arXiv subjects

Gianluigi Rozza

Publications and source records attributed to Gianluigi Rozza.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2026

Reinforcement Learning-Based Filters for Convection-Dominated Flows: Reference-Free and Reference-Guided Training

We propose a reinforcement learning (RL) framework for the dynamic selection of the filter parameter in Evolve-Filter (EF) regularization strategies for incompressible turbulent flows. Instead of prescribing the filter radius heuristically, the RL agent learns to adaptively control the filtering intensity in time, balancing numerical stability and physical accuracy. The methodology is assessed on two benchmark problems with fundamentally different dynamics: flow past a cylinder and decaying homogeneous turbulence. Both reference-guided and reference-free reward formulations are investigated. In the reference-guided setting, the agent is trained using direct numerical simulation (DNS) data over a limited time window and then evaluated in extrapolation. In the reference-free setting, the reward relies exclusively on physics-based quantities, without access to reference solutions, i.e., completely eliminating the computational costs related to DNS simulations. The results show that the proposed RL-EF strategies prevent numerical blow-up while avoiding the excessive dissipation typical of standard EF approaches based on a fixed Kolmogorov length scale. The learned policies accurately reproduce the relevant flow dynamics across scales, preserving the correct balance between large-scale structures and small-scale dissipation. Notably, the reference-free reward achieves performance comparable to the reference-guided approach, demonstrating that stable and spectrally consistent filtering strategies can be learned even without DNS data, drastically reducing the computational costs of the training phase. The proposed framework provides a robust and flexible alternative to manually tuned regularization parameters, enabling adaptive, physically consistent control of filtering in turbulent flow simulations.

math.NA↗

arXiv2026

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↗

arXiv2026

Nonlinear reduction strategies for data compression: a comprehensive comparison from diffusion to advection problems

This work presents an overview of several nonlinear reduction strategies for data compression from various research fields, and a comparison of their performance when applied to problems characterized by diffusion and/or advection terms. We aim to create a common framework by unifying the notation referring to a common two-stage pipeline. At the same time, we underline their main differences and objectives by highlighting the diverse choices made for each stage. We test the considered approaches on three test cases belonging to the family of Advection-Diffusion problems, also focusing on the pure Advection and pure Diffusion benchmarks, studying their reducibility while varying the latent dimension. Finally, we interpret the numerical results under the lens of the discussed theoretical considerations, offering a comprehensive landscape for nonlinear reduction methods for general Advection-Diffusion dynamics.

math.NA↗

arXiv2026

StabOp: A Data-Driven Stabilization Operator for Reduced Order Modeling

Spatial filters have played a central role in large eddy simulation and, more recently, in reduced order model (ROM) stabilization for convection-dominated flows. Nevertheless, important open questions remain: in under-resolved regimes, which filter is most suitable for a given stabilization or closure model? Moreover, once a filter is selected, how should its parameters, such as the filter radius, be determined? Addressing these questions is essential for the reliable design and performance of filter-based stabilization strategies. To answer these questions, we propose a novel strategy that differs fundamentally from current filter-based approaches: we replace traditional spatial filters with a data-driven stabilization operator (StabOp) that yields accurate results for a given resolution, quantity of interest, and stabilization strategy. Although the new StabOp can be used for both classical discretizations and ROMs, and for different types of filter-based stabilization or closure, for clarity we focus on ROMs and the Leray stabilization. To build the new StabOp, we postulate its model form as a linear, quadratic, or nonlinear mapping, and then solve a PDE-constrained optimization problem to minimize a given loss function. Using the resulting StabOp in the Leray ROM (L-ROM) yields a new stabilized ROM, StabOp-L-ROM. To assess the StabOp-L-ROM, we compare it with the L-ROM and the standard ROM in numerical simulations of four flows: 2D flow past a cylinder at Re=500, lid-driven cavity at Re=10000, 3D flow past a hemisphere at Re=2200, and minimal channel flow at Re=5000. Our numerical results show that the StabOp-L-ROM can be orders of magnitude more accurate than the classical L-ROM tuned with an optimal filter radius in the predictive regime. Furthermore, while the new StabOp smooths the input flow fields, its smoothing mechanism differs from that of classical spatial filters.

