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 1238 products
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85/2024 - 11/09/2024
Brivio, S.; Franco, Nicola R.; Fresca, S.; Manzoni, A.
Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition | Abstract | | 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. |
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86/2024 - 11/09/2024
Franco, N.R.; Fraulin, D.; Manzoni, A.; Zunino, P.
On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields | Abstract | | Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs. |
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88/2024 - 11/09/2024
Regazzoni, F.; Poggesi, C.; Ferrantini, C.
Elucidating the cellular determinants of the end-systolic pressure-volume relationship of the heart via computational modelling | Abstract | | The left ventricular end-systolic pressure-volume relationship (ESPVr) is a key indicator of cardiac contractility. Despite its established importance, several studies suggested that the mechanical mode of contraction, such as isovolumetric or ejecting contractions, may affect the ESPVr, challenging the traditional notion of a single, consistent relationship. Furthermore, it remains unclear whether the observed effects of ejection on force generation are inherent to the ventricular chamber itself or are a fundamental property of the myocardial tissue, with the underlying mechanisms remaining poorly understood. We investigated these aspects by using a multiscale in silico model that allowed us to elucidate the links between subcellular mechanisms and organ-level function. Simulations of ejecting and isovolumetric beats with different preload and afterload resistance were performed by modulating calcium and cross-bridge kinetics. The results suggest that the ESPVr is not a fixed curve but depends on the mechanical history of the contraction, with potentially both positive and negative effects of ejection. Isolated tissue simulations suggest that these phenomena are intrinsic to the myocardial tissue, rather than properties of the ventricular chamber. Our results suggest that the ESPVr results from the balance of positive and negative effects of ejection, respectively related to a memory effect of the increased apparent calcium sensitivity at high sarcomere length, and to the inverse relationship between force and velocity. Numerical simulations allowed us to reconcile conflicting results in the literature and suggest translational implications for clinical conditions such as hypertrophic cardiomyopathy, where altered calcium dynamics and cross-bridge kinetics may impact the ESPVr. |
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90/2024 - 11/09/2024
Tomasetto, M.; Arnone, E.; Sangalli, L.M.
Modeling anisotropy and non-stationarity through physics-informed spatial regression | Abstract | | Many spatially dependent phenomena, that are of interest in environmental problems, are characterized by strong anisotropy and non-stationarity. Moreover, the data are often observed over regions with complex conformations, such as water bodies with complicated shorelines, or regions with complex orography. Furthermore, the distribution of the data locations may be strongly inhomogeneous over space. These issues may challenge popular approaches to spatial data analysis. In this work, we show how we can accurately address these issues by spatial regression with differential regularization. We model the spatial variation by a Partial Differential Equation (PDE), defined upon the considered spatial domain. This PDE may depend upon some unknown parameters, that we estimate from the data, through an appropriate profiling estimation approach. The PDE may encode some available problem-specific information on the considered phenomenon, and permits a rich modeling of anisotropy and non-stationarity. The performances of the proposed approach are compared to competing methods, through simulation studies and real data applications. In particular, we analyse rainfall data over Switzerland, characterized by strong anisotropy, and oceanographic data in the Gulf of Mexico, characterized by non-stationarity due to the Gulf Stream.
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83/2024 - 11/08/2024
Conti, P.; Guo, M.; Manzoni, A.; Frangi, A.; Brunton, S. L.; Kutz, J.N.
Multi-fidelity reduced-order surrogate modelling | Abstract | | High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states - time-parameter-dependent expansion coefficients of the POD basis - using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features. |
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82/2024 - 11/08/2024
Rosafalco, L.; Conti, P.; Manzoni, A.; Mariani, S.; Frangi, A.
EKF-SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics | Abstract | | Measured data from a dynamical system can be assimilated into a predictive model by means of Kalman filters. Nonlinear extensions of the Kalman filter, such as the Extended Kalman Filter (EKF), are required to enable the joint estimation of (possibly nonlinear) system dynamics and of input parameters. To construct the evolution model used in the prediction phase of the EKF, we propose to rely on the Sparse Identification of Nonlinear Dynamics (SINDy). SINDy enables to identify the evolution model directly from preliminary acquired data, thus avoiding possible bias due to wrong assumptions and incorrect modelling of the system dynamics. Moreover, the numerical integration of a SINDy model leads to great computational savings compared to alternate strategies based on, e.g., finite elements. Last, SINDy allows an immediate definition of the Jacobian matrices required by the EKF to identify system dynamics and properties, a derivation that is usually extremely involved with physical models. As a result, combining the EKF with SINDy provides a data-driven computationally efficient, easy-to-apply approach for the identification of nonlinear systems, capable of robust operation even outside the range of training of SINDy. To demonstrate the potential of the approach, we address the identification of a linear non-autonomous system consisting of a shear building model excited by real seismograms, and the identification of a partially observed nonlinear system. The challenge arising from the use of SINDy when the system state is not entirely accessible has been relieved by means of time-delay embedding. The great accuracy and the small uncertainty associated with the state identification, where the state has been augmented to include system properties, underscores the great potential of the proposed strategy, paving the way for the setting of predictive digital twins in different fields. |
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80/2024 - 11/04/2024
Crippa, B.; Scotti, A.; Villa, A
Numerical Solution of linear drift-diffusion and pure drift equations on one-dimensional graphs | Abstract | | We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending Finite Volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of an electrical treeing. |
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79/2024 - 10/28/2024
Baioni, P.J.; Benacchio, T.; Capone, L.; de Falco, C.
Portable, Massively Parallel Implementation of a Material Point Method for Compressible Flows | Abstract | | The recent evolution of software and hardware technologies is leading to a renewed computational interest in Particle-In-Cell (PIC) methods such as the Material Point Method (MPM). Indeed, pro- vided some critical aspects are properly handled, PIC methods can be cast in formulations suitable for the requirements of data locality and fine-grained parallelism of modern hardware accelerators such as Graphics Processing Units (GPUs). Such a rapid and continuous technological development increases also the importance of generic and portable implementations. While the capabilities of MPM on a wide range continuum mechanics problem have been already well as- sessed, the use of the method in compressible fluid dynamics has re- ceived less attention. In this paper we present a portable, highly par- allel, GPU based MPM solver for compressible gas dynamics. The implementation aims to reach a good compromise between porta- bility and efficiency in order to provide a first assessment of the potential of this approach in solving strongly compressible gas flow problems, also taking into account solid obstacles. The numerical model considered constitutes a first step towards the development of a monolithic MPM solver for Fluid-Structure Interaction (FSI) problems at all Mach numbers up to the supersonic regime. |
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