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Gianluigi Rozza

Publications and source records attributed to Gianluigi Rozza.

At least 19 recordsLinked to original sources
arXiv2026

A Comparative Study of Finite-Volume-based Coupled and Segregated Reduced-Order Models for Incompressible Flows in Parametrized Domains

This work presents a comparative analysis of Reduced-Order Models (ROMs) applied to incompressible fluid dynamics within geometrically parametrized domains. Two distinct reduced-order solution strategies are investigated and compared: a monolithic coupled solver and a segregated SIMPLE-based algorithm. Their performance is assessed on two steady, two-dimensional benchmark cases: a lid-driven cavity flow and a flow past a cylindrical obstacle. The two algorithms are compared in terms of fields evaluation and aerodynamic coefficients prediction. The computational results highlight a fundamental trade-off between accuracy and numerical efficiency. On the one hand, after an opportune supremizers enrichment, the coupled approach guarantees a faster convergence, despite the need of a larger number of degrees of freedom. On the other hand, the segregated SIMPLE algorithm yields superior reconstruction accuracy, particularly for lower-dimensional reduced spaces, at the cost of a slower convergence rate.

physics.flu-dyn↗
arXiv2026

A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows

We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $μ_0$ through a transport map in the weighted Hilbert space $L^2_{μ_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{μ_0}$, whose orthonormal frames are evolved by a Dirac--Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker--Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.

math.NA↗
arXiv2026

Trainable Spline Representations for Physics-Informed Learning

This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.

cs.LG↗
arXiv2026

A Generative Model-Free Form Deformation Approach for the Generation of Mesh Motions with Applications to PDE

We introduce a topology-agnostic framework for matching deformations of three-dimensional shapes with non-isomorphic mesh graphs by modelling the deformation as the flow of an Ordinary Differential Equation (ODE). The velocity field is parameterised by a time-dependent Free Form Deformation (FFD), expressed through displacements of a coarse control lattice, yielding a smooth and low-dimensional representation that decouples the deformation model from the discretisation of the source and target surfaces. Under mild regularity assumptions, we prove that the induced ODE map is a universal approximator (in the sup norm) for mappings between genus-0 surfaces, providing a theoretical expressivity guarantee. To further compress the representation and enable probabilistic inference, we couple the ODE--FFD model with a flow-based generative approach in the TarFlow framework, learning a compact latent parametrisation over time series of FFD maps. The resulting method supports efficient sampling and optimisation of plausible deformation trajectories while preserving mesh quality, and it enables scalable reduced-order modelling. Experiments on deforming-body flow benchmarks demonstrate improved accuracy and computational efficiency of reduced-order models constructed from the learned latent dynamics.

math.NA↗
arXiv2026

Bifurcation curve detection with deflation for multiparametric PDEs

This work presents a comprehensive framework for capturing bifurcating phenomena and detecting bifurcation curves in nonlinear multiparametric partial differential equations, where the system exhibits multiple coexisting solutions for given values of the parameters. Traditional continuation methods for one-dimensional parameterizations employ the previously computed solution as the initial guess for the next parameter value. These are usually very inefficient, since small step sizes increase computational cost, while larger steps could jeopardize the method convergence jumping to a different solution branch or missing the bifurcation point. To address these challenges, we propose a novel framework that combines: (i) arclength continuation, adaptively selecting new parameter values in higher dimension, and (ii) the deflation technique, discovering multiple branches to construct complete bifurcation diagrams without requiring a costly spectral analysis of the system. In particular, the arclength continuation method is designed to handle multiparametric scenarios, where the parameter vector $λ\in \mathbb{R}^p$ traces a curve $g(λ)$ within a $p$-dimensional parameter space. In addition, we introduce a zigzag path-following strategy to robustly track the bifurcation curves and surfaces, respectively, for two- and three-dimensional parametric spaces. Finally, we demonstrate its performance on three benchmark problems of increasing complexity: from the 1D/2D Bratu and Allen--Cahn equations to the 2D/3D Rayleigh--Benard convection problem.

math.NA↗
arXiv2026

An isotropic recovery-based error estimator algorithm for mesh adaptation in a finite volume environment with application to atmospheric flows

