arXiv subjects

Stefania Fresca

Publications and source records attributed to Stefania Fresca.

At least 19 recordsLinked to original sources

arXiv2026

Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations

Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.

cs.LG↗

arXiv2025

HypeR Adaptivity: Joint $hr$-Adaptive Meshing via Hypergraph Multi-Agent Deep Reinforcement Learning

Adaptive mesh refinement is central to the efficient solution of partial differential equations (PDEs) via the finite element method (FEM). Classical $r$-adaptivity optimizes vertex positions but requires solving expensive auxiliary PDEs such as the Monge-Amp\`ere equation, while classical $h$-adaptivity modifies topology through element subdivision but suffers from expensive error indicator computation and is constrained by isotropic refinement patterns that impose accuracy ceilings. Combined $hr$-adaptive techniques naturally outperform single-modality approaches, yet inherit both computational bottlenecks and the restricted cost-accuracy trade-off. Emerging machine learning methods for adaptive mesh refinement seek to overcome these limitations, but existing approaches address $h$-adaptivity or $r$-adaptivity in isolation. We present HypeR, a deep reinforcement learning framework that jointly optimizes mesh relocation and refinement. HypeR casts the joint adaptation problem using tools from hypergraph neural networks and multi-agent reinforcement learning. Refinement is formulated as a heterogeneous multi-agent Markov decision process (MDP) where element agents decide discrete refinement actions, while relocation follows an anisotropic diffusion-based policy on vertex agents with provable prevention of mesh tangling. The reward function combines local and global error reduction to promote general accuracy. Across benchmark PDEs, HypeR reduces approximation error by up to 6--10$\times$ versus state-of-art $h$-adaptive baselines at comparable element counts, breaking through the uniform refinement accuracy ceiling that constrains subdivision-only methods. The framework produces meshes with improved shape metrics and alignment to solution anisotropy, demonstrating that jointly learned $hr$-adaptivity strategies can substantially enhance the capabilities of automated mesh generation.

cs.CE↗

arXiv2025

Explainable Deep Learning-based Classification of Wolff-Parkinson-White Electrocardiographic Signals

Wolff-Parkinson-White (WPW) syndrome is a cardiac electrophysiology (EP) disorder caused by the presence of an accessory pathway (AP) that bypasses the atrioventricular node, faster ventricular activation rate, and provides a substrate for atrio-ventricular reentrant tachycardia (AVRT). Accurate localization of the AP is critical for planning and guiding catheter ablation procedures. While traditional diagnostic tree (DT) methods and more recent machine learning (ML) approaches have been proposed to predict AP location from surface electrocardiogram (ECG), they are often constrained by limited anatomical localization resolution, poor interpretability, and the use of small clinical datasets. In this study, we present a Deep Learning (DL) model for the localization of single manifest APs across 24 cardiac regions, trained on a large, physiologically realistic database of synthetic ECGs generated using a personalized virtual heart model. We also integrate eXplainable Artificial Intelligence (XAI) methods, Guided Backpropagation, Grad-CAM, and Guided Grad-CAM, into the pipeline. This enables interpretation of DL decision-making and addresses one of the main barriers to clinical adoption: lack of transparency in ML predictions. Our model achieves localization accuracy above 95%, with a sensitivity of 94.32% and specificity of 99.78%. XAI outputs are physiologically validated against known depolarization patterns, and a novel index is introduced to identify the most informative ECG leads for AP localization. Results highlight lead V2 as the most critical, followed by aVF, V1, and aVL. This work demonstrates the potential of combining cardiac digital twins with explainable DL to enable accurate, transparent, and non-invasive AP localization.

cs.LG↗

arXiv2025

Multi-Level Monte Carlo Training of Neural Operators

Operator learning is a rapidly growing field that aims to approximate nonlinear operators related to partial differential equations (PDEs) using neural operators. These rely on discretization of input and output functions and are, usually, expensive to train for large-scale problems at high-resolution. Motivated by this, we present a Multi-Level Monte Carlo (MLMC) approach to train neural operators by leveraging a hierarchy of resolutions of function discretization. Our framework relies on using gradient corrections from fewer samples of fine-resolution data to decrease the computational cost of training while maintaining a high level accuracy. The proposed MLMC training procedure can be applied to any architecture accepting multi-resolution data. Our numerical experiments on a range of state-of-the-art models and test-cases demonstrate improved computational efficiency compared to traditional single-resolution training approaches, and highlight the existence of a Pareto curve between accuracy and computational time, related to the number of samples per resolution.

