Article
Mathematics, Applied
Ludovica Cicci, Stefania Fresca, Mengwu Guo, Andrea Manzoni, Paolo Zunino
Summary: This study explores the use of ROMs based on proper orthogonal decomposition and Gaussian process regression to construct a non-intrusive ROM. By applying this method to solid mechanics examples, it demonstrates remarkable computational speed-ups and high accuracy in global sensitivity analysis and parameter estimation tasks.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Samuel E. Otto, Gregory R. Macchio, Clarence W. Rowley
Summary: Recently developed reduced-order modeling techniques approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. While effective for post-transient regime modeling, modeling transient dynamics near underlying manifolds is complicated by fast dynamics and nonnormal sensitivity mechanisms. To address these challenges, a parametric class of nonlinear projections described by constrained autoencoder neural networks is introduced, along with dynamics-aware cost functions and oblique projection fibers.
Article
Engineering, Multidisciplinary
Chensen Ding, Hussein Rappel, Tim Dodwell
Summary: This paper proposes novel full-field order-reduced Gaussian Processes (GPs) emulators to address the challenging problem of quantifying high-dimensional uncertainty on full-field solution. High-dimensional raw dataset is generated using the isogeometric Monte Carlo simulator, and the map between raw input and output is converted to the map between their reduced basis coefficients assisted by proper orthogonal decomposition. A machine learning Emulator based on Gaussian Process is built and trained using the reduced basis coefficients. The proposed emulators can quickly and directly predict the full-field solution to new inputs, producing accurate non-linear functional approximations with a small number of training samples and offering confidence intervals.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Engineering, Mechanical
Kyusic Park, Matthew S. Allen
Summary: This study proposes a data-driven approach for reduced order modeling that incorporates design variations in finite element models (FEM). The approach uses Gaussian Process Regression (GPR) to create a single reduced order model (ROM) that can account for variations in material properties or geometric parameters. The proposed method is applied to flat and curved beam structures, showing enhanced efficiency and relatively low cost compared to traditional ROM approaches.
MECHANICAL SYSTEMS AND SIGNAL PROCESSING
(2023)
Article
Engineering, Mechanical
Shanwu Li, Yongchao Yang
Summary: This study presents a hierarchical deep learning approach to identify reduced-order models of nonlinear dynamical systems from measurement data only, including nonlinear normal modal subspace and associated dynamics. The approach is validated on unforced and forced nonlinear systems, demonstrating efficient dimensional truncation for optimal low-dimensional ROM. Performance and applicability of this approach are discussed in detail.
NONLINEAR DYNAMICS
(2021)
Article
Materials Science, Multidisciplinary
Lorenzo Del Re, Alessandro Toschi
Summary: This study generalized the formalism of the dynamical vertex approximation (D Gamma A) for treating magnetically ordered phases. By performing ladder resummations of Feynman diagrams in systems with broken SU(2) symmetry, the simplified algorithms and self-energy expressions for ferromagnetic and antiferromagnetic ordered phases were derived, providing reliable guidance for future applications. The results captured fundamental aspects of metallic and insulating ground states in two-dimensional antiferromagnets, with possible routes outlined for further development in diagrammatic-based treatments of magnetic phases in correlated electron systems.
Article
Multidisciplinary Sciences
Omer San, Suraj Pawar, Adil Rasheed
Summary: A central challenge in computational modeling and simulation is achieving robust and accurate closures for coarse-grained representations. Previous efforts focused on supervised learning, while this study introduces a novel variational multiscale reinforcement learning method. Results show that this method can discover robust and accurate scale-aware closure models for complex dynamical systems.
SCIENTIFIC REPORTS
(2022)
Article
Engineering, Multidisciplinary
Zhan Ma, Wenxiao Pan
Summary: The proposed data-driven nonintrusive model order reduction method for dynamical systems with moving boundaries combines several attributes and can accurately forecast solutions beyond the range of snapshot data.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics
Chao Liu, Bin Liu
Summary: In this paper, we investigate the quasilinear two-species chemotaxis model with nonlinear signal production and nonlinear sensitivity. We establish the global boundedness of the classical solution under appropriate regularity assumptions on the initial data. Our results partially generalize and improve some existing results in the literature.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Engineering, Multidisciplinary
Alexander V. Mamonov, Maxim A. Olshanskii
Summary: This paper presents a reduced order model (ROM) for numerical integration of a dynamical system with multiple parameters. The ROM utilizes compressed tensor formats to find a low rank representation for high-fidelity snapshots of the system state. The computational cost of the online phase depends only on tensor compression ranks, making it efficient and accurate.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Engineering, Multidisciplinary
Jinlong Fu, Dunhui Xiao, Rui Fu, Chenfeng Li, Chuanhua Zhu, Rossella Arcucci, Ionel M. Navon
Summary: This paper proposes a physics-data combined machine learning method for non-intrusive parametric reduced-order modeling in small-data regimes. The method combines dimension reduction through proper orthogonal decomposition with establishing reliable mappings between system parameters and reduced coefficients. It demonstrates high prediction accuracy, strong generalization capability, and small data requirements.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Dan Wilson
Summary: A reduced order modeling strategy is proposed to accurately capture the behavior of strongly perturbed nonlinear dynamical systems. This strategy augments the dynamics with an additional variable to select from a family of reference trajectories and limit truncation errors. The proposed reduction strategy can be implemented in situations where external inputs cause the dynamics to transition through a bifurcation, and it has been successfully applied in two examples related to neural control.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Interdisciplinary Applications
Jonggeon Lee, Jaehun Lee, Haeseong Cho, Euiyoung Kim, Maenghyo Cho
Summary: This paper introduces a nonlinear analysis approach that combines element-wise stiffness evaluation procedure with hyper-reduction reduced-order modeling method to improve computational efficiency by reducing costs.
COMPUTATIONAL MECHANICS
(2021)
Article
Mathematics, Applied
Jiabin Xu, Hassan Khan, Rasool Shah, A. A. Alderremy, Shaban Aly, Dumitru Baleanu
Summary: The research paper presents an efficient technique for solving fractional-order nonlinear Swift-Hohenberg equations related to fluid dynamics, showing that the Laplace Adomian decomposition method requires minimal calculations and produces solutions in close agreement with other existing methods. Numerical examples confirm the validity of the suggested method, demonstrating its almost identical solutions with various analytical methods through graphs and tables.
Article
Mathematics, Applied
Bulent Karasozen, Gulden Mulayim, Murat Uzunca, Suleyman Yildiz
Summary: In this study, reduced-order models (ROMs) were developed for a nonlinear cross-diffusion problem involving the SKT equation with Lotka-Volterra kinetics. By separating time into two intervals, more accurate reduced-order solutions were computed, outperforming global proper orthogonal decomposition solutions. The use of proper orthogonal decomposition in a tensorial framework accelerated the computation of reduced-order solutions independently of full-order solutions, showing prediction capabilities for one- and two-dimensional patterns. Additionally, the decrease in entropy by the reduced solutions played a crucial role in ensuring the global existence of solutions for nonlinear cross-diffusion equations like the SKT equation.
APPLIED MATHEMATICS AND COMPUTATION
(2021)