arXiv subjects

Gianluigi Rozza

Publications and source records attributed to Gianluigi Rozza.

At least 73 records · Page 4Linked to original sources

arXiv2025

Generative Adversarial Reduced Order Modelling

In this work, we present GAROM, a new approach for reduced order modelling (ROM) based on generative adversarial networks (GANs). GANs have the potential to learn data distribution and generate more realistic data. While widely applied in many areas of deep learning, little research is done on their application for ROM, i.e. approximating a high-fidelity model with a simpler one. In this work, we combine the GAN and ROM framework, by introducing a data-driven generative adversarial model able to learn solutions to parametric differential equations. The latter is achieved by modelling the discriminator network as an autoencoder, extracting relevant features of the input, and applying a conditioning mechanism to the generator and discriminator networks specifying the differential equation parameters. We show how to apply our methodology for inference, provide experimental evidence of the model generalisation, and perform a convergence study of the method.

cs.LG↗

arXiv2025

Jacobi convolution series for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite interval

Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution series as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution series is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator.

math.NA↗

arXiv2025

A hybrid Reduced Order Model to enforce outflow pressure boundary conditions in computational haemodynamics

This paper deals with the development of a Reduced-Order Model (ROM) to investigate haemodynamics in cardiovascular applications. It employs the use of Proper Orthogonal Decomposition (POD) for the computation of the basis functions and the Galerkin projection for the computation of the reduced coefficients. The main novelty of this work lies in the extension of the lifting function method, which typically is adopted for treating nonhomogeneous inlet velocity boundary conditions, to the handling of nonhomogeneous outlet boundary conditions for the pressure, representing a very delicate point in the numerical simulations of the cardiovascular system. Moreover, we incorporate a properly trained neural network in the ROM framework to approximate the mapping from the time parameter to the outflow pressure, which in the most general case is not available in closed form. We define our approach as "hybrid", because it merges physics-based elements with data-driven ones. At full order level, a Finite Volume method is employed for the discretization of the unsteady Navier-Stokes equations while a two-element Windkessel model is adopted to enforce a valuable estimation of the outflow pressure. Numerical results, firstly related to a 2D idealized blood vessel and then to a 3D patient-specific aortic arch, demonstrate that our ROM is able to accurately approximate the FOM with a significant reduction in the computational cost.

math.NA↗

arXiv2025

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↗

arXiv2024

Data-driven Discovery of Delay Differential Equations with Discrete Delays

The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a robust method for identifying governing equations, successfully applied to ordinary, partial, and stochastic differential equations. In this work we extend SINDy to identify delay differential equations by using an augmented library that includes delayed samples and Bayesian optimization. To identify a possibly unknown delay we minimize the reconstruction error over a set of candidates. The resulting methodology improves the overall performance by remarkably reducing the number of calls to SINDy with respect to a brute force approach. We also address a multivariate setting to identify multiple unknown delays and (non-multiplicative) parameters. Several numerical tests on delay differential equations with different long-term behavior, number of variables, delays, and parameters support the use of Bayesian optimization highlighting both the efficacy of the proposed methodology and its computational advantages. As a consequence, the class of discoverable models is significantly expanded.

math.NA↗

arXiv2024

Deep learning-based reduced-order methods for fast transient dynamics

In recent years, large-scale numerical simulations played an essential role in estimating the effects of explosion events in urban environments, for the purpose of ensuring the security and safety of cities. Such simulations are computationally expensive and, often, the time taken for one single computation is large and does not permit parametric studies. The aim of this work is therefore to facilitate real-time and multi-query calculations by employing a non-intrusive Reduced Order Method (ROM). We propose a deep learning-based (DL) ROM scheme able to deal with fast transient dynamics. In the case of blast waves, the parametrised PDEs are time-dependent and non-linear. For such problems, the Proper Orthogonal Decomposition (POD), which relies on a linear superposition of modes, cannot approximate the solutions efficiently. The piecewise POD-DL scheme developed here is a local ROM based on time-domain partitioning and a first dimensionality reduction obtained through the POD. Autoencoders are used as a second and non-linear dimensionality reduction. The latent space obtained is then reconstructed from the time and parameter space through deep forward neural networks. The proposed scheme is applied to an example consisting of a blast wave propagating in air and impacting on the outside of a building. The efficiency of the deep learning-based ROM in approximating the time-dependent pressure field is shown.

