12 research outputs found
Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models
Deep learning-based reduced order models (DL-ROMs) have been recently
proposed to overcome common limitations shared by conventional ROMs - built,
e.g., exclusively through proper orthogonal decomposition (POD) - when applied
to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can
achieve extreme efficiency in the training stage and faster than real-time
performances at testing, thanks to a prior dimensionality reduction through POD
and a DL-based prediction framework. Nonetheless, they share with conventional
ROMs poor performances regarding time extrapolation tasks. This work aims at
taking a further step towards the use of DL algorithms for the efficient
numerical approximation of parametrized PDEs by introducing the -POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM
framework by adding a two-fold architecture taking advantage of long short-term
memory (LSTM) cells, ultimately allowing long-term prediction of complex
systems' evolution, with respect to the training window, for unseen input
parameter values. Numerical results show that this recurrent architecture
enables the extrapolation for time windows up to 15 times larger than the
training time domain, and achieves better testing time performances with
respect to the already lightning-fast POD-DL-ROMs.Comment: 28 page
A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs
Traditional reduced order modeling techniques such as the reduced basis (RB)
method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from
severe limitations when dealing with nonlinear time-dependent parametrized
PDEs, because of the fundamental assumption of linear superimposition of modes
they are based on. For this reason, in the case of problems featuring coherent
structures that propagate over time such as transport, wave, or
convection-dominated phenomena, the RB method usually yields inefficient
reduced order models (ROMs) if one aims at obtaining reduced order
approximations sufficiently accurate compared to the high-fidelity, full order
model (FOM) solution. To overcome these limitations, in this work, we propose a
new nonlinear approach to set reduced order models by exploiting deep learning
(DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM,
both the nonlinear trial manifold (corresponding to the set of basis functions
in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to
the projection stage in a linear ROM) are learned in a non-intrusive way by
relying on DL algorithms; the latter are trained on a set of FOM solutions
obtained for different parameter values. In this paper, we show how to
construct a DL-ROM for both linear and nonlinear time-dependent parametrized
PDEs; moreover, we assess its accuracy on test cases featuring different
parametrized PDE problems. Numerical results indicate that DL-ROMs whose
dimension is equal to the intrinsic dimensionality of the PDE solutions
manifold are able to approximate the solution of parametrized PDEs in
situations where a huge number of POD modes would be necessary to achieve the
same degree of accuracy.Comment: 28 page
Deep learning-based reduced order models in cardiac electrophysiology
Predicting the electrical behavior of the heart, from the cellular scale to
the tissue level, relies on the formulation and 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 (RB) 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 which exploits deep learning (DL)
algorithms to obtain accurate and efficient ROMs, whose dimensionality matches
the number of system parameters. Our DL approach combines deep feedforward
neural networks (NNs) and convolutional autoencoders (AEs). 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 three challenging test cases in cardiac
electrophysiology and prove that DL-ROM outperforms classical projection-based
ROMs.Comment: 28 page
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
problem
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 onto a functional
output , 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
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
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
Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models
Simulating fluid flows in different virtual scenarios is of key importance in
engineering applications. However, high-fidelity, full-order models relying,
e.g., on the finite element method, are unaffordable whenever fluid flows must
be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on
proper orthogonal decomposition (POD) provide reliable approximations to
parameter-dependent fluid dynamics problems in rapid times. However, they might
require expensive hyper-reduction strategies for handling parameterized
nonlinear terms, and enriched reduced spaces (or Petrov-Galerkin projections)
if a mixed velocity-pressure formulation is considered, possibly hampering the
evaluation of reliable solutions in real-time. Dealing with fluid-structure
interactions entails even higher difficulties. The proposed deep learning
(DL)-based ROMs overcome all these limitations by learning in a non-intrusive
way both the nonlinear trial manifold and the reduced dynamics. To do so, they
rely on deep neural networks, after performing a former dimensionality
reduction through POD enhancing their training times substantially. The
resulting POD-DL-ROMs are shown to provide accurate results in almost real-time
for the flow around a cylinder benchmark, the fluid-structure interaction
between an elastic beam attached to a fixed, rigid block and a laminar
incompressible flow, and the blood flow in a cerebral aneurysm.Comment: 22 page