physics.flu-dyn↗

arXiv2026

Integrating Artificial Intelligence, Physics, and Internet of Things: A Framework for Cultural Heritage Conservation

The conservation of cultural heritage increasingly relies on integrating technological innovation with domain expertise to ensure effective monitoring and predictive maintenance. This paper presents a novel framework to support the preservation of cultural assets, combining Internet of Things (IoT) and Artificial Intelligence (AI) technologies, enhanced with the physical knowledge of phenomena. The framework is structured into four functional layers that permit the analysis of 3D models of cultural assets and elaborate simulations based on the knowledge acquired from data and physics. A central component of the proposed framework consists of Scientific Machine Learning, particularly Physics-Informed Neural Networks (PINNs), which incorporate physical laws into deep learning models. To enhance computational efficiency, the framework also integrates Reduced Order Methods (ROMs), specifically Proper Orthogonal Decomposition (POD), and is also compatible with classical Finite Element (FE) methods. Additionally, it includes tools to automatically manage and process 3D digital replicas, enabling their direct use in simulations. The proposed approach offers three main contributions: a methodology for processing 3D models of cultural assets for reliable simulation; the application of PINNs to combine data-driven and physics-based approaches in cultural heritage conservation; and the integration of PINNs with ROMs to efficiently model degradation processes influenced by environmental and material parameters. The reproducible and open-access experimental phase exploits simulated scenarios on complex and real-life geometries to test the efficacy of the proposed framework in each of its key components, allowing the possibility of dealing with both direct and inverse problems. Code availability: https://github.com/valc89/PhysicsInformedCulturalHeritage

cs.LG↗

arXiv2026

Revisiting Deep Information Propagation: Fractal Frontier and Finite-size Effects

Information propagation characterizes how input correlations evolve across layers in deep neural networks. This framework has been well studied using mean-field theory, which assumes infinitely wide networks. However, these assumptions break down for practical, finite-size networks. In this work, we study information propagation in randomly initialized neural networks with finite width and reveal that the boundary between ordered and chaotic regimes exhibits a fractal structure. This shows the fundamental complexity of neural network dynamics, in a setting that is independent of input data and optimization. To extend this analysis beyond multilayer perceptrons, we leverage recently introduced Fourier-based structured transforms, and show that information propagation in convolutional neural networks also follow the same behavior. In practice, our investigation highlights the importance of finite network depth with respect to the tradeoff between separation and robustness.

cs.LG↗

arXiv2026

Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs

Graph Neural Networks (GNNs) are emerging as powerful tools for nonlinear Model Order Reduction (MOR) of time-dependent parameterized Partial Differential Equations (PDEs). However, existing methodologies struggle to combine geometric inductive biases with interpretable latent behavior, overlooking dynamics-driven features or disregarding spatial information. In this work, we address this gap by introducing Latent Dynamics Graph Convolutional Network (LD-GCN), a purely data-driven, encoder-free architecture that learns a global, low-dimensional representation of dynamical systems conditioned on external inputs and parameters. The temporal evolution is modeled in the latent space and advanced through time-stepping, allowing for time-extrapolation, and the trajectories are consistently decoded onto geometrically parameterized domains using a GNN. Our framework enhances interpretability by enabling the analysis of the reduced dynamics and supporting zero-shot prediction through latent interpolation. The methodology is mathematically validated via a universal approximation theorem for encoder-free architectures, and numerically tested on complex computational mechanics problems involving physical and geometric parameters, including the detection of bifurcating phenomena for Navier-Stokes equations. Code availability: https://github.com/lorenzotomada/ld-gcn-rom

cs.LG↗

arXiv2026

An efficient hyper reduced-order model for segregated solvers for geometrical parametrization problems