We develop an Isotropic Recovery-based Error Estimator (IREE) to drive mesh adaptation within a finite volume framework. Recovery-based error estimators are widely used in practice thanks to their simplicity, relying solely on the available discrete solution and a suitable post-processing step. While recovery-based estimators are well established in the finite element framework, their application to finite volume methods, despite the widespread use of the latter in industrial and commercial codes, remains largely unexplored and motivates the present study. We assess the performance of the proposed IREE-driven mesh adaptation procedure through well-known benchmarks for dry atmospheric flows modeling and compare it against a widely-used plain mesh adaptation algorithm. Both qualitative and quantitative results show that the IREE algorithm is more effective at suppressing numerical instabilities compared with the plain approach and, for most of the simulation time, provides a superior accuracy, though at a moderately higher computational cost. In addition, both the adaptive approaches offer substantial computational savings (between 35% and 94%) relative to simulations based on fixed uniformly fine meshes. These findings demonstrate that IREE-based mesh adaptation is a promising and effective strategy for atmospheric flow simulations and has the potential to substantially reduce the computational effort of generating reanalysis-quality data.

math.NA↗
arXiv2026

A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems

Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives. To address these limitations, we propose a framework for forecasting complex dynamical systems via a kernel autonomous ordinary differential equation approach based on Gaussian Processes and Quadratic Order Model Reduction. Our base method, the Gaussian Process Ordinary Differential Equations model, allows accurate short-term forecasting with uncertainty quantification, and it provably converges to the real autonomous equation in the smooth case. We integrate it with quadratic order reduced-order modelling and sphere projection for learning the latent dynamics efficiently while preserving stability. Numerical experiments demonstrate that our full model outperforms ROM forecasting methods such as Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. These results demonstrate the potential of the framework as a robust and stable tool for forecasting complex dynamical systems with rigorous uncertainty quantification.

math.NA↗
arXiv2026

A Structured Review of Reduced Order Modeling for Domain Decomposition Problems: State of the Art and Perspectives

Reduced Order Models (ROMs) have been regarded as an efficient alternative to conventional high-fidelity Computational Fluid Dynamics (CFD) for accelerating the design and optimization processes in engineering applications. Many industrial geometries feature repeating subdomains or contain sub-regions governed by distinct physical phenomena, making them well-suited to Domain Decomposition (DD) techniques. The integration of ROM and DD is promising to further reduce computational costs by constructing local ROMs and assembling them into global solutions. Due to the complexity and necessity of coupling ROMs, many approaches have been proposed in recent years. This review provides a concise overview of existing methodologies combining ROM and DD. We categorize existing methods into intrusive (projection-based) and non-intrusive (data-driven) frameworks. Various strategies for generating local reduced bases and coupling them across subdomains are illustrated. Particular emphasis is placed on intrusive techniques, including equations, numerical algorithms, and practical implementations. The non-intrusive framework is also discussed, highlighting its general procedures, basic formulations, and underlying principles. Finally, we summarise the state of the literature, identify open challenges, and present perspectives on future implementation from an engineering viewpoint.

math.NA↗
arXiv2026

Constraint-driven Optimization and Parametrization of Industrial NURBS Geometries via Neural Deformation Field

This work presents a differentiable framework for the parametrization and shape optimization of industrial CAD geometries represented by multi-patch NURBS surfaces. The method enables the deformation of complex CAD models through a physics-informed geometric parametrization, allowing direct morphing driven by physical constraints without the need to prescribe a predefined deformation strategy. A neural displacement field, implemented as a multi-layer perceptron acting on the NURBS control points, provides a compact parametrization of the admissible design space while preserving patch connectivity. Global geometric quantities relevant to hydrostatic design, including displaced volume, wetted surface area and buoyancy centroid, are formulated as differentiable integral operators evaluated on the parametric domain. These quantities are computed through Gauss-Legendre quadrature combined with analytical B-spline derivatives for surface metric evaluation, allowing gradient propagation to the deformation parameters while limiting the computational overhead of automatic differentiation. The proposed framework operates directly on CAD representations without intermediate mesh generation. Numerical experiments on a modified KVLCC2 hull demonstrate the capability of the method to satisfy competing hydrostatic constraints while producing smooth CAD-compatible geometries and showing stable convergence across multiple random initializations.

math.NA↗
arXiv2026

Surrogate normal-forms for the numerical bifurcation and stability analysis of navier-stokes flows via machine learning

