MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1317 products
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32/2026 - 04/09/2026
Antonietti, P.F.; Bonizzoni, F.; Perugia, I.; Verani, M.
A Multilevel Monte Carlo Virtual Element Method for Uncertainty Quantification of Elliptic Partial Differential Equations | Abstract | | We introduce a Monte Carlo Virtual Element estimator based on Virtual Element discretizations for stochastic elliptic partial differential equations with random diffusion coefficients. We prove estimates for
the statistical approximation error for both the solution and suitable linear quantities of interest. A Multilevel Monte Carlo Virtual Element method is also developed and analyzed to mitigate the computational
cost of the plain Monte Carlo strategy. The proposed approach exploits the flexibility of the Virtual Element method on general polytopal meshes and employs sequences of coarser spaces constructed via mesh agglomeration, providing a practical realization of the multilevel hierarchy even in complex geometries. This strategy substantially reduces the number of samples required on the finest level to achieve a prescribed accuracy. We prove convergence of the multilevel method and analyze its computational complexity, showing that it yields
significant cost reductions compared to standard Monte Carlo methods for a prescribed accuracy. Extensive numerical experiments support the theoretical results and demonstrate the efficiency of the proposed method. |
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31/2026 - 04/04/2026
Guastamacchia, C.; Piersanti, R; Giardini, F.; Coppini, R.; Ferrantini C.; Dede’ L.; Sacconi L.; Regazzoni F.
The functional impact of myofiber macroscopic organization and disarray in computational models of the murine heart | Abstract | | A major challenge in computational models of cardiac electromechanics is the reconstruction of myocardial fiber architecture, as direct in vivo measurements of fiber orientation are not feasible. Consequently, rule-based methods are commonly adopted as surrogates, relying on empirical descriptions of fiber organization combined with patient-specific geometries. This study investigates the respective roles of macroscopic fiber architecture and microscopic fiber disarray in cardiac electromechanical simulations. A high-fidelity biventricular electromechanical model of a murine heart was developed using a high-resolution myocardial
fiber field obtained via mesoscopic optical imaging, which serves as a reference ground truth. A spatial smoothing strategy is introduced to decouple macroscopic fiber organization from local disarray, and the resulting responses are also compared with those obtained using a rule-based fiber field. The results show that passive mechanics and electrophysiological activation are only weakly affected by fiber disarray, with global chamber compliance and activation times remaining largely unchanged across different fiber descriptions. In contrast, active mechanics is highly sensitive to fiber architecture. Moderate regularization of the
experimentally measured fiber field enhances the ventricular pumping efficiency of the computational model by reducing microscopic disarray while preserving the macroscopic helical organization, whereas excessive smoothing or rule-based fiber reconstructions lead to unphysiologically strong or inefficient contraction.
Within this framework, two commonly adopted surrogate strategies to account for fiber disarray are investigated: (i) a reduction of the effective cross-bridge stiffness in the active tension model, and (ii) the introduction of controlled misalignment between active tension and the local fiber direction. While both approaches reproduce global hemodynamic indicators comparable to the reference case, an effective reduction of contractility – despite its phenomenological nature – provides a closer match to the reference strain patterns than the introduction of orthogonal active stress components. Overall, the results highlight the dominant role of macroscopic fiber architecture in active mechanics and reveal important limitations of commonly adopted surrogate approaches for modeling fiber disarray. |
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30/2026 - 03/31/2026
Regazzoni, F.
