Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1349 prodotti
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30/2025 - 22/05/2025
Rosafalco, L.; Conti, P.; Manzoni, A.; Mariani, S.; Frangi, A.
Online learning in bifurcating dynamic systems via SINDy and Kalman filtering | Abstract | | We propose the use of the Extended Kalman Filter (EKF) for online data assimilation and update of a dynamic model, preliminary identified through the Sparse Identification of Nonlinear Dynamics (SINDy). This data-driven technique may avoid biases due to incorrect modelling assumptions and exploits SINDy to approximate the system dynamics leveraging a predefined library of functions, where active terms are selected and weighted by a sparse set of coefficients. This results in a physically-sound and interpretable dynamic model allowing to reduce epistemic uncertainty often affecting machine learning approaches. Treating the SINDy model coefficients as random variables, we propose to update them while acquiring (possibly noisy) system measurements, thus enabling the online identification of time-varying systems. These changes can stem from, e.g., varying operational conditions or unforeseen events. The EKF performs model adaptation through joint state-parameters estimation, with the Jacobian matrices required to computed the model sensitivity inexpensively evaluated from the SINDy model formulation. The effectiveness of this approach is demonstrated through three case studies: (i) a Lotka-Volterra model in which all parameters simultaneously evolve during the observation period; (ii) a Selkov model where the system undergoes a bifurcation not seen during the SINDy training; (iii) a MEMS arch exhibiting a 1:2 internal resonance. The ability of EKF of recovering inactivated functional terms from the SINDy library, or discarding unnecessary contribution, is also highlighted. Based on the presented applications, this method shows strong promise for handling time-varying nonlinear dynamic systems possibly experiencing bifurcating behaviors. |
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31/2025 - 22/05/2025
Botteghi, N.; Fresca, S.; Guo, M.; Manzoni, A.
HypeRL: Parameter-Informed Reinforcement Learning for Parametric PDEs | Abstract | | In this work, we devise a new, general-purpose reinforcement learning strategy for the optimal control of parametric partial differential equations (PDEs). 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. Indeed, in either way, the optimal control problem has to be solved for each instance of the parameters, and this is out of reach when dealing with high-dimensional time-dependent and parametric PDEs. 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, bypassing the need to numerically solve the HJB equation explicitly for all possible states and parameters, or solving an adjoint problem within an iterative optimization loop for each parameter instance. 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. To do so, HypeRL uses two additional NNs, often 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 NNs. We validate the proposed approach on two PDE-constrained optimal control benchmarks, namely a 1D Kuramoto-Sivashinsky equation with in-domain control and on a 2D Navier Stokes equations with boundary control, by showing that the knowledge of the PDE parameters and how this information is encoded, i.e., via a hypernetwork, is an essential ingredient for learning parameter-dependent control policies that can generalize effectively to unseen scenarios and for improving the sample efficiency of such policies. |
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29/2025 - 18/05/2025
Centofanti, E.; Ziarelli, G.; Parolini, N.; Scacchi, S.; Verani, M. ; Pavarino, L. F.
Learning cardiac activation and repolarization times with operator learning | Abstract | | Solving partial or ordinary differential equation models in cardiac electrophysiology is a computationally
demanding task, particularly when high-resolution meshes are required to capture
the complex dynamics of the heart. Moreover, in clinical applications, it is essential to employ
computational tools that provide only relevant information, ensuring clarity and ease of interpretation.
In this work, we exploit two recently proposed operator learning approaches, namely
Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL), to learn the operator
mapping the applied stimulus in the physical domain into the activation and repolarization time
distributions. These data-driven methods are evaluated on synthetic 2D and 3D domains, as well
as on a physiologically realistic left ventricle geometry. Notably, while the learned map between
the applied current and activation time has its modelling counterpart in the Eikonal model, no
equivalent partial differential equation (PDE) model is known for the map between the applied
current and repolarization time. Our results demonstrate that both FNO and KOL approaches
are robust to hyperparameter choices and computationally efficient compared to traditional PDEbased
Monodomain models. These findings highlight the potential use of these surrogate operators
to accelerate cardiac simulations and facilitate their clinical integration. |
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28/2025 - 10/05/2025
Ciaramella, G.; Gander, M.J.; Mazzieri, I.
