6,887 research outputs found
Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning
In this paper, we put forth a long short-term memory (LSTM) nudging framework
for the enhancement of reduced order models (ROMs) of fluid flows utilizing
noisy measurements. We build on the fact that in a realistic application, there
are uncertainties in initial conditions, boundary conditions, model parameters,
and/or field measurements. Moreover, conventional nonlinear ROMs based on
Galerkin projection (GROMs) suffer from imperfection and solution instabilities
due to the modal truncation, especially for advection-dominated flows with slow
decay in the Kolmogorov width. In the presented LSTM-Nudge approach, we fuse
forecasts from a combination of imperfect GROM and uncertain state estimates,
with sparse Eulerian sensor measurements to provide more reliable predictions
in a dynamical data assimilation framework. We illustrate the idea with the
viscous Burgers problem, as a benchmark test bed with quadratic nonlinearity
and Laplacian dissipation. We investigate the effects of measurements noise and
state estimate uncertainty on the performance of the LSTM-Nudge behavior. We
also demonstrate that it can sufficiently handle different levels of temporal
and spatial measurement sparsity. This first step in our assessment of the
proposed model shows that the LSTM nudging could represent a viable realtime
predictive tool in emerging digital twin systems
Training Echo State Networks with Regularization through Dimensionality Reduction
In this paper we introduce a new framework to train an Echo State Network to
predict real valued time-series. The method consists in projecting the output
of the internal layer of the network on a space with lower dimensionality,
before training the output layer to learn the target task. Notably, we enforce
a regularization constraint that leads to better generalization capabilities.
We evaluate the performances of our approach on several benchmark tests, using
different techniques to train the readout of the network, achieving superior
predictive performance when using the proposed framework. Finally, we provide
an insight on the effectiveness of the implemented mechanics through a
visualization of the trajectory in the phase space and relying on the
methodologies of nonlinear time-series analysis. By applying our method on well
known chaotic systems, we provide evidence that the lower dimensional embedding
retains the dynamical properties of the underlying system better than the
full-dimensional internal states of the network
Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network
Because of their effectiveness in broad practical applications, LSTM networks
have received a wealth of coverage in scientific journals, technical blogs, and
implementation guides. However, in most articles, the inference formulas for
the LSTM network and its parent, RNN, are stated axiomatically, while the
training formulas are omitted altogether. In addition, the technique of
"unrolling" an RNN is routinely presented without justification throughout the
literature. The goal of this paper is to explain the essential RNN and LSTM
fundamentals in a single document. Drawing from concepts in signal processing,
we formally derive the canonical RNN formulation from differential equations.
We then propose and prove a precise statement, which yields the RNN unrolling
technique. We also review the difficulties with training the standard RNN and
address them by transforming the RNN into the "Vanilla LSTM" network through a
series of logical arguments. We provide all equations pertaining to the LSTM
system together with detailed descriptions of its constituent entities. Albeit
unconventional, our choice of notation and the method for presenting the LSTM
system emphasizes ease of understanding. As part of the analysis, we identify
new opportunities to enrich the LSTM system and incorporate these extensions
into the Vanilla LSTM network, producing the most general LSTM variant to date.
The target reader has already been exposed to RNNs and LSTM networks through
numerous available resources and is open to an alternative pedagogical
approach. A Machine Learning practitioner seeking guidance for implementing our
new augmented LSTM model in software for experimentation and research will find
the insights and derivations in this tutorial valuable as well.Comment: 43 pages, 10 figures, 78 reference
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