We propose an efficient hyper-reduced order model (HROM) designed for segregated finite-volume solvers in geometrically parametrized problems. The method follows a discretize-then-project strategy: the full-order operators are first assembled using finite volume or finite element discretizations and then projected onto low-dimensional spaces using a small set of spatial sampling points, selected through hyper-reduction techniques such as DEIM. This approach removes the dependence of the online computational cost on the full mesh size. The method is assessed on three benchmark problems: a linear transport equation, a nonlinear Burgers equation, and the incompressible Navier--Stokes equations. The results show that the hyper-reduced models closely match full-order solutions while achieving substantial reductions in computational time. Since only a sparse subset of mesh cells is evaluated during the online phase, the method is naturally parallelizable and scalable to very large meshes. These findings demonstrate that hyper-reduction can be effectively combined with segregated solvers and geometric parametrization to enable fast and accurate CFD simulations.

math.NA↗

arXiv2026

Data-driven reduced order model for residence time distribution analysis of an industrial-scale continuous casting tundish

The continuous casting tundish plays a critical role as a metallurgical reactor in the continuous casting process, with its flow characteristics serving as a key parameter in the production of high-quality steel. These characteristics are typically assessed through residence time distribution (RTD) curves. This study examines the flow behaviour in a single-strand continuous casting tundish through a combination of numerical simulations and experimental validation. Steady-state full order model (FOM) simulations are performed under both isothermal and non-isothermal conditions to evaluate the influence of thermal buoyancy on the velocity field, which is found to be negligible. The resulting flow fields are used to initialize transient tracer transport simulations for determining the RTD and flow volume partitioning. Subsequently, a data-driven reduced order model (ROM) is developed to predict the RTD response. Comparison of RTD curves obtained from experiments, FOM, and ROM shows excellent agreement, with the ROM accurately capturing the key flow characteristics at a fraction of the computational cost. These results highlight the potential of ROM techniques for efficient real-time analysis, design, and optimization of tundish operations in metallurgical processes.

physics.flu-dyn↗

arXiv2025

ROM for Viscous, Incompressible Flow in Polygons -- exponential $n$-width bounds and convergence rate

We demonstrate exponential convergence of Reduced Order Model (ROM) approximations for mixed boundary value problems of the stationary, incompressible Navier-Stokes equations in plane, polygonal domains $Ω$. Admissible boundary conditions comprise mixed BCs, no-slip, slip and open boundary conditions, subject to corner-weighted analytic boundary data and volume forcing. The small data hypothesis is assumed to ensure existence of a unique weak solution in the sense of Leray-Hopf. Recent results on corner-weighted, analytic regularity of velocity and pressure fields in $Ω$, imply exponential convergence rates of so-called mixed $hp$-Finite Element Methods in $H^1(Ω)^2\times L^2(Ω)$ on sequences of geometric partitions of $Ω$, with corner-refinement. Based on these exponential convergence rate bounds, we infer exponential bounds for the Kolmogorov $n$-widths of solution sets for analytic forcing and boundary data. This implies corresponding exponential convergence rates of POD Galerkin methods that are based on truth solutions which are obtained offline from low-order, divergence stable mixed Finite Element discretizations. Numerical experiments confirm the exponential rates and the theoretical results.

math.NA↗

arXiv2025

A new data-driven energy-stable Evolve-Filter-Relax model for turbulent flow simulation

We present a novel approach to define the filter and relax steps in the evolve-filter-relax (EFR) framework for simulating turbulent flows. The EFR main advantages are its ease of implementation and computational efficiency. However, as it only contains two parameters (one for the filter step and one for the relax step) its flexibility is rather limited. In this work, we propose a data-driven approach in which the optimal filter is found based on DNS data in the frequency domain. The optimization step is computationally efficient and only involves one-dimensional least-squares problems for each wavenumber. Across both decaying turbulence and Kolmogorov flow, our learned filter decisively outperforms the standard differential filter and the Smagorinsky model, yielding significantly improved accuracy in energy spectra and in the temporal evolution of both energy and enstrophy. In addition, the relax parameter is determined by requiring energy and/or enstrophy conservation, which enforces stability of the method and reduces the appearance of numerical wiggles, especially when the filter is built in scarce data regimes. Applying the learned filter is also more computationally efficient compared to traditional differential filters, as it circumvents solving a linear system.