cs.LG↗

arXiv2025

Integrating Fourier Neural Operators with Diffusion Models to improve Spectral Representation of Synthetic Earthquake Ground Motion Response

Nuclear reactor buildings must be designed to withstand the dynamic load induced by strong ground motion earthquakes. For this reason, their structural behavior must be assessed in multiple realistic ground shaking scenarios (e.g., the Maximum Credible Earthquake). However, earthquake catalogs and recorded seismograms may not always be available in the region of interest. Therefore, synthetic earthquake ground motion is progressively being employed, although with some due precautions: earthquake physics is sometimes not well enough understood to be accurately reproduced with numerical tools, and the underlying epistemic uncertainties lead to prohibitive computational costs related to model calibration. In this study, we propose an AI physics-based approach to generate synthetic ground motion, based on the combination of a neural operator that approximates the elastodynamics Green's operator in arbitrary source-geology setups, enhanced by a denoising diffusion probabilistic model. The diffusion model is trained to correct the ground motion time series generated by the neural operator. Our results show that such an approach promisingly enhances the realism of the generated synthetic seismograms, with frequency biases and Goodness-Of-Fit (GOF) scores being improved by the diffusion model. This indicates that the latter is capable to mitigate the mid-frequency spectral falloff observed in the time series generated by the neural operator. Our method showcases fast and cheap inference in different site and source conditions.

cs.LG↗

arXiv2025

Beyond Interpolation: Extrapolative Reasoning with Reinforcement Learning and Graph Neural Networks

Despite incredible progress, many neural architectures fail to properly generalize beyond their training distribution. As such, learning to reason in a correct and generalizable way is one of the current fundamental challenges in machine learning. In this respect, logic puzzles provide a great testbed, as we can fully understand and control the learning environment. Thus, they allow to evaluate performance on previously unseen, larger and more difficult puzzles that follow the same underlying rules. Since traditional approaches often struggle to represent such scalable logical structures, we propose to model these puzzles using a graph-based approach. Then, we investigate the key factors enabling the proposed models to learn generalizable solutions in a reinforcement learning setting. Our study focuses on the impact of the inductive bias of the architecture, different reward systems and the role of recurrent modeling in enabling sequential reasoning. Through extensive experiments, we demonstrate how these elements contribute to successful extrapolation on increasingly complex puzzles.These insights and frameworks offer a systematic way to design learning-based systems capable of generalizable reasoning beyond interpolation.

cs.LG↗

arXiv2025

HypeRL: Hypernetwork-Based Reinforcement Learning for Control of Parametrized Dynamical Systems

In this work, we devise a new, general-purpose reinforcement learning strategy for the optimal control of parametric dynamical systems. Such problems frequently arise in applied sciences and engineering and entail a significant complexity when control and/or state variables are distributed in high-dimensional space or depend on varying parameters. Traditional numerical methods, relying on either iterative minimization algorithms -- exploiting, e.g., the solution of the adjoint problem -- or dynamic programming -- also involving the solution of the Hamilton-Jacobi-Bellman (HJB) equation -- while reliable, often become computationally infeasible. In this paper, we propose HypeRL a deep reinforcement learning (DRL) framework to overcome the limitations shown by traditional methods. HypeRL aims at approximating the optimal control policy directly. Specifically, we employ an actor-critic DRL approach to learn an optimal feedback control strategy that can generalize across the range of variation of the parameters. To effectively learn such optimal control laws for different instances of the parameters, encoding the parameter information into the DRL policy and value function neural networks (NNs) is essential. HypeRL uses two additional NNs, called hypernetworks, to learn the weights and biases of the value function and the policy NNs. In this way, HypeRL effectively embeds the parametric information into the value function and policy. We validate the proposed approach on two parametric control problems, namely (I) a 1D parametric Kuramoto-Sivashinsky equation with in-domain control, and (ii) a navigation problem of particle dynamics in a parametric 2D gyre flow. We show that the knowledge of physical and task-dependent information and the encoding of this information via a hypernetwork, are essential ingredients for learning parameter-dependent control policies.

cs.LG↗

arXiv2024

Handling geometrical variability in nonlinear reduced order modeling through Continuous Geometry-Aware DL-ROMs

