800 research outputs found
Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks
Nonlinear model predictive control (NMPC) often requires real-time solution
to optimization problems. However, in cases where the mathematical model is of
high dimension in the solution space, e.g. for solution of partial differential
equations (PDEs), black-box optimizers are rarely sufficient to get the
required online computational speed. In such cases one must resort to
customized solvers. This paper present a new solver for nonlinear
time-dependent PDE-constrained optimization problems. It is composed of a
sequential quadratic programming (SQP) scheme to solve the PDE-constrained
problem in an offline phase, a proper orthogonal decomposition (POD) approach
to identify a lower dimensional solution space, and a neural network (NN) for
fast online evaluations. The proposed method is showcased on a regularized
least-square optimal control problem for the viscous Burgers' equation. It is
concluded that significant online speed-up is achieved, compared to
conventional methods using SQP and finite elements, at a cost of a prolonged
offline phase and reduced accuracy.Comment: Accepted for publishing at the 58th IEEE Conference on Decision and
Control, Nice, France, 11-13 December, https://cdc2019.ieeecss.org
Parameter Identification by Deep Learning of a Material Model for Granular Media
Classical physical modelling with associated numerical simulation
(model-based), and prognostic methods based on the analysis of large amounts of
data (data-driven) are the two most common methods used for the mapping of
complex physical processes. In recent years, the efficient combination of these
approaches has become increasingly important. Continuum mechanics in the core
consists of conservation equations that -- in addition to the always necessary
specification of the process conditions -- can be supplemented by
phenomenological material models. The latter are an idealized image of the
specific material behavior that can be determined experimentally, empirically,
and based on a wealth of expert knowledge. The more complex the material, the
more difficult the calibration is. This situation forms the starting point for
this work's hybrid data-driven and model-based approach for mapping a complex
physical process in continuum mechanics. Specifically, we use data generated
from a classical physical model by the MESHFREE software to train a Principal
Component Analysis-based neural network (PCA-NN) for the task of parameter
identification of the material model parameters. The obtained results highlight
the potential of deep-learning-based hybrid models for determining parameters,
which are the key to characterizing materials occurring naturally, and their
use in industrial applications (e.g. the interaction of vehicles with sand).Comment: arXiv admin note: text overlap with arXiv:2212.0313
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
Model Reduction and Neural Networks for Parametric PDEs
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature
Multi-fidelity reduced-order surrogate modeling
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 for modeling a
given system. Multi-fidelity surrogate modeling aims to leverage less accurate,
lower-fidelity models that are computationally inexpensive in order to enhance
predictive accuracy when high-fidelity data are limited or scarce. However,
low-fidelity models, while often displaying important qualitative
spatio-temporal features, fail to accurately capture the onset of instability
and critical transients observed in the high-fidelity models, making them
impractical as surrogate models. To address this shortcoming, we present a new
data-driven strategy that combines dimensionality reduction with multi-fidelity
neural network surrogates. The key idea is to generate a spatial basis by
applying the classical 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 (LSTM) 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 and
robustness of this method is demonstrated by a collection of parametrized,
time-dependent PDE problems where the low-fidelity model can be defined by
coarser meshes and/or time stepping, as well as by misspecified physical
features. Importantly, the onset of instabilities and transients are well
captured by this surrogate modeling technique
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