math.NA↗

arXiv2025

Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition

Graph autoencoders have gained attention in nonlinear reduced-order modeling of parameterized partial differential equations defined on unstructured grids. Despite they provide a geometrically consistent way of treating complex domains, applying such architectures to parameterized dynamical systems for temporal prediction beyond the training data, i.e. the extrapolation regime, is still a challenging task due to the simultaneous need of temporal causality and generalizability in the parametric space. In this work, we explore the integration of graph convolutional autoencoders (GCAs) with tensor train (TT) decomposition and Operator Inference (OpInf) to develop a time-consistent reduced-order model. In particular, high-fidelity snapshots are represented as a combination of parametric, spatial, and temporal cores via TT decomposition, while OpInf is used to learn the evolution of the latter. Moreover, we enhance the generalization performance by developing a multi-fidelity two-stages approach in the framework of Deep Operator Networks (DeepONet), treating the spatial and temporal cores as the trunk networks, and the parametric core as the branch network. Numerical results, including heat-conduction, advection-diffusion and vortex-shedding phenomena, demonstrate great performance in effectively learning the dynamic in the extrapolation regime for complex geometries, also in comparison with state-of-the-art approaches e.g. MeshGraphNets.

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↗

arXiv2025

On an adjoint-based numerical approach for time-dependent optimal control problems of biomedical interest

This work develops a rigorous numerical framework for solving time-dependent Optimal Control Problems (OCPs) governed by partial differential equations, with a particular focus on biomedical applications. The approach deals with adjoint-based Lagrangian methodology, which enables efficient gradient computation and systematic derivation of optimality conditions for both distributed and concentrated control formulations. The proposed framework is first verified using a time-dependent advection-diffusion problem endowed with a manufactured solution to assess accuracy and convergence properties. Subsequently, two representative applications involving drug delivery are investigated: (i) a light-triggered drug delivery system for targeted cancer therapy and (ii) a catheter-based drug delivery system in a patient-specific coronary artery. Numerical experiments not only demonstrate the accuracy of the approach, but also its flexibility and robustness in handling complex geometries, heterogeneous parameters, and realistic boundary conditions, highlighting its potential for the optimal design and control of complex biomedical systems.

math.OC↗

arXiv2025

A Review of Equation-Based and Data-Driven Reduced Order Models featuring a Hybrid cardiovascular application

Cardiovascular diseases are a leading cause of death in the world, driving the development of patient-specific and benchmark models for blood flow analysis. This chapter provides a theoretical overview of the main categories of Reduced Order Models (ROMs), focusing on both projection-based and data-driven approaches within a classical setup. We then present a hybrid ROM tailored for simulating blood flow in a patient-specific aortic geometry. The proposed methodology integrates projection-based techniques with neural network-enhanced data-driven components, incorporating a lifting function strategy to enforce physiologically realistic outflow pressure conditions. This hybrid methodology enables a substantial reduction in computational cost while mantaining high fidelity in reconstructing both velocity and pressure fields. We compare the full- and reduced-order solutions in details and critically assess the advantages and limitations of ROMs in patient-specific cardiovascular modeling.

math.NA↗

arXiv2025

Reservoir computing based predictive reduced order model for steel grade intermixing in an industrial continuous casting tundish

Continuous casting is a widely adopted process in the steel industry, where maintaining high steel quality is paramount. Efficient prediction of grade intermixing during ladle changeover operations is critical for maintaining steel quality and minimizing material losses in the continuous casting process. Among various factors influencing grade intermixing, operating parameters play a significant role, in addition to tundish geometry and flow control devices. In this study, three-dimensional, transient, two-phase turbulent flow simulations are conducted to investigate the ladle changeover operation. During this process, the molten steel level in the tundish typically varies over time, significantly affecting the grade intermixing phenomena. The influence of ladle change time on intermixing time has been presented. However, high-fidelity full-order simulations of such complex transient phenomena are computationally expensive and are impractical for real-time monitoring or design-space exploration in industrial-scale applications. To address this issue, a reduced order modelling approach based on proper orthogonal decomposition (POD) and reservoir computing (RC) is employed to efficiently predict intermixing time. The proposed reduced order model (ROM) demonstrates excellent predictive accuracy using limited training data while requiring significantly less computational resources and training time. The results demonstrate the potential of the proposed methodology as a fast, reliable tool for real-time process monitoring and optimization in industrial continuous casting operations.

physics.flu-dyn↗