Inspired by the Equation-Free paradigm, we propose an ``embed-learn-lift'' framework for constructing minimal-dimensional surrogate ROMs for the numerical analysis of high-fidelity Navier-Stokes simulations, even in the presence of symmetries that standard machine-learning surrogates often fail to preserve. The framework consists of four main stages. First, manifold learning (here both POD and Diffusion Maps) is used to uncover the intrinsic geometry and dimensionality of the latent space underlying the high-dimensional spatio-temporal Navier-Stokes dynamics across the parameter space. Second, we construct ROMs (here, via Gaussian Process regression (GPR)) of minimal dimension -- by learning the evolution equations directly on the identified latent space. Third, we exploit the toolkit of numerical bifurcation analysis to construct bifurcation diagrams and perform systematic stability analysis directly in the latent coordinates. This enables, for example, the efficient continuation of branches of limit cycles emerging from Andronov-Hopf and Neimark-Sacker bifurcations, together with the computation of limit-cycles periods and stability properties via Floquet multipliers. Such analysis is effectively intractable for the full Navier-Stokes equations. Finally, by solving the pre-image problem in manifold learning, we reconstruct the bifurcating steady and time-periodic states in the original high-dimensional physical space, thus closing the ``lift'' step of the pipeline. We show that DMs-based ROMs allow for a computationally efficient and accurate numerical bifurcation and stability analysis, thus outperforming the widely used POD-ROMs by providing a geometrically consistent parametrization and correctly identifying the intrinsic dimension even in the presence of secondary instabilities, highlighting the need for nonlinear manifold learning methods in CFD.

physics.flu-dyn↗
arXiv2026

End-to-end optimization of subgrid scale models for discontinuous spectral element schemes based on the discrete adjoint method

In computational fluid dynamics, Large Eddy Simulation (LES) offers a compelling balance between accuracy and computational cost by resolving large-scale flow structures while modeling unresolved subgrid scales. However, its predictive capacity is critically dependent on the choice and calibration of subgrid-scale (SGS) models, which often involve problem-dependent parameters and exhibit intricate interactions with the numerical discretization. In this work, we propose a discrete-adjoint framework to optimize SGS model parameters in the loop, leveraging automatic differentiation within a high-order Spectral Difference (SD) solver. Coarse-grained simulations of Forced Homogeneous Isotropic Turbulence (FHIT), together with filtered Direct Numerical Simulation (DNS) data, are used to optimize a limited set of parameters for classical SGS models, including the Smagorinsky model and non-linear tensor-basis formulations. For chaotic systems such as LES, the choice of objective function plays a crucial role in the stability and accuracy of the optimization. Here, we consider the spatio-temporally averaged decay of the Legendre modal coefficients as the quantity of interest for the SD scheme. The optimization is performed across different grid resolutions and polynomial orders, highlighting the impact of numerical discretization on model performance. The methodology is applied to both one-dimensional Burgers turbulence and fully three-dimensional turbulence. The trained models are subsequently assessed on out-of-sample configurations, including Decaying Homogeneous Isotropic Turbulence (DHIT) and the Taylor-Green vortex. Variations in polynomial order, grid resolution, and Reynolds number are considered to evaluate robustness and generalization. In all test cases, the optimized models demonstrate significant improvements over baseline SGS closures.

physics.flu-dyn↗
arXiv2026

Stochastic bifurcation analysis via polynomial chaos: consistency and convergence of branch-approximating solutions

Parameter-dependent dynamical systems that exhibit bifurcations pose significant computational challenges, as traditional continuation methods require repeated, costly simulations across large ranges of parameter values to capture sudden qualitative changes in the solution. In this work, we propose a systematic approach to reconstruct the branches of the entire bifurcation diagram in a single numerical solver leveraging generalized Polynomial Chaos (PC) expansion. By treating the parameter as a random variable, we cast the deterministic parameter-dependent model in a weak stochastic form, and then use a Galerkin projection to recover bifurcation branches globally across the parameter domain without iterative pointwise continuation. We show that the resulting Galerkin system, in the non-uniqueness regime, produces many discrete algebraic roots that naturally split into two classes: highly oscillatory solutions and branch-approximating ones. We develop a rigorous theoretical framework that establishes consistency, proves convergence of the branch-approximating solutions to the true steady states, and guarantees uniqueness of the Galerkin solution under suitable assumptions. Finally, we confirm these theoretical results with numerical experiments on several parameter-dependent ordinary differential equations (ODEs), demonstrating the accuracy and computational efficiency of our single-run framework in capturing complex bifurcation diagrams for both scalar and vector-valued systems.

math.NA↗
arXiv2026

Evaluating passing decision-making in professional football: An enhanced MPNN approach to Receiver Selection

The process of decision-making in football is characterized by a complex interplay between spatial positioning, opponent pressure, and player intent. This work introduces a Graph Neural Network (GNN) framework designed to predict Receiver Selection, the optimal passing target, by modeling on-field interactions as dynamic graphs. Each player is represented as a node with positional and contextual features, while potential passing lines form weighted edges characterized by distance, angle, and pressure metrics. A Message-Passing Neural Network (MPNN) has been developed and trained using a combination of tracking data and event data from professional matches, synchronized through a robust pipeline based on an optimized version of the Needleman-Wunsch Algorithm. The model achieves competitive accuracy in identifying the actual chosen receiver and state-of-the-art accuracy within its top three suggestions. Our model further offers quantification of each option's likelihood, threat, and creativity, enabling performance analysts to evaluate over 1,000 passes in seconds.

