Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures

Fresca, Stefania; Gobat, Giorgio; Fedeli, Patrick; Frangi, Attilio; Manzoni, Andrea

doi:10.1002/nme.7054

Cited by 30 publications

(18 citation statements)

References 71 publications

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“…We will focus on mechanical structures subjected to periodic forcing that undergo large transformations but still only experience small strains, a condition that is well described by the Saint Venant–Kirchhoff constitutive model. As detailed in [ 26 ], the space discretization of the governing equations by means of finite elements yields a system of coupled nonlinear differential equations representing the full order model (FOM):

where the vector

collects the

unknown displacements nodal values;

is the mass matrix;

is the Rayleigh mass-proportional damping matrix with

reference eigenfrequency and Q quality factor; and

is the nodal force vector which depends on the vector of

parameters

and expresses the actuation due, in general, to a multiphysics coupling, e.g., with piezolectricity or electrostatics. In particular, the vector

depends on the angular frequency

of the actuation and is nonlinearly modulated in amplitude by the coefficient

.…”

Section: Problem Formulationmentioning

confidence: 99%

“…When nonlinearities are present, this property is generally lost. For instance, in systems behaving as simple duffing oscillators with hardening (or softening) properties, the phase of the response with respect to the forcing signal has been exploited in [ 26 ] as an order parameter.…”

Section: Data-driven Reduced Order Modeling Through Dl-romsmentioning

confidence: 99%

“…Four regions are identified, as illustrated in Figure 2 , and the arc-length of each region is rescaled from 0 to 1, yielding a total arc-length of 4. Even if this procedure requires some insight into the physics of the response, it is, however, very general and powerful, thus representing a major improvement with respect to other procedures that would not be applicable in this context [ 26 ].…”

Section: Data-driven Reduced Order Modeling Through Dl-romsmentioning

confidence: 99%

“…In [ 26 ], an extension of the DL-ROM technique has been proposed and applied to microstructures, showing very good predictive capabilities. In particular, a first dimensionality reduction in the data is achieved by means of a POD-Galerkin (POD-G) ROM in order to reduce the costly FOM data generation phase, thus defining the POD-G DL-ROM technique.…”

Section: Introductionmentioning

confidence: 99%

“…As a natural consequence of these remarks, the aim of this contribution is twofold. After a short introduction to the governing equations in Section 2 and of the DL-based ROM method in Section 3 , we further improve the POD-DL-ROM technique [ 24 , 33 ] in Section 3 through the use of an arc-length abscissa defined on the frequency response functions (FRFs) to enable the analysis of complex interaction problems in real nonlinear microstructures showing, e.g., internal resonances and coupled electromechanical interactions inducing autoparametric effects; this feature represents a key difference with respect to previous works [ 26 ] on similar architecture. In parallel, we explore in-depth the relationship between the POD-DL-ROM and the DPIM approaches in Section 4 .…”

Section: Introductionmentioning

confidence: 99%

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Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

Gobat

Fresca

Manzoni

et al. 2023

Sensors

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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 for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches, and gyroscopes, as well as displaying intricate dynamical evolutions such as 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 the extraction of 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.

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where the vector

collects the

unknown displacements nodal values;

is the mass matrix;

is the Rayleigh mass-proportional damping matrix with

reference eigenfrequency and Q quality factor; and

is the nodal force vector which depends on the vector of

parameters

and expresses the actuation due, in general, to a multiphysics coupling, e.g., with piezolectricity or electrostatics. In particular, the vector

depends on the angular frequency

of the actuation and is nonlinearly modulated in amplitude by the coefficient

.…”

Section: Problem Formulationmentioning

confidence: 99%

Section: Data-driven Reduced Order Modeling Through Dl-romsmentioning

confidence: 99%

Section: Data-driven Reduced Order Modeling Through Dl-romsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

Gobat

Fresca

Manzoni

et al. 2023

Sensors

Self Cite

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Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

Fresca,

Dedè,

Manzoni

2024

Big Data Analysis and Artificial Intelligence for Medical Sciences

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Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms-such as deep feedforward neural networks and convolutional autoencodersto obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.

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

Cicci,

Fresca,

Manzoni

et al. 2023

Numer Methods Biomed Eng

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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. Indeed, while FOMs, such as those based on the finite element method, provide valuable information on the cardiac mechanical function, accurate numerical results can be obtained at the price of very fine spatio‐temporal discretizations. As a matter of fact, simulating even just a few heartbeats can require up to hours of wall time on high‐performance computing architectures. In addition, cardiac models usually depend on a set of input parameters that are calibrated 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 employed in reduced‐order models (ROMs) for nonlinear parametrized systems. This method can provide extremely accurate approximations to parametrized cardiac mechanics problems, such as in the case of the complete cardiac cycle in a patient‐specific left ventricle geometry. In this respect, a 3D model for tissue mechanics is coupled with a 0D model for external blood circulation; active force generation is provided through an adjustable parameter‐dependent surrogate model as input to the tissue 3D model. The proposed strategy is shown to outperform classical projection‐based ROMs, in terms of orders of magnitude of computational speed‐up, and to return accurate pressure‐volume loops in both physiological and pathological cases. Finally, an application to a forward uncertainty quantification analysis, unaffordable if relying on a FOM, is considered, involving output quantities of interest such as, for example, the ejection fraction or the maximal rate of change in pressure in the left ventricle.

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Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures

Cited by 30 publications

References 71 publications

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

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

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