The internal law of a material can be discovered from its boundary | Abstract | | Since the earliest stages of human civilization, advances in technology have been tightly linked to our ability to understand and predict the mechanical behavior of materials. In recent years, this challenge has increasingly been framed within the broader paradigm of data-driven scientific discovery, where governing laws are inferred directly from observations. However, existing methods require either stress-strain pairs or full-field displacement measurements, which are often inaccessible in practice. We introduce Neural-DFEM, a method that enables unsupervised discovery of hyperelastic material laws even from partial observations, such as boundary-only measurements. The method embeds a differentiable finite element solver within the learning loop, directly linking candidate energy functionals to available measurements. To guarantee thermodynamic consistency and mathematical well-posedness throughout training, the method employs Hyperelastic Neural Networks, a novel structure-preserving neural architecture that enforces frame indifference, material symmetry, polyconvexity, and coercivity by design. The resulting framework enables robust material model discovery in both two- and three-dimensional settings, including scenarios with boundary-only measurements. Neural-DFEM allows for generalization across geometries and loading conditions, and exhibits unprecedented accuracy and strong resilience to measurement noise. Our results demonstrate that reliable identification of material laws is achievable even under partial observability when strong physical inductive biases are embedded in the learning architecture. |
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28/2026 - 03/19/2026
Daniele, F.; Leimer Saglio, C. B.; Pagani, S.; Antonietti, P. F.
Mathematical and numerical modeling of coupled oxygen dynamics and neuronal electrophysiology | Abstract | | Modeling and simulating how oxygen supply shapes neuronal excitability is crucial for advancing the understanding of brain function in pathological scenarios, such as ischemia. This condition is caused by a reduced blood supply, leading to the deprivation of oxygen and other metabolites; this energy deficit impairs ionic pumps and causes cellular swelling. In this work, this phenomenon is modeled through a volumetric variation law that links cell swelling to local oxygen concentration and the percentage of blood flow reduction. The swelling law links volume changes to local oxygen and the degree of blood-flow depression, providing a simple mechanistic pathway from hypoxia to tortuosity-driven transport impairment. The interplay between oxygen supply and excitability in brain tissue is described by coupling the monodomain model with specific neuronal ionic and metabolic models that characterize ion and metabolite concentration dynamics.
The numerical approximation of this coupled multiscale problem is particularly challenging, owing to the presence of sharp and fast-propagating wavefronts and complex geometrical domains.
To address these challenges, suitable space- and time-adaptive schemes are employed to capture the action potential dynamics accurately.
This multiscale model is discretized in space with the high-order p-adaptive discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and integrated in time with a Crank-Nicolson scheme. We numerically investigate different pathological scenarios on a two-dimensional idealized square domain and on a realistic geometry, both discretized with a polygonal grid, analyzing how subclinical and severe ischemia can affect brain electrophysiology and metabolic concentrations. |
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27/2026 - 03/09/2026
Antonietti, P. F.; Abdalla, O. M. O.; Garroni, M. G.; Mazzieri, I.; Parolini, N.
A hybrid reduced-order and high-fidelity discontinuous Galerkin Spectral Element framework for large-scale PMUT array simulations | Abstract | | Piezoelectric Micromachined Ultrasonic Transducers (PMUTs) are essential for next-generation ultrasonic sensing and imaging due to their bidirectional electromechanical behavior, compact design, and compatibility with low-voltage electronics. As PMUT arrays grow in size and complexity, efficiently modeling their coupled electromechanical-acoustic behavior becomes increasingly challenging. This work presents a novel computational framework that combines model order reduction with a Discontinuous Galerkin Spectral Element Method (DGSEM) paradigm to simulate large PMUT arrays. Each PMUT’s mechanical behavior is represented using a reduced set of vibration modes, which are coupled to an acoustic domain model to describe the full array. To further improve efficiency, a secondary acoustic domain is connected via DG interfaces, enabling non-conforming mesh refinement, with variable approximation order, and accurate wave propagation. The framework is implemented
in the SPectral Elements in Elastodynamics with Discontinuous Galerkin (SPEED) software, an open-source, parallelized platform leveraging domain decomposition, high-order polynomials, METIS graph partitioning, and MPI for scalable performance. The proposed methodology addresses key challenges in meshing, supporting high-fidelity simulations for both PMUT transmission and reception phases. Numerical results demonstrate the framework’s accuracy, scalability, and efficiency for large PMUT array simulations. |
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23/2026 - 03/04/2026
Ballini, E.; Muscarnera, L.; Fumagalli, A.; Scotti, A.; Regazzoni, F.
