surrogate models with a finite library of representative local patterns. Since scientific data often occupy limited and recurring regions of the input space, we generate predictions by measuring similarities between a query and stored patterns, then combining their associated responses through weighted reconstruction. We apply this approach to nonlinear dynamical systems, data-driven regression, and physics-informed learning using suitable embeddings and similarity measures. For dynamical systems, our offline-online workflow extracts patterns from data or governing equations during the offline phase, while online prediction requires only similarity evaluation and response aggregation. This structure helps us reduce computational and memory demands while providing explicit control over the trade-off among accuracy, storage, and inference cost.

cs.LG↗

arXiv2026

Assessing Emulator Design and Training for Modal Aerosol Microphysics Parameterizations in E3SMv2

Toward the goal of using Scientific Machine Learning (SciML) emulators to improve the numerical representation of aerosol processes in global atmospheric models, we explore the emulation of aerosol microphysics processes under cloud-free conditions in the 4-mode Modal Aerosol Module (MAM4) within the Energy Exascale Earth System Model version 2 (E3SMv2). To develop an in-depth understanding of the challenges and opportunities in applying SciML to aerosol processes, we begin with a simple feedforward neural network architecture that has been used in earlier studies, but we systematically examine key emulator design choices, including architecture complexity and variable normalization, while closely monitoring training convergence behavior. Our results show that optimization convergence, scaling strategy, and network complexity strongly influence emulation accuracy. When effective scaling is applied and convergence is achieved, the relatively simple architecture, used together with a moderate network size, can reproduce key features of the microphysics-induced aerosol concentration changes with promising accuracy. These findings provide practical clues for the next stages of emulator development; they also provide general insights that are likely applicable to the emulation of other aerosol processes, as well as other atmospheric physics involving multi-scale variability.

physics.ao-ph↗

arXiv2025

Ensemble Kalman Filter for Data Assimilation coupled with low-resolution computations techniques applied in Fluid Dynamics

This paper presents an innovative Reduced-Order Model (ROM) for merging experimental and simulation data using Data Assimilation (DA) to estimate the "True" state of a fluid dynamics system, leading to more accurate predictions. Our methodology introduces a novel approach implementing the Ensemble Kalman Filter (EnKF) within a reduced-dimensional framework, grounded in a robust theoretical foundation and applied to fluid dynamics. To address the substantial computational demands of DA, the proposed ROM employs low-resolution (LR) techniques to drastically reduce computational costs. This approach involves downsampling datasets for DA computations, followed by an advanced reconstruction technique based on low-cost Singular Value Decomposition (lcSVD). The lcSVD method, a key innovation in this paper, has never been applied to DA before and offers a highly efficient way to enhance resolution with minimal computational resources. Our results demonstrate significant reductions in both computation time and RAM usage through the LR techniques without compromising the accuracy of the estimations. For instance, in a turbulent test case, the LR approach with a compression rate of 15.9 can achieve a speed-up of 13.7 and a RAM compression of 90.9% while maintaining a low Relative Root Mean Square Error (RRMSE) of 2.6%, compared to 0.8% in the high-resolution (HR) reference. Furthermore, we highlight the effectiveness of the EnKF in estimating and predicting the state of fluid flow systems based on limited observations and low-fidelity numerical data. This paper highlights the potential of the proposed DA method in fluid dynamics applications, particularly for improving computational efficiency in CFD and related fields. Its ability to balance accuracy with low computational and memory costs makes it suitable for large-scale and real-time applications, such as environmental monitoring or aerospace.

cs.CE↗

arXiv2024

Scientific machine learning for closure models in multiscale problems: a review

Closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation's accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machine-learned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed. In addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned models is a major issue that needs to be addressed further.

math.NA↗

arXiv2023

PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments

This paper describes PyOED, a highly extensible scientific package that enables developing and testing model-constrained optimal experimental design (OED) for inverse problems. Specifically, PyOED aims to be a comprehensive Python toolkit for model-constrained OED. The package targets scientists and researchers interested in understanding the details of OED formulations and approaches. It is also meant to enable researchers to experiment with standard and innovative OED technologies with a wide range of test problems (e.g., simulation models). OED, inverse problems (e.g., Bayesian inversion), and data assimilation (DA) are closely related research fields, and their formulations overlap significantly. Thus, PyOED is continuously being expanded with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, observation error models, and observation operators. These pieces are added such that they can be permuted to enable testing OED methods in various settings of varying complexities. The PyOED core is completely written in Python and utilizes the inherent object-oriented capabilities; however, the current version of PyOED is meant to be extensible rather than scalable. Specifically, PyOED is developed to enable rapid development and benchmarking of OED methods with minimal coding effort and to maximize code reutilization. This paper provides a brief description of the PyOED layout and philosophy and provides a set of exemplary test cases and tutorials to demonstrate the potential of the package.