cs.LG↗
arXiv2026

Machine Learning-based quadratic closures for non-intrusive Reduced Order Models

In the present work, we introduce a data-driven approach to enhance the accuracy of non-intrusive Reduced Order Models (ROMs). In particular, we focus on ROMs built using Proper Orthogonal Decomposition (POD) in an under-resolved and marginally-resolved regime, i.e. when the number of modes employed is not enough to capture the system dynamics. We propose a method to re-introduce the contribution of neglected modes through a quadratic correction term, given by the action of a quadratic operator on the POD coefficients. Differently from the state-of-the-art methodologies, where the operator is learned via least-squares optimisation, we propose to parametrise the operator by a Multi-Input Operators Network (MIONet). This way, we are able to build models with higher generalisation capabilities, where the operator itself is continuous in space -- thus agnostic of the domain discretisation -- and parameter-dependent. We test our model on two standard benchmarks in fluid dynamics and show that the correction term improves the accuracy of standard POD-based ROMs.

math.NA↗
arXiv2026

Enhancing the accuracy of under-resolved numerical simulations of atmospheric flows with super resolution

Super-resolution (SR) techniques based on deep learning have recently emerged as a promising approach to enhance the spatial resolution of computational fluid dynamics simulations while containing computational cost. In this paper, we investigate several SR architectures to improve coarse-grid simulations of mesoscale atmospheric flows, with training data generated from simulations of the weakly compressible Euler equations. We compare a baseline convolutional neural network (CNN), an attention-enhanced CNN, a multi-scale CNN designed to capture flow structures across different spatial scales, and a diffusion-based SR model. The methods are evaluated on two standard atmospheric benchmarks: the rising thermal bubble and the density current. Results show that the baseline CNN can accurately reconstruct simpler flow features, while more complex flows require multi-scale architectures. Overall, SR based on the multi-scale CNN provides the best balance of accuracy, robustness, and computational efficiency, outperforming even a state-of-the-art diffusion-based approach. We also analyze the sensitivity of the models to the size of the training dataset, highlighting limitations and trade-offs of the proposed SR strategies.

physics.flu-dyn↗
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↗
arXiv2026

Model Order Reduction of Cerebrovascular Hemodynamics Using POD_Galerkin and Reservoir Computing_based Approach

We investigate model order reduction (MOR) strategies for simulating unsteady hemodynamics within cerebrovascular systems, contrasting a physics-based intrusive approach with a data-driven non-intrusive framework. High-fidelity 3D Computational Fluid Dynamics (CFD) snapshots of an idealised basilar artery bifurcation are first compressed into a low-dimensional latent space using Proper Orthogonal Decomposition (POD). We evaluate the performance of a POD-Galerkin (POD-G) model, which projects the Navier-Stokes equations onto the reduced basis, against a POD-Reservoir Computing (POD-RC) model that learns the temporal evolution of coefficients through a recurrent architecture. A multi-harmonic and multi-amplitude training signal is introduced to improve training efficiency. Both methodologies achieve computational speed-ups on the order of 10^2 to 10^3 compared to full-order simulations, demonstrating their potential as efficient and accurate surrogates for predicting flow quantities such as wall shear stress.

math.NA↗
arXiv2026

Efficient and Accurate Surrogate Modeling of Turbulent Flows via Space-Dependent Aggregation and Reduced Order Models

Reynolds-Averaged Navier-Stokes (RANS) models are widely used for turbulent flow simulations due to their computational efficiency, but their accuracy strongly depends on the selected turbulence closure and may vary across the flow domain. Space-dependent model aggregation has been shown to improve RANS predictions by combining multiple turbulence models, although at the cost of repeated high-fidelity simulations. The first novelty of this work is a unified framework that combines different turbulence models, space-dependent aggregation, and non-intrusive reduced order models to achieve both accuracy and efficiency. Two aggregation pipelines are proposed: a Mixed FOM-ROM (MFR) approach, where a reduced order model is trained on aggregated RANS solutions, and a Mixed-ROM (MR) approach, which directly aggregates multiple reduced order models built on top of different RANS full-order models. The second novelty is that the aggregation weights are learned via a neural-network that provides smooth, space-continuous weights and improves generalization with respect to standard weighting techniques. The resulting surrogate models are validated on the two-dimensional periodic hill benchmark and on the flow over a height-dependent bump, demonstrating improved accuracy over individual RANS and ROM predictions at near real-time computational cost.

math.NA↗