math.NA↗

arXiv2024

Stochastic Parameter Prediction in Cardiovascular Problems

Patient-specific modeling of cardiovascular flows with high-fidelity is challenging due to its dependence on accurately estimated velocity boundary profiles, which are essential for precise simulations and directly influence wall shear stress calculations - key in predicting cardiovascular diseases like atherosclerosis. This data, often derived from in vivo modalities like 4D flow MRI, suffers from low resolution and noise. To address this, we employ a stochastic data assimilation technique that integrates computational fluid dynamics with an advanced Ensemble-based Kalman filter, enhancing model accuracy while accounting for uncertainties. Our approach sequentially collects velocity data over time within the vascular model, enabling real-time refinement of unknown boundary estimations. The mathematical model uses the incompressible Navier-Stokes equation to simulate aortic blood flow. We consider unknown boundaries as constant, time-dependent, and space-time dependent in two- and three-dimensional models. In our 2-dimensional model, relative errors were as low as 0.996\% for constant boundaries and up to 2.63\% and 2.61\% for time-dependent and space-time dependent boundaries, respectively, over an observation span of two-time steps. For the 3-dimensional patient-specific model, the relative error was 7.37\% for space-time dependent boundaries. By refining the velocity boundary profile, our method improves wall shear stress predictions, enhancing the accuracy and reliability of models specific to individual cardiovascular patients. These advancements could contribute to better diagnosis and treatment of cardiovascular diseases.

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 data-driven study on Implicit LES using a spectral difference method

In this paper, we introduce a data-driven filter to analyze the relationship between Implicit Large-Eddy Simulations (ILES) and Direct Numerical Simulations (DNS) in the context of the Spectral Difference method. The proposed filter is constructed from a linear combination of sharp-modal filters where the weights are given by a convolutional neural network trained to replicate ILES results from filtered DNS data. In order to preserve the compactness of the discretization, the filter is local in time and acts at the elementary cell level. The neural network is trained on the data generated from the Taylor-Green Vortex test-case at Re=1600. In order to mitigate the temporal effects and highlight the influence of the spatial discretization, the ILES are periodically restarted from DNS data for different time windows. Smaller time windows result in higher cross-correlations between ILES and the filtered DNS snapshots using the data-driven filters. The modal decay of the filter for the smallest time window considered aligns with classical eigenanalysis, showing better energy conservation for higher orders of approximation. Similarly, an analysis of the filter's kernel in the Fourier space confirms that higher polynomial orders are less dissipative compared to lower orders. As large time windows are considered, the trained filter encounters difficulties in representing the data due to significant non linear effects. Additionally, the impact of the data-driven filter on the resolved kinetic energy has been assessed through the evaluation of the sub-grid production term which results in both direct and inverse cascades with the former being more likely on average. The presence of backscatter suggests that ILES based on Discontinuous Spectral Element Methods might be equipped with an intrinsic mechanisms to transfer energy in both directions with a predominance of direct kinetic energy cascade.

physics.flu-dyn↗

arXiv2024

On the accuracy and efficiency of reduced order models: towards real-world applications

This chapter provides an extended overview about Reduced Order Models (ROMs), with a focus on their features in terms of efficiency and accuracy. In particular, the aim is to browse the more common ROM frameworks, considering both intrusive and data-driven approaches. We present the validation of such techniques against several test cases. The first one is an academic benchmark, the thermal block problem, where a Poisson equation is considered. Here a classic intrusive ROM framework based on a Galerkin projection scheme is employed. The second and third test cases come from real-world applications, the one related to the investigation of the blood flow patterns in a patient specific coronary arteries configuration where the Navier Stokes equations are addressed and the other one concerning the granulation process within pharmaceutical industry where a fluid-particle system is considered. Here we employ two data-driven ROM approaches showing a very relevant trade-off between accuracy and efficiency. In the last part of the contribution, two novel technological platforms, ARGOS and ATLAS, are presented. They are designed to provide a user-friendly access to data-driven models for real-time predictions for complex biomedical and industrial problems.

math.NA↗

arXiv2024

Data-driven reduced order modeling of a two-layer quasi-geostrophic ocean model

The two-layer quasi-geostrophic equations (2QGE) is a simplified model that describes the dynamics of a stratified, wind-driven ocean in terms of potential vorticity and stream function. Its numerical simulation is plagued by a high computational cost due to the size of the typical computational domain and the need for high resolution to capture the full spectrum of turbulent scales. In this paper, we present a data-driven reduced order model (ROM) for the 2QGE that drastically reduces the computational time to predict ocean dynamics, especially when there are variable physical parameters. The main building blocks of our ROM are: i) proper orthogonal decomposition (POD) and ii) long short-term memory (LSTM) recurrent neural networks. Snapshots data are collected from a high-resolution simulation for part of the time interval of interest and for given parameter values in the case of variable parameters. POD is applied to each field variable to extract the dominant modes and a LSTM model is trained on the modal coefficients associated with the snapshots for each variable. Then, the trained LSTM models predict the modal coefficients for the remaining part of the time interval of interest and for a new parameter value. To illustrate the predictive performance of our POD-LSTM ROM and the corresponding time savings, we consider an extension of the so-called double-gyre wind forcing test. We show that the POD-LSTM ROM is accurate in predicting both time-averaged fields and time-dependent quantities (modal coefficients, enstrophy, and kinetic energy), even when retaining only 10-20\% of the singular value energy of the system. The computational speed up for the prediction is about up to 1E+07 compared to a finite volume based full order method.