Deep Learning-based Reduced Order Models (DL-ROMs) provide nowadays a well-established class of accurate surrogate models for complex physical systems described by parametrized PDEs, by nonlinearly compressing the solution manifold into a handful of latent coordinates. Until now, design and application of DL-ROMs mainly focused on physically parameterized problems. Within this work, we provide a novel extension of these architectures to problems featuring geometrical variability and parametrized domains, namely, we propose Continuous Geometry-Aware DL-ROMs (CGA-DL-ROMs). In particular, the space-continuous nature of the proposed architecture matches the need to deal with multi-resolution datasets, which are quite common in the case of geometrically parametrized problems. Moreover, CGA-DL-ROMs are endowed with a strong inductive bias that makes them aware of geometrical parametrizations, thus enhancing both the compression capability and the overall performance of the architecture. Within this work, we justify our findings through a thorough theoretical analysis, and we practically validate our claims by means of a series of numerical tests encompassing physically-and-geometrically parametrized PDEs, ranging from the unsteady Navier-Stokes equations for fluid dynamics to advection-diffusion-reaction equations for mathematical biology.

math.NA↗

arXiv2024

On latent dynamics learning in nonlinear reduced order modeling

In this work, we present the novel mathematical framework of latent dynamics models (LDMs) for reduced order modeling of parameterized nonlinear time-dependent PDEs. Our framework casts this latter task as a nonlinear dimensionality reduction problem, while constraining the latent state to evolve accordingly to an (unknown) dynamical system. A time-continuous setting is employed to derive error and stability estimates for the LDM approximation of the full order model (FOM) solution. We analyze the impact of using an explicit Runge-Kutta scheme in the time-discrete setting, resulting in the $\Delta\text{LDM}$ formulation, and further explore the learnable setting, $\Delta\text{LDM}_\theta$, where deep neural networks approximate the discrete LDM components, while providing a bounded approximation error with respect to the FOM. Moreover, we extend the concept of parameterized Neural ODE - recently proposed as a possible way to build data-driven dynamical systems with varying input parameters - to be a convolutional architecture, where the input parameters information is injected by means of an affine modulation mechanism, while designing a convolutional autoencoder neural network able to retain spatial-coherence, thus enhancing interpretability at the latent level. Numerical experiments, including the Burgers' and the advection-reaction-diffusion equations, demonstrate the framework's ability to obtain, in a multi-query context, a time-continuous approximation of the FOM solution, thus being able to query the LDM approximation at any given time instance while retaining a prescribed level of accuracy. Our findings highlight the remarkable potential of the proposed LDMs, representing a mathematically rigorous framework to enhance the accuracy and approximation capabilities of reduced order modeling for time-dependent parameterized PDEs.

math.NA↗

arXiv2024

PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs

The coupling of Proper Orthogonal Decomposition (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process -- that is, by making them physics-informed -- to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem's solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection-diffusion-reaction equations, to nonlinear problems like the Navier-Stokes equations for fluid flows.

math.NA↗

arXiv2023

Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks

Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution of parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency. This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs in the presence of geometrical variability. In particular, we propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme where a GNN architecture is used to efficiently evolve the system. With respect to the majority of surrogate models, the proposed approach stands out for its ability of tackling problems with parameter dependent spatial domains, while simultaneously generalizing to different geometries and mesh resolutions. We assess the effectiveness of the proposed approach through a series of numerical experiments, involving both two- and three-dimensional problems, showing that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios. We also assess, from a numerical standpoint, the importance of using GNNs, rather than classical dense deep neural networks, for the proposed framework.

math.NA↗

arXiv2023

Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition

POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solution accuracy. This decomposition, combined with the constructive nature of the proofs, allows us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.

math.NA↗

arXiv2023

Uncertainty quantification for nonlinear solid mechanics using reduced order models with Gaussian process regression

