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Aerodynamic reduced-order model (ROM) is a useful tool to predict nonlinear unsteady aerodynamics with reasonable accuracy and very low computational cost. The efficacy of this method has been validated by many recent studies. However, the generalization capability of aerodynamic ROMs with respect to different flow conditions and different aeroelastic parameters should be further improved. In order to enhance the predicting capability of ROM for varying operating conditions, this paper presents an unsteady aerodynamic model based on long short-term memory (LSTM) network from deep learning theory for large training dataset and sampling space. This type of network has attractive potential in modeling temporal sequence data, which is well suited for capturing the time-delayed effects of unsteady aerodynamics. Different from traditional reduced-order models, the current model based on LSTM network does not require the selection of delay orders. The performance of the proposed model is evaluated by a NACA 64A010 airfoil pitching and plunging in the transonic flow across multiple Mach numbers. It is demonstrated that the model can accurately capture the dynamic characteristics of aerodynamic and aeroelastic systems for varying flow and structural parameters.

Over the past two decades, data-driven reduced-order modeling (ROM) strategies have gained significant traction in the nonlinear dynamics community. Currently, several challenges in physical interpretation and data availability remain overlooked in current methodologies. This work proposes a novel ROM methodology based on a newly proposed generalized characteristic value decomposition (GCVD) to address these obstacles. The GCVD-ROM approach proposes a new perspective toward data-driven ROMs via characterization of the dynamics before any ROM considerations are made. In doing so, a significant degree of versatility is inherited in the GCVD-ROM strategy, allowing our models to reproduce the full-scale dynamics in different regions of the parameter space at the cost of a single training data set. Our approach utilizes computationally efficient free-decay data sets alongside a windowed-decomposition scheme, allowing us to extract energy-dependent modal structures for use in model-order reduction. This is accomplished using the physically insightful characteristic values provided by the GCVD, which are shown to be directly related to the system poles at a particular response amplitude. This natural metric, paired with a resonance tracking scheme, allows us to address the difficulties associated with physical interpretation and data availability without sacrificing the convenient aspects of linear projection-based model order reduction. A computational framework for the continuation and bifurcation analysis using linear projection-based ROMs is also presented, permitting us to deploy rigorous analysis and bifurcation studies to verify that our ROMs reproduce the intrinsic complexity of full-scale systems. A detailed walk-through of the GCVD-ROM approach is demonstrated on a simple system where important practical considerations and implementation details are discussed using a concrete example. The discretized von Kármán beam and shallow arch partial differential equations are also used to explore complicated scenarios involving modal coupling across disparate time scales and internal resonances.

In this paper, a formulation of reduced-order finite element (FE) model is presented for geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation and harmonic balance method. The main focus is to formulate the nonlinear reduced-order models (ROMs) in the time and frequency domain without involving the corresponding full-order FE models, and it is carried out by means of a special factorization of the nonlinear strain–displacement matrix. Furthermore, a methodology for the enrichment of reduction basis (RB) over that obtained from conventional approaches is presented where the proper orthogonal decomposition method is utilized by comprising the correlation matrix as the union of stiffness-normalized reduction basis vectors and the corresponding static derivatives. The results reveal a significantly reduced computational time due to the formulation of the nonlinear ROMs without involving the full-order FE model. A good accuracy of the nonlinear ROMs of viscoelastic structures is also achieved through the present method of enrichment of RB.

Subsurface geometries, such as faults and subducting slab interfaces, are often poorly constrained, yet they exert first‐order control on key geophysical processes, including subduction zone thermal structure and earthquake rupture dynamics. Quantifying model sensitivity to geometric variability remains challenging for high‐fidelity simulations that require generated meshes, due to the manual effort of mesh generation and the computational cost of exploring high‐dimensional parameter spaces. We present a mesh morphing approach that deforms a reference mesh into geometrically varying configurations while preserving mesh connectivity. This enables the automated generation of large ensembles of geometrically variable meshes with minimal user input. Importantly, the preserved connectivity allows for the application of data‐driven, non‐intrusive reduced‐order models (ROMs) to perform robust sensitivity analysis and uncertainty quantification. We demonstrate mesh morphing in two geophysical applications: (a) 3D dynamic rupture simulations with fault dip angles varying across a 40° range, and (b) 2D thermal models of subduction zones incorporating realistic slab interface curvature and depth uncertainties. The morphed meshes retain high quality and lead to accurate simulation results that closely match those obtained using generated meshes. For the dynamic rupture case, we construct ROMs that efficiently predict surface displacement and velocity time series as functions of fault geometry, achieving speedups of up to 109 $1{0}^{9}$ times relative to full simulations. Our results show that mesh morphing can be a powerful and generalizable tool for incorporating geometric uncertainty into physics‐based modeling. The method supports efficient ensemble modeling for rigorous sensitivity studies applicable across a range of problems in computational geophysics.