math.NA↗

arXiv2024

Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs

In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet number, we consider a Streamline Upwind Petrov-Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

math.NA↗

arXiv2024

Mathematical modelling and computational reduction of molten glass fluid flow in a furnace melting basin

In this work, we present the modelling and numerical simulation of a molten glass fluid flow in a furnace melting basin. We first derive a model for a molten glass fluid flow and present numerical simulations based on the Finite Element Method (FEM). We further discuss and validate the results obtained from the simulations by comparing them with experimental results. Finally, we also present a non-intrusive Proper Orthogonal Decomposition (POD) based on Artificial Neural Networks (ANN) to efficiently handle scenarios which require multiple simulations of the fluid flow upon changing parameters of relevant industrial interest. This approach lets us obtain solutions of a complex 3D model, with good accuracy with respect to the FEM solution, yet with negligible associated computational times.

physics.flu-dyn↗

arXiv2024

Optimisation-Based Coupling of Finite Element Model and Reduced Order Model for Computational Fluid Dynamics

Using Domain Decomposition (DD) algorithm on non--overlapping domains, we compare couplings of different discretisation models, such as Finite Element (FEM) and Reduced Order (ROM) models for separate subcomponents. In particular, we consider an optimisation-based DD model where the coupling on the interface is performed using a control variable representing the normal flux. We use iterative gradient-based optimisation algorithms to decouple the subdomain state solutions as well as to locally generate ROMs on each subdomain. Then, we consider FEM or ROM discretisation models for each of the DD problem components, namely, the triplet state1-state2-control. On the backward-facing step Navier-Stokes (NS) problem, we investigate the efficacy of the presented couplings in terms of optimisation iterations, optimal functional values and relative errors.

math.NA↗

arXiv2024

A time-adaptive algorithm for pressure dominated flows: a heuristic estimator

This work aims to introduce a heuristic timestep-adaptive algorithm for Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems where the flow is dominated by the pressure. In such scenarios, many time-adaptive algorithms based on the interplay of implicit and explicit time schemes fail to capture the fast transient dynamics of pressure fields. We present an algorithm that relies on a temporal error estimator using Backward Differentiation Formulae (BDF$k$) of order $k=2,3$. Specifically, we demonstrate that the implicit BDF$3$ solution can be well approximated by applying a single Newton-type nonlinear solver correction to the implicit BDF$2$ solution. The difference between these solutions determines our adaptive temporal error estimator. The effectiveness of our approach is confirmed by numerical experiments conducted on a backward-facing step flow CFD test case with Reynolds number $300$ and on a two-dimensional haemodynamics FSI benchmark.

math.NA↗

arXiv2024

A Numerical Proof of Shell Model Turbulence Closure

The development of turbulence closure models, parametrizing the influence of small non-resolved scales on the dynamics of large resolved ones, is an outstanding theoretical challenge with vast applicative relevance. We present a closure, based on deep recurrent neural networks, that quantitatively reproduces, within statistical errors, Eulerian and Lagrangian structure functions and the intermittent statistics of the energy cascade, including those of subgrid fluxes. To achieve high-order statistical accuracy, and thus a stringent statistical test, we employ shell models of turbulence. Our results encourage the development of similar approaches for 3D Navier-Stokes turbulence.

physics.flu-dyn↗

arXiv2024

A hybrid reduced-order model for segregated fluid-structure interaction solvers in an ALE approach at high Reynolds number

This study introduces a first step for constructing a hybrid reduced-order models (ROMs) for segregated fluid-structure interaction in an Arbitrary Lagrangian-Eulerian (ALE) approach at a high Reynolds number using the Finite Volume Method (FVM). The ROM is driven by proper orthogonal decomposition (POD) with hybrid techniques that combines the classical Galerkin projection and two data-driven methods (radial basis networks , and neural networks/ long short term memory). Results demonstrate the ROM ability to accurately capture the physics of fluid-structure interaction phenomena. This approach is validated through a case study focusing on flow-induced vibration (FIV) of a pitch-plunge airfoil at a high Reynolds number 10000000.

physics.flu-dyn↗

arXiv2024

Parametric Intrusive Reduced Order Models enhanced with Machine Learning Correction Terms

In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in standard ROMs are not taken into account. In particular, in this work we consider two types of contributions: the turbulence modeling, added through a reduced-order approximation of the eddy viscosity field, and the correction model, aimed to re-introduce the contribution of the discarded modes. Both approaches have been investigated in previous works and the goal of this paper is to extend the model to a parametric setting making use of ad-hoc machine learning procedures. More in detail, we investigate different neural networks' architectures, from simple dense feed-forward to Long-Short Term Memory neural networks, in order to find the most suitable model for the re-introduced contributions. We tested the methods on two test cases with different behaviors: the periodic turbulent flow past a circular cylinder and the unsteady turbulent flow in a channel-driven cavity. In both cases, the parameter considered is the Reynolds number and the machine learning-enhanced ROM considerably improved the pressure and velocity accuracy with respect to the standard ROM.

math.NA↗