Data-driven Reduced Order Model Research Articles

Data-Driven Reduced-Order Model for Multiphase Flow in Annulus with Air and Cavitation Effect

Summary For multiphase (water-liquid, water-vapor, and air) flow in a sharp-edge converging-diverging annulus, when the downstream/upstream pressure ratio is low, the cavitation phenomenon could occur in the contraction area if the liquid pressure drops below its vaporization pressure. The occurrence of the cavitation will contribute to choked flow, affect the local flow velocity field, and limit the flow rate through the annulus, resulting in damages to the annulus surfaces. In this paper, we use the turbulence model (k-ω) and the multiphase models [volume-of-fluid (VOF) and Eulerian] to simulate the liquid-vapor-gas flow with cavitation in the near-closing annulus. The closing percentage of the annulus varies from 50 to 90% in this numerical investigation. This study compares the VOF and the Eulerian models in liquid-vapor-gas multiphase flow computational fluid dynamics (CFD) simulation results. Furthermore, the effect of gas volume fraction (GVF) at the inlet boundary of the computational domain on the cavitation characteristics is analyzed. The analysis of the simulation results suggests that the choked flow rate and the discharge coefficient gradually decrease with the increase in GVF. Conversely, as the GVF grows within the required range (0–20%), the critical pressure ratio for cavitation increases. In this study, we developed data-driven nondimensional models based on such numerical investigations to predict the choked flow rate and the critical pressure ratio in the multiphase (water-liquid, water-vapor, and air) annulus flow. The findings can provide numerical guidance for the experimental design of annulus multiphase cavitation flow and serve as an engineering tool for flow prediction in offshore oil and gas applications.

Physics-informed data-driven reduced-order models for Dynamic Induction Control

In this work, we find a reduced-order model for the wake of a wind turbine controlled with dynamic induction control. We use a physics-informed dynamic mode decomposition algorithm to reduce the model complexity in a way such that the physics of the wake mixing can be investigated and that the model itself can be easily embedded into control-oriented frameworks.After discussing the advantage of forcing the linear system resulting from the algorithm to be conservative (as a consequence of the periodicity of the pitch excitation) and the choice of observables, we describe a procedure for calculating the energy associated with individual modes. The considered data-set is composed of large eddy simulation (LES) results for a single DTU 10 MW wind turbine in uniform flow. Simulations were performed first with baseline control (for reference) and then with the Pulse and the Helix approaches with constant excitation amplitude and different excitation frequencies. The frequencies and energies associated with the resulting modes are discussed.

Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learning

We propose a unified data-driven reduced order model (ROM) that bridges the performance gap between linear and nonlinear manifold approaches. Deep learning ROM (DL-ROM) using deep-convolutional autoencoders (DC–AE) has been shown to capture nonlinear solution manifolds but fails to perform adequately when linear subspace approaches such as proper orthogonal decomposition (POD) would be optimal. Besides, most DL-ROM models rely on convolutional layers, which might limit its application to only a structured mesh. The proposed framework in this study relies on the combination of an autoencoder (AE) and Barlow Twins (BT) self-supervised learning, where BT maximizes the information content of the embedding with the latent space through a joint embedding architecture. Through a series of benchmark problems of natural convection in porous media, BT–AE performs better than the previous DL-ROM framework by providing comparable results to POD-based approaches for problems where the solution lies within a linear subspace as well as DL-ROM autoencoder-based techniques where the solution lies on a nonlinear manifold; consequently, bridges the gap between linear and nonlinear reduced manifolds. We illustrate that a proficient construction of the latent space is key to achieving these results, enabling us to map these latent spaces using regression models. The proposed framework achieves a relative error of 2% on average and 12% in the worst-case scenario (i.e., the training data is small, but the parameter space is large.). We also show that our framework provides a speed-up of 7 times 10^{6} times, in the best case, and 7 times 10^{3} times on average compared to a finite element solver. Furthermore, this BT–AE framework can operate on unstructured meshes, which provides flexibility in its application to standard numerical solvers, on-site measurements, experimental data, or a combination of these sources.

Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media

In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media. The Koopman operator has received extensive attention as an effective data-driven technique, which can transform the nonlinear dynamical systems into linear systems through acting on observation function spaces. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale media often vary in very different scales. In order to capture multiscale information well, the dimension of collected data may be high. To accurately construct the models of dynamical systems in multiscale media, we use high spatial dimension of observation data. It is challenging to compute the Koopman operators using the very high dimensional data. Thus, the strategy of reduced-order modeling is proposed to treat the difficulty. The proposed reduced-order modeling includes two stages: offline stage and online stage. In offline stage, a block-wise low rank decomposition is used to reduce the spatial dimension of initial snapshot data. For the nonautonomous dynamical systems, real-time observation data may be required to update the Koopman operators. The online reduced-order modeling is proposed to correct the offline reduced-order modeling. Three methods are developed for the online reduced-order modeling: fully online, semi-online and adaptive online. The adaptive online method automatically selects the fully online or semi-online and can achieve a good trade-off between modeling accuracy and efficiency. A few numerical examples are presented to illustrate the performance of the different reduced-order modeling methods.

Seismic wavefield reconstruction based on compressed sensing using data-driven reduced-order model

SUMMARYReconstruction of the distribution of ground motion due to an earthquake is one of the key technologies for the prediction of seismic damage to infrastructure. Particularly, the immediate reconstruction of the spatially continuous wavefield is valuable for decision-making of disaster response decisions in the initial phase. For a fast and accurate reconstruction, utilization of prior information is essential. In fluid mechanics, full-state recovery, which recovers the full state from sparse observation using a data-driven model reduced-order model, is actively used. In this study, the framework developed in the field of fluid mechanics is applied to seismic wavefield reconstruction. A seismic wavefield reconstruction framework based on compressed sensing using the data-driven reduced-order model (ROM) is proposed and its characteristics are investigated through numerical experiments. The data-driven ROM is generated from the data set of the wavefield using the singular value decomposition. The spatially continuous seismic wavefield is reconstructed from the sparse and discrete observation and the data-driven ROM. The observation sites used for reconstruction are effectively selected by the sensor optimization method for linear inverse problems based on a greedy algorithm. The proposed framework was applied to simulation data of theoretical waveform with the subsurface structure of the horizontally stratified three layers. The validity of the proposed method was confirmed by the reconstruction based on the noise-free observation. Since the ROM of the wavefield is used as prior information, the reconstruction error is reduced to an approximately lower error bound of the present framework, even though the number of sensors used for reconstruction is limited and randomly selected. In addition, the reconstruction error obtained by the proposed framework is much smaller than that obtained by the Gaussian process regression. For the numerical experiment with noise-contaminated observation, the reconstructed wavefield is degraded due to the observation noise, but the reconstruction error obtained by the present framework with all available observation sites is close to a lower error bound, even though the reconstructed wavefield using the Gaussian process regression is fully collapsed. Although the reconstruction error is larger than that obtained using all observation sites, the number of observation sites used for reconstruction can be reduced while minimizing the deterioration and scatter of the reconstructed data by combining it with the sensor optimization method. Hence, a better and more stable reconstruction of the wavefield than randomly selected observation sites can be realized, even if the reconstruction is carried out with a smaller number of observations with observation noise, by combining it with the sensor optimization method.

Comparing different nonlinear dimensionality reduction techniques for data-driven unsteady fluid flow modeling

Computational fluid dynamics (CFD) is known for producing high-dimensional spatiotemporal data. Recent advances in machine learning (ML) have introduced a myriad of techniques for extracting physical information from CFD. Identifying an optimal set of coordinates for representing the data in a low-dimensional embedding is a crucial first step toward data-driven reduced-order modeling and other ML tasks. This is usually done via principal component analysis (PCA), which gives an optimal linear approximation. However, fluid flows are often complex and have nonlinear structures, which cannot be discovered or efficiently represented by PCA. Several unsupervised ML algorithms have been developed in other branches of science for nonlinear dimensionality reduction (NDR), but have not been extensively used for fluid flows. Here, four manifold learning and two deep learning (autoencoder)-based NDR methods are investigated and compared to PCA. These are tested on two canonical fluid flow problems (laminar and turbulent) and two biomedical flows in brain aneurysms. The data reconstruction capabilities of these methods are compared, and the challenges are discussed. The temporal vs spatial arrangement of data and its influence on NDR mode extraction is investigated. Finally, the modes are qualitatively compared. The results suggest that using NDR methods would be beneficial for building more efficient reduced-order models of fluid flows. All NDR techniques resulted in smaller reconstruction errors for spatial reduction. Temporal reduction was a harder task; nevertheless, it resulted in physically interpretable modes. Our work is one of the first comprehensive comparisons of various NDR methods in unsteady flows.