Discontinuous Galerkin time integration for second-order differential problems: formulations, analysis, and analogies | Abstract | | We thoroughly investigate Discontinuous Galerkin (DG) discretizations as time integrators for second-order oscillatory systems, considering both second-order and first-order formulations of the original problem. Key contributions include new convergence analyses for the second-order formulation and equivalence proofs between DG and classical time-stepping schemes (such as Newmark schemes and general linear methods). In addition, the chapter provides a detailed review and convergence analysis for the first-order formulation, alongside comparisons of the proposed schemes in terms of accuracy, consistency, and computational cost |
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27/2025 - 10/05/2025
Antonietti P.F.; Artoni, A.; Ciaramella, G.; Mazzieri, I.
A review of discontinuous Galerkin time-stepping methods for wave propagation problems | Abstract | | This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave propagation problems. Two formulations, tailored for second- and first-order systems of ordinary differential equations, are discussed within a generalized framework, assessing their stability, accuracy, and computational efficiency. Theoretical results are supported by various illustrative examples that validate the findings, enhancing the understanding and applicability of these methods in practical scenarios. |
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24/2025 - 08/05/2025
Bartsch, J.; Borzi, A.; Ciaramella, G.; Reichle, J.
Adjoint-based optimal control of jump-diffusion processes | Abstract | | Stochastic differential equations (SDEs) using jump-diffusion processes describe many natural phenomena at the microscopic level. Since they are commonly used to model economic and financial evolutions, the calibration and optimal control of such processes are of interest to many communities and have been the subject of extensive research. In this work, we develop an optimization method working at the microscopic level. This allows us also to reduce computational time since we can parallelize the calculations and do not encounter the so-called curse of dimensionality that occurs when lifting the problem to its macroscopic counterpart using partial differential equations (PDEs). Using a discretize-then- optimize approach, we derive an adjoint process and an optimality system in the Lagrange framework. Then, we apply Monte Carlo methods to solve all the arising equations. We validate our optimization strategy by extensive numerical experiments. We also successfully test a optimization procedure that avoids storing the information of the forward equation.
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22/2025 - 30/04/2025
Leimer Saglio, C. B.; Pagani, S.; Antonietti P. F.
A p-adaptive polytopal discontinuous Galerkin method for high-order
approximation of brain electrophysiology | Abstract | | Multiscale mathematical models have shown great promise in computational brain electrophysiology but are still hindered by high computational costs due to fast dynamics and complex brain geometries, requiring very fine spatio-temporal resolution. This paper introduces a novel p-adaptive discontinuous Galerkin method on polytopal grids (PolyDG) coupled with Crank–Nicolson time integration to approximate such models efficiently. The p-adaptive method enhances local accuracy via dynamic, element-wise polynomial refinement/de-refinement guided by a-posteriori error estimators. A novel clustering algorithm automatizes the selection of elements for adaptive updates, further improving efficiency. A wide set of numerical tests, including epileptic seizure simulations in a sagittal section of a human brain stem, demonstrate the method’s ability to reduce computational load while maintaining the accuracy of the numerical solution in capturing the dynamics of multiple wavefronts. |
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23/2025 - 30/04/2025
Antonietti, P. F.; Caldana, M.; Mazzieri, I.; Re Fraschini, A.
MAGNET: an open-source library for mesh agglomeration by Graph Neural Networks | Abstract | | We introduce MAGNET, an open-source Python library designed for mesh agglomeration in both two- and three-dimensions, based on employing Graph Neural Networks (GNN). MAGNET serves as a comprehensive solution for training a variety of GNN models, integrating deep learning and other advanced algorithms such as METIS and k-means to facilitate mesh agglomeration and quality metric computation. The library's introduction is outlined through its code structure and primary features. The GNN framework adopts a graph bisection methodology that capitalizes on connectivity and geometric mesh information via SAGE convolutional layers, in line with the methodology proposed by Antonietti et al. (2024). Additionally, the proposed MAGNET library incorporates reinforcement learning to enhance the accuracy and robustness of the model for predicting coarse partitions within a multilevel framework. A detailed tutorial is provided to guide the user through the process of mesh agglomeration and the training of a GNN bisection model. We present several examples of mesh agglomeration conducted by MAGNET, demonstrating the library's applicability across various scenarios. Furthermore, the performance of the newly introduced models is contrasted with that of METIS and k-means, illustrating that the proposed GNN models are competitive regarding partition quality and computational efficiency. Finally, we exhibit the versatility of MAGNET's interface through its integration with lymph, an open-source library implementing discontinuous Galerkin methods on polytopal grids for the numerical discretization of multiphysics differential problems. |
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