We explore the possibility of combining a knowledge-based reduced order model (ROM) with a reservoir computing approach to learn and predict the dynamics of chaotic systems. The ROM is based on proper orthogonal decomposition (POD) with Galerkin projection to capture the essential dynamics of the chaotic system while the reservoir computing approach used is based on echo state networks (ESNs). Two different hybrid approaches are explored: one where the ESN corrects the modal coefficients of the ROM (hybrid-ESN-A) and one where the ESN uses and corrects the ROM prediction in full state space (hybrid-ESN-B). These approaches are applied on two chaotic systems: the Charney–DeVore system and the Kuramoto–Sivashinsky equation and are compared to the ROM obtained using POD/Galerkin projection and to the data-only approach based uniquely on the ESN. The hybrid-ESN-B approach is seen to provide the best prediction accuracy, outperforming the other hybrid approach, the POD/Galerkin projection ROM, and the data-only ESN, especially when using ESNs with a small number of neurons. In addition, the influence of the accuracy of the ROM on the overall prediction accuracy of the hybrid-ESN-B is assessed rigorously by considering ROMs composed of different numbers of POD modes. Further analysis on how hybrid-ESN-B blends the prediction from the ROM and the ESN to predict the evolution of the system is also provided.

Multi-dimensional aerodynamic database technology is widely used, but its model often has the curse of dimensionality. In order to solve this problem, we need projection to reduce the dimension. In addition, due to the lack of traditional method, we have improved the traditional flow field reconstruction method based on artificial neural networks, and we proposed an array neural network method.

In this paper, a set of flow field data for the target problem of the fixed Mach number is obtained by the existing CFD method. Then we arrange all the sampled flow field data into a matrix and use proper orthogonal decomposition (POD) to reduce the dimension, whose size is determined by the first few modals of energy. Therefore, significantly reduced data are obtained. Then we use an arrayed neural network to map the flow field data of simplified target problem and the flow field characteristics. Finally, the unknown flow field data can be effectively predicted through the flow field characteristic and the trained array neural network.

At the end of this paper, the effectiveness of the method is verified by airfoil flow fields. The calculation results show that the array neural network can reconstruct the flow field of the target problem more accurately than the traditional method, and its convergence speed is significantly faster. In addition, for the case of high angle flow field, the array neural network also performs well. There are no obvious jumps, and huge errors are found in results. In general, the proposed method is better than the traditional method.

A numerical time-stepping algorithm for differential or partial differential equations is proposed that adaptively modifies the dimensionality of the underlying modal basis expansion. Specifically, the method takes advantage of any underlying low-dimensional manifolds or subspaces in the system by using dimensionality-reduction techniques, such as the proper orthogonal decomposition, in order to adaptively represent the solution in the optimal basis modes. The method can provide significant computational savings for systems where low-dimensional manifolds are present since the reduction can lower the dimensionality of the underlying high-dimensional system by orders of magnitude. A comparison of the computational efficiency and error for this method are given showing the algorithm to be potentially of great value for high-dimensional dynamical systems simulations, especially where slow-manifold dynamics are known to arise. The method is envisioned to automatically take advantage of any potential computational saving associated with dimensionality-reduction, much as adaptive time-steppers automatically take advantage of large step sizes whenever possible.

In this paper, a surrogate model approach for non-linear aerodynamics is presented in order to reduce the computational effort of coupled aeroelastic analyses. The usability of the approach is demonstrated in static as well as transient aeroelastic analyses of the HIRENASD wing-fuselage configuration. Furthermore, it is shown that the surrogate model approach is able to cover variations of flow conditions at a fixed Mach and Reynolds number.

Galerkin reduced-order models for the semi-discrete wave equation, that preserve thesecond-order structure, are studied. Error bounds for the full state variables are derivedin the continuous setting (when the whole trajectory is known) and in the discrete settingwhen the Newmark average-acceleration scheme is used on the second-order semi-discreteequation. When the approximating subspace is constructed using the proper orthogonaldecomposition, the error estimates are proportional to the sums of the neglected singularvalues. Numerical experiments illustrate the theoretical results.

