Gianluigi Rozza

Gabriele Codega, Anna Ivagnes, Nicola Demo et al.

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.

Nicola Demo, Marco Tezzele, Andrea Mola et al.

In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is a common benchmark in ship hydrodynamics simulations.

Giovanni Stabile, Gianluigi Rozza

In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier-Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

Giovanni Stabile, Saddam Hijazi, Andrea Mola et al.

Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. A Reduced Order Model (ROM) of the incompressible flow around a circular cylinder is presented in this work. The ROM is built performing a Galerkin projection of the governing equations onto a lower dimensional space. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) the projection of the Governing equations (momentum equation and Poisson equation for pressure) performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework. The accuracy of the reduced order model is assessed against full order results.

Lorenzo Tomada, Federico Pichi, Gianluigi Rozza

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

Martin Hess, Alessandro Alla, Annalisa Quaini et al.

Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

Giovanni Stabile, Francesco Ballarin, Giacomo Zuccarino et al.

The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

Francesco Ballarin, Alessandro D'Amario, Simona Perotto et al.

Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without significantly compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a dimensionality reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.

Marco Tezzele, Francesco Ballarin, Gianluigi Rozza

In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.

Gianluigi Rozza, Haris Malik, Nicola Demo et al.

Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.

Francesco Ballarin, Gianluigi Rozza, Yvon Maday

POD--Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

Maria Strazzullo, Francesco Ballarin, Renzo Mosetti et al.

We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations, in its linear and nonlinear version, describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.

Giuseppe Pitton, Gianluigi Rozza

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

Nicola Demo, Marco Tezzele, Gianluca Gustin et al.

Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship.

Efthymios N. Karatzas, Giovanni Stabile, Leo Nouveau et al.

We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

Marco Tezzele, Nicola Demo, Andrea Mola et al.

In this work we present an integrated computational pipeline involving several model order reduction techniques for industrial and applied mathematics, as emerging technology for product and/or process design procedures. Its data-driven nature and its modularity allow an easy integration into existing pipelines. We describe a complete optimization framework with automated geometrical parameterization, reduction of the dimension of the parameter space, and non-intrusive model order reduction such as dynamic mode decomposition and proper orthogonal decomposition with interpolation. Moreover several industrial examples are illustrated.

Saddam Hijazi, Shafqat Ali, Giovanni Stabile et al.

We present two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.

Nicola Demo, Marco Tezzele, Gianluigi Rozza

In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of high-fidelity information, using it for building the reduced order model and for learning the residual. In this work, we explore the integration of proper orthogonal decomposition (POD), and gappy POD for sensors data, with the recent DeepONet architecture. Numerical investigations for a parametric benchmark function and a nonlinear parametric Navier-Stokes problem are presented.

Luca Venturi, Francesco Ballarin, Gianluigi Rozza

In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a $L^2$ norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.

Francesco Romor, Giovanni Stabile, Gianluigi Rozza

Non-affine parametric dependencies, nonlinearities and advection-dominated regimes of the model of interest can result in a slow Kolmogorov n-width decay, which precludes the realization of efficient reduced-order models based on linear subspace approximations. Among the possible solutions, there are purely data-driven methods that leverage autoencoders and their variants to learn a latent representation of the dynamical system, and then evolve it in time with another architecture. Despite their success in many applications where standard linear techniques fail, more has to be done to increase the interpretability of the results, especially outside the training range and not in regimes characterized by an abundance of data. Not to mention that none of the knowledge on the physics of the model is exploited during the predictive phase. In order to overcome these weaknesses, we implement the non-linear manifold method introduced by Carlberg et al [37] with hyper-reduction achieved through reduced over-collocation and teacher-student training of a reduced decoder. We test the methodology on a 2d non-linear conservation law and a 2d shallow water models, and compare the results obtained with a purely data-driven method for which the dynamics is evolved in time with a long-short term memory network.

Martin Hess, Gianluigi Rozza

We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation [1] in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

Dario Coscia, Laura Meneghetti, Nicola Demo et al.