Validation and parameterization of a novel physics-constrained neural dynamics model applied to turbulent fluid flow

In fluid physics, data-driven models to enhance or accelerate time to solution are becoming increasingly popular for many application domains, such as alternatives to turbulence closures, system surrogates, or for new physics discovery. In the context of reduced order models of high-dimensional time-dependent fluid systems, machine learning methods grant the benefit of automated learning from data, but the burden of a model lies on its reduced-order representation of both the fluid state and physical dynamics. In this work, we build a physics-constrained, data-driven reduced order model for Navier–Stokes equations to approximate spatiotemporal fluid dynamics in the canonical case of isotropic turbulence in a triply periodic box. The model design choices mimic numerical and physical constraints by, for example, implicitly enforcing the incompressibility constraint and utilizing continuous neural ordinary differential equations for tracking the evolution of the governing differential equation. We demonstrate this technique on a three-dimensional, moderate Reynolds number turbulent fluid flow. In assessing the statistical quality and characteristics of the machine-learned model through rigorous diagnostic tests, we find that our model is capable of reconstructing the dynamics of the flow over large integral timescales, favoring accuracy at the larger length scales. More significantly, comprehensive diagnostics suggest that physically interpretable model parameters, corresponding to the representations of the fluid state and dynamics, have attributable and quantifiable impact on the quality of the model predictions and computational complexity.

Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning

Deep reinforcement learning (RL) is a data-driven method capable of discovering complex control strategies for high-dimensional systems, making it promising for flow control applications. In particular, the present work is motivated by the goal of reducing energy dissipation in turbulent flows, and the example considered is the spatiotemporally chaotic dynamics of the Kuramoto–Sivashinsky equation (KSE). A major challenge associated with RL is that substantial training data must be generated by repeatedly interacting with the target system, making it costly when the system is computationally or experimentally expensive. We mitigate this challenge in a data-driven manner by combining dimensionality reduction via an autoencoder with a neural ODE framework to obtain a low-dimensional dynamical model from just a limited data set. We substitute this data-driven reduced-order model (ROM) in place of the true system during RL training to efficiently estimate the optimal policy, which can then be deployed on the true system. For the KSE actuated with localized forcing (‘jets’) at four locations, we demonstrate that we are able to learn a ROM that accurately captures the actuated dynamics as well as the underlying natural dynamics just from snapshots of the KSE experiencing random actuations. Using this ROM and a control objective of minimizing dissipation and power cost, we extract a control policy from it using deep RL. We show that the ROM-based control strategy translates well to the true KSE and highlight that the RL agent discovers and stabilizes an underlying forced equilibrium solution of the KSE system. We show that this forced equilibrium captured in the ROM and discovered through RL is related to an existing known equilibrium solution of the natural KSE.

Predicting solar wind streams from the inner-heliosphere to Earth via shifted operator inference

Solar wind conditions are predominantly predicted via three-dimensional numerical magnetohydrodynamic (MHD) models. Despite their ability to produce highly accurate predictions, MHD models require computationally intensive high-dimensional simulations. This renders them inadequate for making time-sensitive predictions and for large-ensemble analysis required in uncertainty quantification. This paper presents a new data-driven reduced-order model (ROM) capability for forecasting heliospheric solar wind speeds. Traditional model reduction methods based on Galerkin projection have difficulties with advection-dominated systems—such as solar winds—since they require a large number of basis functions and can become unstable. A core contribution of this work addresses this challenge by extending the non-intrusive operator inference ROM framework to exploit the translational symmetries present in the solar wind caused by the Sun's rotation. The numerical results show that our method can adequately emulate the MHD simulations and is more accurate than a reduced-physics surrogate model, the Heliospheric Upwind Extrapolation model.

