Model order reduction (MOR) can provide low-dimensional numerical models for fast simulation. Unlike intrusive methods, nonintrusive methods are attractive because they can be applied even without access to full order models (FOMs). Since nonintrusive MOR methods strongly rely on snapshots of the FOMs, constructing good snapshot sets becomes crucial. In this work, we propose a novel active-learning-based approach for use in conjunction with nonintrusive MOR methods. It is based on two crucial novelties. First, our approach uses joint space sampling to prepare a data pool of the training data. The training data are selected from the data pool using a greedy strategy supported by an error estimator based on Gaussian process regression. Second, we introduce a case-independent validation strategy based on probably approximately correct learning. While the methods proposed here can be applied to different MOR methods, we test them here with artificial neural networks and operator inference.

Model order reduction (MOR) methods enable the generation of real-time-capable digital twins, with the potential to unlock various novel value streams in industry. While traditional projection-based methods are robust and accurate for linear problems, incorporating machine learning to deal with nonlinearity becomes a new choice for reducing complex problems. These kinds of methods are independent to the numerical solver for the full order model and keep the nonintrusiveness of the whole workflow. Such methods usually consist of two steps. The first step is the dimension reduction by a projection-based method, and the second is the model reconstruction by a neural network (NN). In this work, we apply some modifications for both steps respectively and investigate how they are impacted by testing with three different simulation models. In all cases Proper orthogonal decomposition is used for dimension reduction. For this step, the effects of generating the snapshot database with constant input parameters is compared with time-dependent input parameters. For the model reconstruction step, three types of NN architectures are compared: multilayer perceptron (MLP), explicit Euler NN (EENN), and Runge–Kutta NN (RKNN). The MLPs learn the system state directly, whereas EENNs and RKNNs learn the derivative of system state and predict the new state as a numerical integrator. In the tests, RKNNs show their advantage as the network architecture informed by higher-order numerical strategy.

We present a new iterative model order reduction method for large-scale linear time-invariant dynamical systems, based on a combined singular value decomposition–adaptive-order rational Arnoldi (SVD-AORA) approach. This method is an extension of the SVD-rational Krylov method. It is based on two-sided projections: the SVD side depends on the observability Gramian by the resolution of the Lyapunov equation, and the Krylov side is generated by the adaptive-order rational Arnoldi based on moment matching. The use of the SVD provides stability for the reduced system, and the use of the AORA method provides numerical efficiency and a relative lower computation complexity. The reduced model obtained is asymptotically stable and minimizes the error ($H_{2}$ and $H_{\infty }$) between the original and the reduced system. Two examples are given to study the performance of the proposed approach.

In this paper, we propose a Static Condensation Reduced Basis Element (SCRBE) approach for the Reynolds Lubrication Equation (RLE). The SCRBE method is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components. Here, we extend the methodology to allow for a more general domain decomposition, a typical example being a checkerboard-pattern assembled from similar components. To this end, we extend the formulation and associated a posteriori error bound procedure. Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE. However, the SCRBE approach presented is not limited to bearings and the RLE, but directly extends to other component-based systems. We show numerical results for plain bearings to demonstrate the validity of the proposed approach.

We consider the efficient and reliable solution of linear-quadratic optimal controlproblems governed by parametrized parabolic partial differential equations. To this end,we employ the reduced basis method as a low-dimensional surrogate model to solve theoptimal control problem and develop a posteriori error estimationprocedures that provide rigorous bounds for the error in the optimal control and theassociated cost functional. We show that our approach can be applied to problems involvingcontrol constraints and that, even in the presence of control constraints, the reducedorder optimal control problem and the proposed bounds can be efficiently evaluated in anoffline-online computational procedure. We also propose two greedy sampling procedures toconstruct the reduced basis space. Numerical results are presented to confirm the validityof our approach.

In this paper, we employ the reduced basis method as a surrogate model for the solutionof linear-quadratic optimal control problems governed by parametrized elliptic partialdifferential equations. We present a posteriori error estimation and dualprocedures that provide rigorous bounds for the error in several quantities of interest:the optimal control, the cost functional, and general linear output functionals of thecontrol, state, and adjoint variables. We show that, based on the assumption of affineparameter dependence, the reduced order optimal control problem and the proposed boundscan be efficiently evaluated in an offline-online computational procedure. Numericalresults are presented to confirm the validity of our approach.

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.

In this paper we consider simplifying a model of the nitrogen cycle in Port Phillip Bay, Victoria, Australia. The approach taken is to aggregate state variables that are linearly related using a projection in state space. The technique involved is a modification of proper orthogonal decomposition and was developed so that a resulting simplified model retains an ecological interpretation. It can be applied automatically, and enables insights into the system to be gained that were not obvious beforehand. In the case of the Port Phillip Bay model, we find that the variables representing water and sand are unaffected by the remaining variables, while only variables on the same trophic level can be grouped together. The validity of the aggregation under several nutrient loads is also discussed.

Dynamic simulation of a flexible manipulator performing physical contact (including low-speed impact) tasks with stiff environment is very time consuming because very small integration step sizes have to be used for numerical stability. Existing model order reduction techniques cannot be readily applied due to the nonlinear nature of the contact dynamics. In this paper, a method is introduced to deal with this problem. The method first linearizes the contact force model on the right-hand side of the dynamics equations periodically. It then identifies the linear “stiffness” and “damping” terms from the linearized contact force model and combines them with the existing structural stiffness and damping matrices of the associated multibody system on the left-hand side of the equations. After such a process, the traditional modal analysis and reduction techniques for linear dynamic systems can be applied to reduce the order of the resulting dynamic system. Two numerical examples of flexible manipulators performing a contact task are presented to demonstrate the significant gain in computational efficiency and the improved output results.

In this paper, an efficient macromodel extraction technique for dynamical MEMS gas damping effects is presented. The technique applies an Arnoldi-based model-order-reduction algorithm to generate low-order macromodels from a FEM approximation of the governing equation of the squeeze-film fluidic damping effect, the Reynolds equation. We demonstrate that this approach is more than 100 times efficient than previous approaches, which solve the Reynolds equation using transient finite-element/finite-difference methods. The generated gas-damping macromodels can be easily inserted into system-level modeling packages, such as SPICE, Saber and Simulink, for transient and frequency coupled-domain analysis. We also demonstrated that the simulated results are in good agreement with experimental results for various MEMS devices.