We address the issue of parameter variations in POD approximations of time-dependentproblems, without any specific restriction on the form of parameter dependence.Considering a parabolic model problem, we propose a POD construction strategy allowing usto obtain some a priori error estimates controlled by the POD remainder –in the construction procedure – and some parameter-wise interpolation errors for the modelsolutions. We provide a thorough numerical assessment of this strategy with theFitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in twoparameter estimation applications, the first in a variational estimation framework withthe FitzHugh − Nagumo model, and the second with a beating heart mechanical model forwhich we employ a sequential estimation method to characterize model parameters using realimage data in a clinical case.

Flow around a cavity is characterized by a self-sustained mechanism in which the shear layer impinges on the downstream edge of the cavity resulting in a feedback mechanism. Direct Numerical Simulations of the flow at low Reynolds number has been carried out to get pressure and velocity fluctuations, for the case of un-actuated and multi frequency actuation. A Reduced Order Model for the isentropic compressible equations based on the method of Proper Orthogonal Decomposition has been constructed. The model has been extended to include the effect of control. The Reduced Order dynamical system shows a divergence in time integration. A method of calibration based on the minimization of a linear functional of error, to the sensitivity of the modes, is proposed. The calibrated low order model is used to design a feedback control of cavity flows based on an observer design. For the experimental implementation of the controller, a state estimate based on the observed pressure measurements is obtained through a linear stochastic estimation. Finally the obtained control is introduced into the Direct Numerical Simulation to obtain a decrease in spectra of the cavity acoustic mode.

Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD–Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD–Greedy algorithm.

We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

Mathematical models for option pricing often result in partial differential equations.Recent enhancements are models driven by Lévy processes, which lead to a partialdifferential equation with an additional integral term. In the context of modelcalibration, these partial integro differential equations need to be solved quitefrequently. To reduce the computational cost the implementation of a reduced order modelhas shown to be very successful numerically. In this paper we give a priorierror estimates for the use of the proper orthogonal decomposition technique inthe context of option pricing models.

An optimal control problem governed by a bilinear elliptic equation is considered. Thisproblem is solved by the sequential quadratic programming (SQP) method in aninfinite-dimensional framework. In each level of this iterative method the solution oflinear-quadratic subproblem is computed by a Galerkin projection using proper orthogonaldecomposition (POD). Thus, an approximate (inexact) solution of the subproblem isdetermined. Based on a POD a-posteriori error estimator developed byTröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115]the difference of the suboptimal to the (unknown) optimal solution of the linear-quadraticsubproblem is estimated. Hence, the inexactness of the discrete solution is controlled insuch a way that locally superlinear or even quadratic rate of convergence of the SQP isensured. Numerical examples illustrate the efficiency for the proposed approach.

The construction of reduced order models for dynamical systems usingproper orthogonal decomposition (POD) is based on the informationcontained in so-called snapshots. These provide the spatialdistribution of the dynamical system at discrete time instances.This work is devoted to optimizing the choice of these timeinstances in such a manner that the error between the POD-solutionand the trajectory of the dynamical system is minimized. First andsecond order optimality systems are given. Numerical examplesillustrate that the proposed criterion is sensitive with respect tothe choice of the time instances and further they demonstrate thefeasibility of the method in determining optimal snapshot locationsfor concrete diffusion equations.

Proper orthogonal decomposition (POD) is apowerful technique for model reduction of non-linear systems. Itis based on a Galerkin type discretization with basis elementscreated from the dynamical system itself. In the context ofoptimal control this approach may suffer from the fact that thebasis elements are computed from a reference trajectory containingfeatures which are quite different from those of the optimallycontrolled trajectory. A method is proposed which avoids thisproblem of unmodelled dynamics in the proper orthogonaldecomposition approach to optimal control. It is referred to asoptimality system proper orthogonal decomposition (OS-POD).