cs.MS↗

arXiv2023

A Multifidelity deep operator network approach to closure for multiscale systems

Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed "in-the-loop" training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier-Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

physics.comp-ph↗

arXiv2023

Forward Sensitivity Analysis and Mode Dependent Control for Closure Modeling of Galerkin Systems

Model reduction by projection-based approaches is often associated with losing some of the important features that contribute towards the dynamics of the retained scales. As a result, a mismatch occurs between the predicted trajectories of the original system and the truncated one. We put forth a framework to apply a continuous time control signal in the latent space of the reduced order model (ROM) to account for the effect of truncation. We set the control input using parameterized models by following energy transfer principles. Our methodology relies on observing the system behavior in the physical space and using the projection operator to restrict the feedback signal into the latent space. Then, we leverage the forward sensitivity method (FSM) to derive relationships between the feedback and the desired mode-dependent control. We test the performance of the proposed approach using two test cases, corresponding to viscous Burgers and vortex merger problems at high Reynolds number. Results show that the ROM trajectory with the applied FSM control closely matches its target values in both the data-dense and data-sparse regimes.

physics.flu-dyn↗

arXiv2022

Nonintrusive reduced order modeling of convective Boussinesq flows

In this paper, we formulate three nonintrusive methods and systematically explore their performance in terms of the ability to reconstruct the quantities of interest and their predictive capabilities. The methods include deterministic dynamic mode decomposition (DMD), randomized DMD and nonlinear proper orthogonal decomposition (NLPOD). We apply these methods to a convection dominated fluid flow problem governed by the Boussinesq equations. We analyze the reconstruction results primarily at two different times for considering different noise levels synthetically added into the data snapshots. Overall, our results indicate that, with a proper selection of the number of retained modes and neural network architectures, all three approaches make predictions that are in a good agreement with the full order model solution. However, we find that the NLPOD approach seems more robust for higher noise levels compared to both DMD approaches.

physics.flu-dyn↗

arXiv2022

Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling

We propose a new physics guided machine learning (PGML) paradigm that leverages the variational multiscale (VMS) framework and available data to dramatically increase the accuracy of reduced order models (ROMs) at a modest computational cost. The hierarchical structure of the ROM basis and the VMS framework enable a natural separation of the resolved and unresolved ROM spatial scales. Modern PGML algorithms are used to construct novel models for the interaction among the resolved and unresolved ROM scales. Specifically, the new framework builds ROM operators that are closest to the true interaction terms in the VMS framework. Finally, machine learning is used to reduce the projection error and further increase the ROM accuracy. Our numerical experiments for a two-dimensional vorticity transport problem show that the novel PGML-VMS-ROM paradigm maintains the low computational cost of current ROMs, while significantly increasing the ROM accuracy.

physics.flu-dyn↗

arXiv2022

Sketching Methods for Dynamic Mode Decomposition in Spherical Shallow Water Equations

Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.

math.NA↗

arXiv2021

Nonlinear proper orthogonal decomposition for convection-dominated flows

Autoencoder techniques find increasingly common use in reduced order modeling as a means to create a latent space. This reduced order representation offers a modular data-driven modeling approach for nonlinear dynamical systems when integrated with a time series predictive model. In this letter, we put forth a nonlinear proper orthogonal decomposition (POD) framework, which is an end-to-end Galerkin-free model combining autoencoders with long short-term memory networks for dynamics. By eliminating the projection error due to the truncation of Galerkin models, a key enabler of the proposed nonintrusive approach is the kinematic construction of a nonlinear mapping between the full-rank expansion of the POD coefficients and the latent space where the dynamics evolve. We test our framework for model reduction of a convection-dominated system, which is generally challenging for reduced order models. Our approach not only improves the accuracy, but also significantly reduces the computational cost of training and testing.