Real-Time Prediction of Transarterial Drug Delivery Based on a Deep Convolutional Neural Network

This study develops a data-driven reduced-order model based on a deep convolutional neural network (CNN) for real-time and accurate prediction of the drug trajectory and concentration field in transarterial chemoembolization therapy to assist in directing the drug to the tumor site. The convolutional and deconvoluational layers are used as the encoder and the decoder, respectively. The input of the network model is designed to contain the information of drug injection location and the blood vessel geometry and the output consists of the drug trajectory and the concentration field. We studied drug delivery in two-dimensional straight, bifurcated blood vessels and the human hepatic artery system and showed that the proposed model can quickly and accurately predict the spatial–temporal drug concentration field. For the human hepatic artery system, the most complex case, the average prediction accuracy was 99.9% compared with the CFD prediction. Further, the prediction time for each concentration field was less than 0.07 s, which is four orders faster than the corresponding CFD simulation. The high performance, accuracy and speed of the CNN model shows the potential for effectively assisting physicians in directing chemoembolization drugs to tumor-bearing segments, thus improving its efficacy in real-time.

Data-driven reduced order model and simplicial homology global optimization for reliability analysis and application

A novel framework of reliability analysis was developed in this study to consider the uncertainty of geomaterials and geological conditions by combining the reduced-order model (ROM), reliability analysis, and numerical model. The reliability method was used to determine the reliability index using the simplicial homology global optimization (SHGO) based on the ROM. The developed method was verified and illustrated using three numerical examples and a simple slope. The limit state curve in all three numerical examples was in excellent agreement with the actual curve. The reliability index and failure probability were also in excellent agreement with those of the actual limit state function using the first-order reliability method (FORM) and Monte Carlo simulation, respectively, indicating that the ROM method can present the limit state function well. The results showed that the developed method is feasible and effective for reliability analysis of geotechnical and geological engineering problems with a complex, nonlinear, and implicit limit state function. Furthermore, the developed method is effective, efficient, and accurate for reliability analysis. It provides an excellent way to approximate the limit state function to avoid the time-consuming numerical model in a practical engineering system.

A Gaussian process regression reduced order model for geometrically nonlinear structures

Reduced order models, such as Hollkamp and Gordon’s Implicit Condensation and Expansion (ICE) model, are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. However, a reduced order model (ROM) is typically only valid for one FEM. It does not capture how each ROM coefficient changes due to variations in the FEM (e.g., design parameters or uncertainties), so if the FEM is updated then the ROM needs to be re-computed with a new set of static load–displacement data. This study presents a data-driven reduced order modeling approach that creates a single ROM that incorporates design variations in FEM. The proposed method applies Gaussian Process Regression (GPR) to the ICE approach, making each coefficient in an ICE ROM a regression model with respect to a collection of FEMs with varying material properties or geometric parameters. Once the GPR ROM has been identified, one can immediately produce an ICE ROM for a set of FEM parameters without a need to solve any static load–displacement cases on the full FEM. This dramatically enhances the computational efficiency and could be helpful when model uncertainty needs to be considered or when seeking to update a model to correlate with measurements. Additionally, the coefficients of a ROM can often change considerably if the scale on the load–displacement data changes, so it can be difficult to know whether the scaling that was chosen has really identified an accurate ROM. The proposed GPR ROM estimates the mean ROM coefficients for a range of load scaling as well as the uncertainty on each ROM coefficient with respect to the load level. This can be used to gauge the success of the ROM identification and to eliminate ROM coefficients that are unimportant and hence highly variable. The proposed GPR ROM approach is evaluated by applying it to flat and curved beam structures, revealing that the advantages outlined above can be realized with a relatively modest increase in cost relative to a traditional ICE ROM.

