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Machine Learning and Projection-Based Model Reduction in Hydrology and Geosciences
Published in Anuj Karpatne, Ramakrishnan Kannan, Vipin Kumar, Knowledge-Guided Machine Learning, 2023
Mojtaba Forghani, Yizhou Qian, Jonghyun Lee, Matthew Farthing, Tyler Hesser, Peter K. Kitanidis, Eric F. Darve
POD, also known as the principal component analysis (PCA) in other contexts, is a mathematical technique that is employed to extract the dominant statistical characteristics of a system by identifying its most energetic modes [67]. POD works via approximating the dynamics of a computationally expensive system of equations by finding its reduced-order orthogonal basis via minimization of the projection error [24]. The most energetic modes are kept to generate the reduced-order system, while the other modes are truncated [67]. DMD is another approach for mode decomposition which has emerged as an alternative to POD for the reduction of the system size, in particular, for transient, non-linear systems [11, 47]. The DMD is based on constructing linear approximations of the dynamics in the low-order model and generates modes based on their dynamics rather than just energy content. In other words, DMD spatial modes have an associated frequency and temporal dynamics based on the growth and decay of the approximated system [47, 48].
Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators
Published in Nuclear Science and Engineering, 2023
Patrick Behne, Jan Vermaak, Jean Ragusa
The projection-based ROM seeks to reduce the dimensionality of the FOM by seeking the discretized solution as a linear combination of basis vectors. The subspace that the basis vectors span is determined during a data-driven offline learning phase. During the data collection phase, the parameter space is adequately sampled and the FOM solutions, or snapshots, are computed for all sample (training) points. Next, POD is performed by computing the singular value decomposition (SVD) of the snapshot data to obtain the aforementioned basis vectors, or POD modes. SVD minimizes the distance between the FOM and ROM solutions for the snapshot data set in an least-squares sense.59 The POD modes are ordered by decreasing variance, with each mode describing some characteristics of the system.35 With the basis vectors determined, the POD mode expansion approximation is substituted into the transport equation and projected onto the learned subspace, yielding a small (relative to the FOM) parametric system to be solved for the modal coefficients in the online prediction phase. With the coefficients in hand, the approximation to the FOM is obtained by evaluating the linear combination of basis vectors.
Data-driven approximation of geotechnical dynamics to an equivalent single-degree-of-freedom vibration system based on dynamic mode decomposition
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Akihiro Shioi, Yu Otake, Ikumasa Yoshida, Shogo Muramatsu, Susumu Ohno
Here, we focus on dynamic mode decomposition (DMD) (e.g. SCHMID 2010, 2011, Arai et al. 2021, Kaneko et al. 2019), stemming from the fluid mechanics community. DMD has been successfully used for prediction, state estimation, and complex system control in fluid mechanics (SCHMID 2011). SCHMID (2010) first defined the DMD algorithm by demonstrating that the theory could describe the internal mechanism based on the numerical simulation and measured experimental data in a high-dimensional flow field. The core of the DMD algorithm can be regarded as a space dimension reduction technique, similar to the algorithm of proper orthogonal decomposition (POD). While POD decomposes data into multiple orthogonal modal frequency components and ranks them in terms of energy content, the DMD algorithm can describe the dynamic characteristics in a series of single non-orthogonal frequency modes and superpose them with coefficients according to the time scale (Hemati, Williams, and Rowley 2014, Dang et al. 2018). Therefore, these properties may apply to the authors' future goal.
Response of flames with different degrees of premixedness to acoustic oscillations
Published in Combustion Science and Technology, 2018
A.M. Kypraiou, P.M. Allison, A. Giusti, E. Mastorakos
The POD method is a spatio-temporal statistical method used to extract the most dominant spatial features of the system (POD modes) and the respective frequency content (Power Spectral Density (PSD) of the POD time coefficients). The OH* chemiluminescence images were analyzed using the POD method, as described in previous studies (Ayache and Mastorakos, 2013; Berkooz et al., 1993) based on the Sirovich’s method of snapshots (Sirovich, 1987). The POD method applied in this system has been presented previously (Kypraiou et al., 2015). Since the results of the POD analysis for the studied datasets begin to converge after 500 snapshots, the POD method was applied to 1000 snapshots in order to achieve a good statistical representation.