physics.flu-dyn↗

arXiv2021

On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr-Sommerfeld stability equation and numerous vortex models, of which the von Kármán vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [N. Aubry, P. Holmes, J. Lumley, and E. Stone, Journal of Fluid Mechanics, 192, 115-173 (1988)] pioneered data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data was used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM. But data has made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

physics.flu-dyn↗

arXiv2021

Multifidelity Computing for Coupling Full and Reduced Order Models

Hybrid physics-machine learning models are increasingly being used in simulations of transport processes. Many complex multiphysics systems relevant to scientific and engineering applications include multiple spatiotemporal scales and comprise a multifidelity problem sharing an interface between various formulations or heterogeneous computational entities. To this end, we present a robust hybrid analysis and modeling approach combining a physics-based full order model (FOM) and a data-driven reduced order model (ROM) to form the building blocks of an integrated approach among mixed fidelity descriptions toward predictive digital twin technologies. At the interface, we introduce a long short-term memory network to bridge these high and low-fidelity models in various forms of interfacial error correction or prolongation. The proposed interface learning approaches are tested as a new way to address ROM-FOM coupling problems solving nonlinear advection-diffusion flow situations with a bifidelity setup that captures the essence of a broad class of transport processes.

physics.flu-dyn↗

arXiv2021

Forward sensitivity analysis of the FitzHugh-Nagumo system: Parameter estimation

The FitzHugh-Nagumo (FHN) model, from computational neuroscience, has attracted attention in nonlinear dynamics studies as it describes the behavior of excitable systems and exhibits interesting bifurcation properties. The accurate estimation of the model parameters is vital to understand how the solution trajectory evolves in time. To this end, we provide a forward sensitivity method (FSM) approach to quantify the main model parameters using sparse measurement data. FSM constitutes a variational data assimilation technique which integrates model sensitivities into the process of fitting the model to the observations. We analyse the applicability of FSM to update the FHN model parameters and predict its dynamical characteristics. Furthermore, we highlight a few guidelines for observations placement to control the shape of the cost functional and improve the parameter inference iterations.

nlin.PS↗

arXiv2021

Hybrid analysis and modeling for next generation of digital twins

The physics-based modeling has been the workhorse for many decades in many scientific and engineering applications ranging from wind power, weather forecasting, and aircraft design. Recently, data-driven models are increasingly becoming popular in many branches of science and engineering due to their non-intrusive nature and online learning capability. Despite the robust performance of data-driven models, they are faced with challenges of poor generalizability and difficulty in interpretation. These challenges have encouraged the integration of physics-based models with data-driven models, herein denoted hybrid analysis and modeling (HAM). We propose two different frameworks under the HAM paradigm for applications relevant to wind energy in order to bring the physical realism within emerging digital twin technologies. The physics-guided machine learning (PGML) framework reduces the uncertainty of neural network predictions by embedding physics-based features from a simplified model at intermediate layers and its performance is demonstrated for the aerodynamic force prediction task. Our results show that the proposed PGML framework achieves approximately 75\% reduction in uncertainty for smaller angle of attacks. The interface learning (IL) framework illustrates how different solvers can be coupled to produce a multi-fidelity model and is successfully applied for the Boussinesq equations that govern a broad class of transport processes. The IL approach paves the way for seamless integration of multi-scale, multi-physics and multi-fidelity models (M^3 models).

physics.comp-ph↗

arXiv2020

Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction

In this paper, we propose a variational approach to estimate eddy viscosity using forward sensitivity method (FSM) for closure modeling in nonlinear reduced order models. FSM is a data assimilation technique that blends model's predictions with noisy observations to correct initial state and/or model parameters. We apply this approach on a projection based reduced order model (ROM) of the one-dimensional viscous Burgers equation with a square wave defining a moving shock, and the two-dimensional vorticity transport equation formulating a decay of Kraichnan turbulence. We investigate the capability of the approach to approximate an optimal value for eddy viscosity with different measurement configurations. Specifically, we show that our approach can sufficiently assimilate information either through full field or sparse noisy measurements to estimate eddy viscosity closure to cure standard Galerkin reduced order model (GROM) predictions. Therefore, our approach provides a modular framework to correct forecasting error from a sparse observational network on a latent space. We highlight that the proposed GROM-FSM framework is promising for emerging digital twin applications, where real-time sensor measurements can be used to update and optimize surrogate model's parameters.

math.DS↗

arXiv2020

Interface learning of multiphysics and multiscale systems

Complex natural or engineered systems comprise multiple characteristic scales, multiple spatiotemporal domains, and even multiple physical closure laws. To address such challenges, we introduce an interface learning paradigm and put forth a data-driven closure approach based on memory embedding to provide physically correct boundary conditions at the interface. To enable the interface learning for hyperbolic systems by considering the domain of influence and wave structures into account, we put forth the concept of upwind learning towards a physics-informed domain decomposition. The promise of the proposed approach is shown for a set of canonical illustrative problems. We highlight that high-performance computing environments can benefit from this methodology to reduce communication costs among processing units in emerging machine learning ready heterogeneous platforms toward exascale era.

physics.comp-ph↗