A Data-Driven Reduced-Order Model for Estimating the Stimulated Reservoir Volume (SRV)

The main goal of hydraulic fracturing stimulation in unconventional and tight reservoirs is to maximize hydrocarbon production by creating an efficient stimulated reservoir volume (SRV) around the horizontal wells. To zreach this goal, a physics-based model is typically used to design and optimize the hydraulic fracturing process before executing the job. However, two critical issues make this approach insufficient for achieving the mentioned goal. First, the physics-based models are based on several simplified assumptions and do not correctly represent the physics of unconventional reservoirs; hence, they often fail to match the observed SRVs in the field. Second, the success of the executed stimulation job is evaluated after it is completed in the field, leaving no room to modify some parameters such as proppant concentration in the middle of the job. To this end, this paper proposes data-driven and global sensitivity approaches to address these two issues. It introduces a novel workflow for estimating SRV in near real-time using some hydraulic fracturing parameters that can be inferred before or during the stimulation process. It also utilizes a robust global sensitivity framework known as the Sobol Method to rank the input parameters and create a reduced-order (mathematically simple) model for near real-time estimation of SRV (referred to as DSRV). The proposed framework in this paper has two main advantages and novelties. First, it is based on a pure data-based approach, with no simplified assumptions due to the use of a simulator for generating the training and test dataset, which is often the case in similar studies. Second, it treats SRV generation as a rock mechanics problem (rather than a reservoir engineering problem with fixed fracture lengths), accounting for changes in hydraulic fracture topology and SRV changes with time. A dataset from the Marcellus Shale Energy and Environment Laboratory (MSEEL) project is used. The model’s input parameters include stimulation variables of 58 stages of two wells. These parameters are stage number, step, pump rate and duration, proppant concentration and mass, and treating pressure. The model output consists of the corresponding microseismic (MS) cloud size at each step (i.e., time window) during the job. Based on the model, guidelines are provided to help operators design more efficient fracturing jobs for maximum recovery and to monitor the effectiveness of the hydraulic fracturing process. A few future improvements to this approach are also provided.

Bayesian operator inference for data-driven reduced-order modeling

This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inverse problem with Gaussian prior and likelihood. The resulting posterior distribution characterizes the operators defining the reduced-order model, hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The statistical moments of these predictions are estimated via a Monte Carlo sampling of the posterior distribution. Since the reduced models are fast to solve, this sampling is computationally efficient. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the regularization term that is present in the deterministic operator inference problem, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated on two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.

On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays

Data-driven reduced order models (ROMs) recently emerged as powerful tool for the solution of inverse scattering problems. The main drawback of this approach is that it was limited to measurement arrays with reciprocally collocated transmitters and receivers, that is, square symmetric matrix (data) transfer functions. To relax this limitation, we use our previous work Druskin et al (2021 Inverse Problems 37 075003), where the ROMs were combined with the Lippmann–Schwinger integral equation to produce a direct nonlinear inversion method. In this work we extend this approach to more general transfer functions, including those that are non-symmetric, e.g., obtained by adding only receivers or sources. The ROM is constructed based on the symmetric subset of the data and is used to construct all internal solutions. Remaining receivers are then used directly in the Lippmann–Schwinger equation. We demonstrate the new approach on a number of 1D and 2D examples with non-reciprocal arrays, including a single input/multiple outputs inverse problem, where the data is given by just a single-row matrix transfer function. This allows us to approach the flexibility of the Born approximation in terms of acceptable measurement arrays; at the same time significantly improving the quality of the inversion compared to the latter for strongly nonlinear scattering effects.

Extending the Capabilities of Data-Driven Reduced-Order Models to Make Predictions for Unseen Scenarios: Applied to Flow Around Buildings

