Abstract:The reconstruction of fine-scale information from sparse data measured at irregular locations is often needed in many diverse applications, including numerous instances of practical fluid dynamics observed in natural environments. This need is driven by tasks such as data assimilation or the recovery of fine-scale knowledge including models from limited data. Sparse reconstruction is inherently badly represented when formulated as a linear estimation problem. Therefore, the most successful linear estim… Show more
“…orthogonal, may be helpful in identifying more sparse dynamics (Champion et al. 2019 a ; Ladjal, Newson & Pham 2019; Jayaraman & Mamun 2020). Figure 13.Results with Alasso of the transient example.…”
Section: Resultsmentioning
confidence: 99%
“…2019; Ladjal et al. 2019; Jayaraman & Mamun 2020). Finally, we believe that the results of our examples, above, enables us to have a strong motivation for future work and notice the remarkable potential of SINDy for fluid dynamics.…”
We perform a sparse identification of nonlinear dynamics (SINDy) for low-dimensionalized complex flow phenomena. We first apply the SINDy with two regression methods, the thresholded least square algorithm and the adaptive least absolute shrinkage and selection operator which show reasonable ability with a wide range of sparsity constant in our preliminary tests, to a two-dimensional single cylinder wake at
$Re_D=100$
, its transient process and a wake of two-parallel cylinders, as examples of high-dimensional fluid data. To handle these high-dimensional data with SINDy whose library matrix is suitable for low-dimensional variable combinations, a convolutional neural network-based autoencoder (CNN-AE) is utilized. The CNN-AE is employed to map a high-dimensional dynamics into a low-dimensional latent space. The SINDy then seeks a governing equation of the mapped low-dimensional latent vector. Temporal evolution of high-dimensional dynamics can be provided by combining the predicted latent vector by SINDy with the CNN decoder which can remap the low-dimensional latent vector to the original dimension. The SINDy can provide a stable solution as the governing equation of the latent dynamics and the CNN-SINDy-based modelling can reproduce high-dimensional flow fields successfully, although more terms are required to represent the transient flow and the two-parallel cylinder wake than the periodic shedding. A nine-equation turbulent shear flow model is finally considered to examine the applicability of SINDy to turbulence, although without using CNN-AE. The present results suggest that the proposed scheme with an appropriate parameter choice enables us to analyse high-dimensional nonlinear dynamics with interpretable low-dimensional manifolds.
“…orthogonal, may be helpful in identifying more sparse dynamics (Champion et al. 2019 a ; Ladjal, Newson & Pham 2019; Jayaraman & Mamun 2020). Figure 13.Results with Alasso of the transient example.…”
Section: Resultsmentioning
confidence: 99%
“…2019; Ladjal et al. 2019; Jayaraman & Mamun 2020). Finally, we believe that the results of our examples, above, enables us to have a strong motivation for future work and notice the remarkable potential of SINDy for fluid dynamics.…”
We perform a sparse identification of nonlinear dynamics (SINDy) for low-dimensionalized complex flow phenomena. We first apply the SINDy with two regression methods, the thresholded least square algorithm and the adaptive least absolute shrinkage and selection operator which show reasonable ability with a wide range of sparsity constant in our preliminary tests, to a two-dimensional single cylinder wake at
$Re_D=100$
, its transient process and a wake of two-parallel cylinders, as examples of high-dimensional fluid data. To handle these high-dimensional data with SINDy whose library matrix is suitable for low-dimensional variable combinations, a convolutional neural network-based autoencoder (CNN-AE) is utilized. The CNN-AE is employed to map a high-dimensional dynamics into a low-dimensional latent space. The SINDy then seeks a governing equation of the mapped low-dimensional latent vector. Temporal evolution of high-dimensional dynamics can be provided by combining the predicted latent vector by SINDy with the CNN decoder which can remap the low-dimensional latent vector to the original dimension. The SINDy can provide a stable solution as the governing equation of the latent dynamics and the CNN-SINDy-based modelling can reproduce high-dimensional flow fields successfully, although more terms are required to represent the transient flow and the two-parallel cylinder wake than the periodic shedding. A nine-equation turbulent shear flow model is finally considered to examine the applicability of SINDy to turbulence, although without using CNN-AE. The present results suggest that the proposed scheme with an appropriate parameter choice enables us to analyse high-dimensional nonlinear dynamics with interpretable low-dimensional manifolds.
“…Considerable previous studies proposed multiple algorithms to determine sensor locations within the gappy framework. The simplest approach is to sample randomly (hereafter referred to as Random algorithm). ,,,, Some studies , leveraged the properties of the linear system and designed a method that minimizes the matrix condition number M or κ( M ) (hereafter referred to as MCN algorithm). Mathematically, the condition number of a matrix M refers to the ratio of the maximum to minimum singular values of M , which reveals the orthogonality of the matrix.…”
A myriad of studies
have attempted to use ground-level observations
to obtain gap-free spatiotemporal variations of PM2.5,
in support of air quality management and impact studies. Statistical
methods (machine learning, etc.) or numerical methods by combining
chemical transport modeling and observations with data assimilation
techniques have been typically applied, yet the significance of site
placement has not been well recognized. In this study, we apply five
proper orthogonal decomposition (POD)-based sensor placement algorithms
to identify optimal site locations and systematically evaluate their
reconstruction ability. We demonstrate that the QR pivot is relatively
more reliable in deciding optimal monitoring site locations. When
the number of planned sites (sensors) is limited, using a lower number
of modes would yield lower estimation errors. However, the dimension
of POD modes has little impact on reconstruction quality when sufficient
sensors are available. The locations of sites guided by the QR pivot
algorithm are mainly located in regions where PM2.5 pollution
is severe. We compare reconstructed PM2.5 pollution based
on QR pivot-guided sites and existing China National Environmental
Monitoring Center (CNEMC) sites and find that the QR pivot-guided
sites are superior to existing sites with respect to reconstruction
accuracy. The current planning of monitoring stations is likely to
miss sources of pollution in less-populated regions, while our QR
pivot-guided sites are planned based on the severity of PM2.5 pollution. This planning methodology has additional potentials in
chemical data assimilation studies as duplicate information from current
CNEMC-concentrated stations is not likely to boost performance.
“…QR has been widely used in applications including fluid flows, sea surface temperature monitoring, and face image recognition [22,23,24,25]. It is computationally efficient and is one of the leading paradigms for monitoring engineering and physical systems.…”
Magnetoencephalography (MEG) is a noninvasive method for measuring magnetic flux signals caused by brain activity using sensor arrays located on or above the scalp. A common strategy for monitoring brain activity is to place sensors on a nearly uniform grid, or sensor array, around the head. By increasing the total number of sensors, higher spatial-frequency components of brain activity can be resolved as dictated by Nyquist sampling theory. Currently, there are few principled mathematical architectures for sensor placement aside from Nyquist considerations. However, global brain activity often exhibits low-dimensional patterns of spatio-temporal dynamics. The low-dimensional global patterns can be computed from the singular value decomposition and can be leveraged to select a small number of sensors optimized for reconstructing brain signals and localizing brain sources. Moreover, a smaller number of sensors which are systematically chosen can outperform the entire sensor array when considering noisy measurements. We propose a greedy selection algorithm based upon the QR decomposition that is computationally efficient to implement for MEG. We demonstrate the performance of the sensor selection algorithm for the tasks of signal reconstruction and localization. The performance is dependent upon source localization, with shallow sources easily identified and reconstructed, and deep sources more difficult to locate. Our findings suggest that principled methods for sensor selection can improve MEG capabilities and potentially add cost savings for monitoring brain-wide activity.
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