Convolutional Neural Network (CNN) is one of the most important architectures in deep learning. The fundamental building block of a CNN is a trainable filter, represented as a discrete grid, used to perform convolution on discrete input data. In this work, we propose a continuous version of a trainable convolutional filter able to work also with unstructured data. This new framework allows exploring CNNs beyond discrete domains, enlarging the usage of this important learning technique for many more complex problems. Our experiments show that the continuous filter can achieve a level of accuracy comparable to the state-of-the-art discrete filter, and that it can be used in current deep learning architectures as a building block to solve problems with unstructured domains as well.

Anna Ivagnes, Nicola Demo, Gianluigi Rozza

In this work, we propose a model order reduction framework to deal with inverse problems in a non-intrusive setting. Inverse problems, especially in a partial differential equation context, require a huge computational load due to the iterative optimization process. To accelerate such a procedure, we apply a numerical pipeline that involves artificial neural networks to parametrize the boundary conditions of the problem in hand, compress the dimensionality of the (full-order) snapshots, and approximate the parametric solution manifold. It derives a general framework capable to provide an ad-hoc parametrization of the inlet boundary and quickly converges to the optimal solution thanks to model order reduction. We present in this contribution the results obtained by applying such methods to two different CFD test cases.

Dinh Bao Phuong Huynh, Federico Pichi, Gianluigi Rozza

In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations - rapid convergence, an a posteriori error estimation procedures - rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor, and Offline-Online computational decomposition strategies - minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations - with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

Giacomo Venier, Isabella Carla Gonnella, Federico Pichi et al.

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.

Sajad Salavatidezfouli, Giovanni Stabile, Gianluigi Rozza

This research study explores the applicability of Deep Reinforcement Learning (DRL) for thermal control based on Computational Fluid Dynamics. To accomplish that, the forced convection on a hot plate prone to a pulsating cooling jet with variable velocity has been investigated. We begin with evaluating the efficiency and viability of a vanilla Deep Q-Network (DQN) method for thermal control. Subsequently, a comprehensive comparison between different variants of DRL is conducted. Soft Double and Duel DQN achieved better thermal control performance among all the variants due to their efficient learning and action prioritization capabilities. Results demonstrate that the soft Double DQN outperforms the hard Double DQN. Moreover, soft Double and Duel can maintain the temperature in the desired threshold for more than 98% of the control cycle. These findings demonstrate the promising potential of DRL in effectively addressing thermal control systems.

Denis Devaud, Gianluigi Rozza

In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA) is a growing research theme in scientific computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. This work wants to demonstrate that it is also possible for some class of problems to deal with affine geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA. In this work we show a certification of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation.

Ferdinando Auricchio, Michele Conti, Adrian Lefieux et al.

The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

Moaad Khamlich, Federico Pichi, Gianluigi Rozza

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.

Toni Lassila, Gianluigi Rozza

We propose a two-fold approach to model reduction of fluid-structure interaction. The state equations for the fluid are solved with reduced basis methods. These are model reduction methods for parametric partial differential equations using well-chosen snapshot solutions in order to build a set of global basis functions. The other reduction is in terms of the geometric complexity of the moving fluid-structure interface. We use free-form deformations to parameterize the perturbation of the flow channel at rest configuration. As a computational example we consider a steady fluid-structure interaction problem: an incmpressible Stokes flow in a channel that has a flexible wall.

Alessandro Della Pia, Dimitrios G. Patsatzis, Gianluigi Rozza et al.

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.

Isabella Carla Gonnella, Martin W. Hess, Giovanni Stabile et al.

Parametric time-dependent systems are of a crucial importance in modeling real phenomena, often characterized by non-linear behaviors too. Those solutions are typically difficult to generalize in a sufficiently wide parameter space while counting on limited computational resources available. As such, we present a general two-stages deep learning framework able to perform that generalization with low computational effort in time. It consists in a separated training of two pipe-lined predictive models. At first, a certain number of independent neural networks are trained with data-sets taken from different subsets of the parameter space. Successively, a second predictive model is specialized to properly combine the first-stage guesses and compute the right predictions. Promising results are obtained applying the framework to incompressible Navier-Stokes equations in a cavity (Rayleigh-Bernard cavity), obtaining a 97% reduction in the computational time comparing with its numerical resolution for a new value of the Grashof number.