Testing wave energy converters in the ocean could be expensive and complex, which necessitates the use of numerical modeling. However, accurately modeling the response of wave energy converters with high-fidelity simulations can be computationally intensive in the design stage where different configurations must be considered. Reduced-order models based on simplified equations of motion can be very useful in the design, optimization, or control of wave energy converters. Given the complex dynamics of wave energy converters, accurate representation, and evaluation of relative contributions by different forces are required. This effort is concerned with a performance characterization of the hydrodynamic response of an oscillating surge wave energy converter that is based on a reduced-order model. A state-space model is used to represent the radiation damping term. Morison’s representation of unsteady forces is used to account for the nonlinear damping. Wave tank tests are performed to validate simulations. A free response simulation is used to determine the coefficients of the state-space model. Torque-forced simulations are used to identify the coefficients of the nonlinear damping term for different amplitudes and wave frequencies. The impact of varying these coefficients on the response is investigated. An assessment of the capability of the model in predicting the hydrodynamic response under irregular forcing is performed. The results show that the maximum error is 3% when compared with high-fidelity simulations. It is determined that the nonlinear damping is proportional to the torque amplitude and its effects are more pronounced as the amplitude of the flap oscillations increases.

We show how spectral submanifold theory can be used to construct reduced-order models for harmonically excited mechanical systems with internal resonances. Efficient calculations of periodic and quasi-periodic responses with the reduced-order models are discussed in this paper and its companion, Part II, respectively. The dimension of a reduced-order model is determined by the number of modes involved in the internal resonance, independently of the dimension of the full system. The periodic responses of the full system are obtained as equilibria of the reduced-order model on spectral submanifolds. The forced response curve of periodic orbits then becomes a manifold of equilibria, which can be easily extracted using parameter continuation. To demonstrate the effectiveness and efficiency of the reduction, we compute the forced response curves of several high-dimensional nonlinear mechanical systems, including the finite-element models of a von Kármán beam and a plate.

Guidewire-based pressure measurement is essential for diagnosing coronary artery disease. However, the impact of the guidewire on local hemodynamics and diagnostic outcomes is not fully understood. In this study, we propose a generalized reduced-order model (ROM) to accurately predict the trans-stenotic pressure drop in arteries. A key advantage of this model is that the viscous term does not rely on empirical parameters, making it applicable to both scenarios with and without guidewire insertion, and across varying stenosis severities. The proposed model demonstrates good accuracy compared to 3D idealized numerical models, achieving an average prediction error of 3.61% for cases without a guidewire and 4.53% for cases with a guidewire. Furthermore, when applied to a patient-specific model, it achieves comparable or better results than previously published ROMs. Finally, this ROM is employed to investigate the shifting relative importance of different components of the trans-stenotic pressure drop at various stenosis severities, and to provide further insights into the guidewire’s influence on FFR measurements.

Identifying accurate and yet interpretable low-order models from data has gained a renewed interest over the past decade. In the present work, we illustrate how the combined use of dimensionality reduction and sparse system identification techniques allows us to obtain an accurate model of the chaotic thermal convection in a two-dimensional annular thermosyphon. Taking as guidelines the derivation of the Lorenz system, the chaotic thermal convection dynamics simulated using a high-fidelity computational fluid dynamics solver are first embedded into a low-dimensional space using dynamic mode decomposition. After having reviewed the physical properties the reduced-order model should exhibit, the latter is identified using SINDy, an increasingly popular and flexible framework for the identification of nonlinear continuous-time dynamical systems from data. The identified model closely resembles the canonical Lorenz system, having the same structure and exhibiting the same physical properties. It moreover accurately predicts a bifurcation of the high-dimensional system (corresponding to the onset of steady convection cells) occurring at a much lower Rayleigh number than the one considered in this study.