Elimination-compensation pruning for fully-connected neural networks | Abstract | | The unmatched ability of deep neural networks to capture complex patterns in large and noisy datasets is often associated with their large hypothesis space and the vast number of parameters characterizing modern architectures. Pruning techniques have emerged as effective tools to extract sparse representations of neural network parameters while preserving accuracy. However, a fundamental assumption behind pruning is that expendable weights have a small impact on the network error, whereas highly important weights exert a larger influence on inference.
We argue that this idea could be generalized; what if a weight is not simply removed but also compensated with a perturbation of the adjacent bias, which does not contribute to the network sparsity? Our work introduces a novel pruning method in which the importance measure of each weight is computed considering the output behavior after an optimal perturbation of its adjacent bias.
These perturbations can be then applied directly after the removal of each weight, independently of each other. After deriving analytical expressions for the aforementioned quantities, numerical experiments are conducted to benchmark this technique against some of the most popular pruning strategies, demonstrating an intrinsic efficiency of the proposed approach in very diverse machine learning scenarios. |
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26/2026 - 03/04/2026
Dokuchaev, A.; Bonizzoni, F.; Pagani, S.; Regazzoni, F.; Pezzuto, S.
Learning geometry-dependent lead-field operators for forward ECG modeling | Abstract | | Modern forward electrocardiogram (ECG) computational models rely on an accurate representation of the torso domain. The lead-field method enables fast ECG simulations while preserving full geometric fidelity. Achieving high anatomical accuracy in torso representation is, however, challenging in clinical practice, as imaging protocols are typically focused on the heart and often do not include the entire torso. In addition, the computational cost of the lead-field method scales linearly with the number of electrodes, limiting its applicability in high-density recording settings. To date, no existing approach simultaneously achieves high anatomical fidelity, low data requirements and computational efficiency. In this work, we propose a shape-informed surrogate model of the lead-field operator that serves as a drop-in replacement for the full-order model in forward ECG simulations. The proposed framework consists of two components: a geometry-encoding module that maps anatomical shapes into a low-dimensional latent space, and a geometry-conditioned neural surrogate that predicts lead-field gradients from spatial coordinates, electrode positions and latent codes. The proposed method achieves high accuracy in approximating lead fields both within the torso (mean angular error <5°) and inside the heart, resulting in highly accurate ECG simulations (relative mean squared error <2.5%). The surrogate consistently outperforms the widely used pseudo lead-field approximation while preserving negligible inference cost. Owing to its compact latent representation, the method does not require a fully detailed torso segmentation and can therefore be deployed in data-limited settings while preserving high-fidelity ECG simulations. |
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25/2026 - 03/04/2026
Carrara, D.; Hirschvogel, M.; Bonizzoni, F.; Pagani, S.; Pezzuto, S.; Regazzoni, F.
Shape-informed cardiac mechanics surrogates in data-scarce regimes via geometric encoding and generative augmentation | Abstract | | High-fidelity computational models of cardiac mechanics provide mechanistic insight into the heart function but are computationally prohibitive for routine clinical use. Surrogate models can accelerate simulations, but generalization across diverse anatomies is challenging, particularly in data-scarce settings. We propose a two-step framework that decouples geometric representation from learning the physics response, to enable shape-informed surrogate modeling under data-scarce conditions. First, a shape model learns a compact latent representation of left ventricular geometries. The learned latent space effectively encodes anatomies and enables synthetic geometries generation for data augmentation. Second, a neural field-based surrogate model, conditioned on this geometric encoding, is trained to predict ventricular displacement under external loading. The proposed architecture performs positional encoding by using universal ventricular coordinates, which improves generalization across diverse anatomies. Geometric variability is encoded using two alternative strategies, which are systematically compared: a PCA-based approach suitable for working with point cloud representations of geometries, and a DeepSDF-based implicit neural representation learned directly from point clouds. Overall, our results, obtained on idealized and patient-specific datasets, show that the proposed approaches allow for accurate predictions and generalization to unseen geometries, and robustness to noisy or sparsely sampled inputs. |
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