Uncertainty quantification (UQ) tasks, such as sensitivity analysis and parameter estimation, entail a huge computational complexity when dealing with input-output maps involving the solution of nonlinear differential problems, because of the need to query expensive numerical solvers repeatedly. Projection-based reduced order models (ROMs), such as the Galerkin-reduced basis (RB) method, have been extensively developed in the last decades to overcome the computational complexity of high fidelity full order models (FOMs), providing remarkable speedups when addressing UQ tasks related with parameterized differential problems. Nonetheless, constructing a projection-based ROM that can be efficiently queried usually requires extensive modifications to the original code, a task which is non-trivial for nonlinear problems, or even not possible at all when proprietary software is used. Non-intrusive ROMs - which rely on the FOM as a black box - have been recently developed to overcome this issue. In this work, we consider ROMs exploiting proper orthogonal decomposition to construct a reduced basis from a set of FOM snapshots, and Gaussian process regression (GPR) to approximate the RB projection coefficients. Two different approaches, namely a global GPR and a tensor-decomposition-based GPR, are explored on a set of 3D time-dependent solid mechanics examples. Finally, the non-intrusive ROM is exploited to perform global sensitivity analysis (relying on both screening and variance-based methods) and parameter estimation (through Markov chain Monte Carlo methods), showing remarkable computational speedups and very good accuracy compared to high-fidelity FOMs.

math.NA↗

arXiv2022

Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.

cs.LG↗

arXiv2022

Approximation bounds for convolutional neural networks in operator learning

Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature still lack a rigorous justification on their mathematical foundations. Motivated by this fact, in this paper we derive rigorous error bounds for the approximation of nonlinear operators by means of CNN models. More precisely, we address the case in which an operator maps a finite dimensional input $\boldsymbol{\mu}\in\mathbb{R}^{p}$ onto a functional output $u_{\boldsymbol{\mu}}:[0,1]^{d}\to\mathbb{R}$, and a neural network model is used to approximate a discretized version of the input-to-output map. The resulting error estimates provide a clear interpretation of the hyperparameters defining the neural network architecture. All the proofs are constructive, and they ultimately reveal a deep connection between CNNs and the Fourier transform. Finally, we complement the derived error bounds by numerical experiments that illustrate their application.

math.NA↗

arXiv2022

Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches

Micro-Electro-Mechanical-Systems are complex structures, often involving nonlinearites of geometric and multiphysics nature, that are used as sensors and actuators in countless applications. Starting from full-order representations, we apply deep learning techniques to generate accurate, efficient and real-time reduced order models to be used as virtual twin for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches and gyroscopes, also displaying intricate dynamical evolutions like internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows extracting the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems

math.DS↗

arXiv2022

Efficient approximation of cardiac mechanics through reduced order modeling with deep learning-based operator approximation

Reducing the computational time required by high-fidelity, full order models (FOMs) for the solution of problems in cardiac mechanics is crucial to allow the translation of patient-specific simulations into clinical practice. While FOMs, such as those based on the finite element method, provide valuable information of the cardiac mechanical function, up to hundreds of thousands degrees of freedom may be needed to obtain accurate numerical results. As a matter of fact, simulating even just a few heartbeats can require hours to days of CPU time even on powerful supercomputers. In addition, cardiac models depend on a set of input parameters that we could let vary in order to explore multiple virtual scenarios. To compute reliable solutions at a greatly reduced computational cost, we rely on a reduced basis method empowered with a new deep-learning based operator approximation, which we refer to as Deep-HyROMnet technique. Our strategy combines a projection-based POD-Galerkin method with deep neural networks for the approximation of (reduced) nonlinear operators, overcoming the typical computational bottleneck associated with standard hyper-reduction techniques. This method is shown to provide reliable approximations to cardiac mechanics problems outperforming classical projection-based ROMs in terms of computational speed-up of orders of magnitude, and enhancing forward uncertainty quantification analysis otherwise unaffordable.

math.NA↗

arXiv2022

Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs

To speed-up the solution to parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as surrogate models obtained via a machine learning approach. Thanks to its physics-based structure, ensured by the use of a Galerkin projection of the full order model (FOM) onto a linear low-dimensional subspace, RB methods yield approximations that fulfill the physical problem at hand. However, to make the assembling of a ROM independent of the FOM dimension, intrusive and expensive hyper-reduction stages are usually required, such as the discrete empirical interpolation method (DEIM), thus making this strategy less feasible for problems characterized by (high-order polynomial or nonpolynomial) nonlinearities. To overcome this bottleneck, we propose a novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs). The resulting hyper-reduced order model enhanced by deep neural networks, to which we refer to as Deep-HyROMnet, is then a physics-based model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed. Numerical results dealing with fast simulations in nonlinear structural mechanics show that Deep-HyROMnets are orders of magnitude faster than POD-Galerkin-DEIM ROMs, keeping the same level of accuracy.

math.NA↗