To explore the impact of pest-control strategy on integrated pest management, a three-dimensional (3D) fractional- order slow–fast prey–predator model is introduced in this article. The prey community (assumed as pest) represents fast dynamics and two predators exhibit slow dynamical variables in the three-species interacting prey–predator model. In addition, common enemies of that pest are assumed as predators of two different species. Pest community causes serious damage to the economy. Fractional-order systems can better describe the real scenarios than classical-order dynamical systems, as they show previous history-dependent properties. We establish the ability of a fractional-order model with Caputo’s fractional derivative to capture the dynamics of this prey–predator system and analyze its qualitative properties. To investigate the importance of fractional-order dynamics on the behavior of the pest, we perform the local stability analysis of possible equilibrium points, using certain assumptions for different sets of parameters and reveal that the fractional-order exponent has an impact on the stability and the existence of Hopf bifurcations in the prey–predator model. Next, we discuss the existence, uniqueness and boundedness of the fractional-order system. We also observe diverse oscillatory behavior of different amplitude modulations including mixed mode oscillations (MMOs) for the fractional-order prey–predator model. Higher amplitude pest periods are interspersed with the outbreaks of small pest concentration. With the decrease of fractional-order exponent, small pest concentration increases with decaying long pest periods. We further notice that the reduced-order model is biologically significant and sensitive to the fractional-order exponent. Additionally, the dynamics captures adaptation that occurs over multiple timescales and we find consistent differences in the characteristics of the model for various fractional exponents.

A hybrid data assimilation algorithm is developed for complex dynamical systems with partial observations. The method starts with applying a spectral decomposition to the entire spatiotemporal fields, followed by creating a machine learning model that builds a nonlinear map between the coefficients of observed and unobserved state variables for each spectral mode. A cheap low‐order nonlinear stochastic parameterized extended Kalman filter (SPEKF) model is employed as the forecast model in the ensemble Kalman filter to deal with each mode associated with the observed variables. The resulting ensemble members are then fed into the machine learning model to create an ensemble of the corresponding unobserved variables. In addition to the ensemble spread, the training residual in the machine learning‐induced nonlinear map is further incorporated into the state estimation, advancing the diagnostic quantification of the posterior uncertainty. The hybrid data assimilation algorithm is applied to a precipitating quasi‐geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two‐level QG model. The complicated nonlinearities in the PQG equations prevent traditional methods from building simple and accurate reduced‐order forecast models. In contrast, the SPEKF forecast model is skillful in recovering the intermittent observed states, and the machine learning model effectively estimates the chaotic unobserved signals. Utilizing the calibrated SPEKF and machine learning models under a moderate cloud fraction, the resulting hybrid data assimilation remains reasonably accurate when applied to other geophysical scenarios with nearly clear skies or relatively heavy rainfall, implying the robustness of the algorithm for extrapolation. Plain Language Summary Data assimilation of complex turbulent systems by only observing a subset of state variables is a notoriously challenging topic. In this paper, a hybrid data assimilation method is developed to efficiently estimate the model states. The method exploits cheap stochastic parameterized processes as surrogate models for filtering the observed state variables, significantly reducing the computational cost. It also uses machine learning to build a nonlinear map between observed and unobserved state variables for each spectral mode, which advances the computation of the ensemble members of the unobserved states. The hybrid data assimilation algorithm is applied to a precipitating quasi‐geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two‐level QG model. Utilizing the calibrated models under a moderate cloud fraction, the resulting hybrid data assimilation remains reasonably accurate when applied to other geophysical scenarios with nearly clear skies or relatively heavy rainfall, implying the robustness of the algorithm for extrapolation. Key Points Cheap stochastic parameterized reduced‐order models are designed as efficient forecast models in the ensemble Kalman filter Machine learning is used to map each ensemble member from the observed variables to the corresponding unobserved variables The training residual in the machine learning‐induced nonlinear map is incorporated to advance the quantification of posterior uncertainty