Laura Meneghetti, Nicola Demo, Gianluigi Rozza

As a major breakthrough in artificial intelligence and deep learning, Convolutional Neural Networks have achieved an impressive success in solving many problems in several fields including computer vision and image processing. Real-time performance, robustness of algorithms and fast training processes remain open problems in these contexts. In addition object recognition and detection are challenging tasks for resource-constrained embedded systems, commonly used in the industrial sector. To overcome these issues, we propose a dimensionality reduction framework based on Proper Orthogonal Decomposition, a classical model order reduction technique, in order to gain a reduction in the number of hyperparameters of the net. We have applied such framework to SSD300 architecture using PASCAL VOC dataset, demonstrating a reduction of the network dimension and a remarkable speedup in the fine-tuning of the network in a transfer learning context.

Carmine Valentino, Federico Pichi, Francesco Colace et al.

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

Rahul Halder, Arash Hajisharifi, Kabir Bakhshaei et al.

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.

Piero Zappi, Anna Ivagnes, Niccolò Tonicello et al.

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.

Gabriel Masella, Giuseppe Alessio D'Inverno, Max Goldsmith et al.

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.

Moaad Khamlich, Niccolò Tonicello, Federico Pichi et al.

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.

Marco Tezzele, Filippo Salmoiraghi, Andrea Mola et al.

We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA math- Lab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.

Davide Torlo, Francesco Ballarin, Gianluigi Rozza

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline--Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

Luca Venturi, Davide Torlo, Francesco Ballarin et al.

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

Moaad Khamlich, Federico Pichi, Michele Girfoglio et al.

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.

Sajad Salavatidezfouli, Saeid Rakhsha, Armin Sheidani et al.

This paper aims to comprehensively investigate the efficacy of various Model Order Reduction (MOR) and deep learning techniques in predicting heat transfer in a pulsed jet impinging on a concave surface. Expanding on the previous experimental and numerical research involving pulsed circular jets, this investigation extends to evaluate Predictive Surrogate Models (PSM) for heat transfer across various jet characteristics. To this end, this work introduces two predictive approaches, one employing a Fast Fourier Transformation augmented Artificial Neural Network (FFT-ANN) for predicting the average Nusselt number under constant-frequency scenarios. Moreover, the investigation introduces the Proper Orthogonal Decomposition and Long Short-Term Memory (POD-LSTM) approach for random-frequency impingement jets. The POD-LSTM method proves to be a robust solution for predicting the local heat transfer rate under random-frequency impingement scenarios, capturing both the trend and value of temporal modes. The comparison of these approaches highlights the versatility and efficacy of advanced machine learning techniques in modelling complex heat transfer phenomena.

Dario Coscia, Max Welling, Nicola Demo et al.

Autoregressive and recurrent networks have achieved remarkable progress across various fields, from weather forecasting to molecular generation and Large Language Models. Despite their strong predictive capabilities, these models lack a rigorous framework for addressing uncertainty, which is key in scientific applications such as PDE solving, molecular generation and Machine Learning Force Fields. To address this shortcoming we present BARNN: a variational Bayesian Autoregressive and Recurrent Neural Network. BARNNs aim to provide a principled way to turn any autoregressive or recurrent model into its Bayesian version. BARNN is based on the variational dropout method, allowing to apply it to large recurrent neural networks as well. We also introduce a temporal version of the "Variational Mixtures of Posteriors" prior (tVAMP-prior) to make Bayesian inference efficient and well-calibrated. Extensive experiments on PDE modelling and molecular generation demonstrate that BARNN not only achieves comparable or superior accuracy compared to existing methods, but also excels in uncertainty quantification and modelling long-range dependencies.

Giuseppe Alessio D'Inverno, Zhiyuan Hu, Leo Davy et al.

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. Our investigation highlights the importance of finite network depth with respect to the tradeoff between separation and robustness.

Yuanhong Chen, Federico Pichi, Zhen Gao et al.

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.

Dario Coscia, Nicola Demo, Gianluigi Rozza

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.

Giulio Ortali, Alessandro Corbetta, Gianluigi Rozza et al.

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.

Salvatore Cuomo, Vincenzo Schiano di Cola, Fabio Giampaolo et al.

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

Nicola Demo, Maria Strazzullo, Gianluigi Rozza

In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.