We present a data-driven or non-intrusive reduced-order model (NIROM) which is capable of making predictions for a significantly larger domain than the one used to generate the snapshots or training data. This development relies on the combination of a novel way of sampling the training data (which frees the NIROM from its dependency on the original problem domain) and a domain decomposition approach (which partitions unseen geometries in a manner consistent with the sub-sampling approach). The method extends current capabilities of reduced-order models to generalise, i.e., to make predictions for unseen scenarios. The method is applied to a 2D test case which simulates the chaotic time-dependent flow of air past buildings at a moderate Reynolds number using a computational fluid dynamics (CFD) code. The procedure for 3D problems is similar, however, a 2D test case is considered sufficient here, as a proof-of-concept. The reduced-order model consists of a sampling technique to obtain the snapshots; a convolutional autoencoder for dimensionality reduction; an adversarial network for prediction; all set within a domain decomposition framework. The autoencoder is chosen for dimensionality reduction as it has been demonstrated in the literature that these networks can compress information more efficiently than traditional (linear) approaches based on singular value decomposition. In order to keep the predictions realistic, properties of adversarial networks are exploited. To demonstrate its ability to generalise, once trained, the method is applied to a larger domain which has a different arrangement of buildings. Statistical properties of the flows from the reduced-order model are compared with those from the CFD model in order to establish how realistic the predictions are.

Data-driven reduced-order modeling of spatiotemporal chaos with neural ordinary differential equations.

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced-order modeling method that capitalizes on this fact by finding a coordinate representation for this manifold and then a system of ordinary differential equations (ODEs) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder-a neural network (NN) that reduces and then expands dimension. Then, the ODE, in these coordinates, is determined by a NN using the neural ODE framework. Both of these steps only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. Time-derivative information is not needed. We apply this framework to the Kuramoto-Sivashinsky equation for domain sizes that exhibit chaotic dynamics with again estimated manifold dimensions ranging from 8 to 28. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ∼ 0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction and find a "sweet spot" in terms of performance vs dimension.

Data-driven reduced order modeling for parametrized time-dependent flow problems

This paper proposes a nonintrusive reduced basis (RB) method based on dynamic mode decomposition (DMD) for parameterized time-dependent flows. In the offline stage, the reduced basis functions are extracted by a two-step proper orthogonal decomposition algorithm. Then, a novel hybrid DMD regression model that combines windowed DMD and optimized DMD is introduced for the temporal evolution of the RB coefficients. To improve the stability of this method for complex nonlinear problems, we introduce a threshold value to modify the DMD eigenvalues and eigenvectors. Moreover, the interpolation of the coefficients in parameter space is conducted by a feedforward neural network or random forest algorithm. The prediction of the RB solution at a new time/parameter value can be recovered at a low computational cost in the online stage, which is completely decoupled from the high-fidelity dimension. We demonstrate the performance of the proposed model with two cases: (i) laminar flow past a two-dimensional cylinder and (ii) turbulent flow around a three-dimensional SD7003 airfoil. The results show reasonable efficiency and robustness of this novel reduced-order model.

An Adaptive Data-Driven Reduced Order Model Based on Higher Order Dynamic Mode Decomposition

An Adaptive Data-Driven Reduced Order Model Based on Higher Order Dynamic Mode Decomposition

Towards robust data-driven reduced-order modelling for turbulent flows: application to vortex-induced vibrations

This work presents a robust method that minimises the impact of user-selected parameter on the identification of generic models to study the coherent dynamics in turbulent flows. The objective is to gain insight into the flow dynamics from a data-driven reduced order model (ROM) that is developed from measurement data of the respective flow. For an efficient separation of the coherent dynamics, spectral proper orthogonal decomposition (SPOD) is used, projecting the flow field onto a low-dimensional subspace, so that the dominating dynamics can be represented with a minimal number of modes. A function library is defined using polynomial combinations of the temporal modal coefficients to describe the flow dynamics with a system of nonlinear ordinary differential equations. The most important library functions are identified in a two-stage cross-validation procedure (conservative and restrictive sparsification) and combined in the final model. In the first stage, the process uses a simple approximation of the derivative to match the model with the data. This stage delivers a reduced set of possible library function candidates for the model. In the second, more complex stage, the model of the entire flow is integrated over a short time and compared with the progression of the measured data. This restrictive stage allows a robust identification of nonlinearities and modal interactions in the data and their representation in the model. The method is demonstrated using data from particle image velocimetry (PIV) measurements of a circular cylinder undergoing vortex-induced vibration (VIV) at mathrm{Re}=4000. It delivers a reduced order model that reproduces the average dynamics of the flow and reveals the interaction of coexisting flow dynamics by the model structure.