Projection-based Reduced Order Models Research Articles

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Projection-based reduced order modeling and data-driven artificial viscosity closures for incompressible fluid flows

Projection-based reduced order models rely on offline–online model decomposition, where the data-based energetic spatial basis is used in the expensive offline stage to obtain equations of reduced states that evolve in time during the inexpensive online stage. The online stage requires a solution method for the dynamic evolution of the coupled system of pressure and velocity states for incompressible fluid flows. The first contribution of this article is to demonstrate the applicability of the incremental pressure correction scheme for the dynamic evolution of pressure and velocity states. The evolution of a large number of these reduced states in the online stage can be expensive. In contrast, the accuracy significantly decreases if only a few reduced states are considered while not accounting for the interactions between unresolved and resolved states. The second contribution of this article is to compare three closure model forms based on global, modal and tensor artificial viscosity approximation to account for these interactions. The unknown model parameters are determined using two calibration techniques: least squares minimization of error in energy approximation and closure term approximation. This article demonstrates that an appropriate selection of solution methods and data-driven artificial viscosity closure models is essential for consistently accurate dynamics forecasting of incompressible fluid flows.

Model order reduction with novel discrete empirical interpolation methods in space–time

This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space–time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations.

Projection-based reduced order modeling of an iterative scheme for linear thermo-poroelasticity

This paper explores an iterative approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot’s poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.

Nonlinear model order reduction for problems with microstructure using mesh informed neural networks

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment.

Explicit synchronous partitioned scheme for coupled reduced order models based on composite reduced bases

This paper formulates, analyzes and demonstrates numerically a method for the explicit partitioned solution of coupled interface problems involving combinations of projection-based reduced order models (ROM) and/or full order models (FOMs). The method builds on the partitioned scheme developed in Peterson et al. (2019), which starts from a well-posed formulation of the coupled interface problem and uses its dual Schur complement to obtain an approximation of the interface flux. Explicit time integration of this problem decouples its subdomain equations and enables their independent solution on each subdomain. Extension of this partitioned scheme to coupled ROM–ROM or ROM–FOM problems requires formulations with non-singular Schur complements. To obtain these problems, we project a well-posed coupled FOM–FOM problem onto a composite reduced basis comprising separate sets of basis vectors for the interface and interior variables, and use the interface reduced basis as a Lagrange multiplier. Our analysis confirms that the resulting coupled ROM–ROM and ROM–FOM problems have provably non-singular Schur complements, independent of the mesh size and the reduced basis size. In the ROM–FOM case, analysis shows that one can also use the interface FOM space as a Lagrange multiplier. We illustrate the theoretical and computational properties of the partitioned scheme through reproductive and predictive tests for a model advection–diffusion transmission problem.

Hyper-reduction for Petrov–Galerkin reduced order models

Projection-based Reduced Order Models are based on the idea of minimizing the discrete residual of a “full order model” (FOM) while at the same time constraining the unknowns to live in a space of reduced dimension. For problems with symmetric positive definite (SPD) Jacobians, this minimization can be achieved optimally by projecting the full order residual onto the approximation basis (Galerkin Projection). This approach is sub-optimal for problems with non-SPD Jacobians since it only guarantees that the projection of the residual onto the chosen basis is minimized and not the residual itself. One possible alternative in such cases is to directly minimize the 2-norm of the residual. This minimization can be achieved either by using QR factorization to solve the resulting least-squares problem or by employing the method of the normal equations (LSPG) to the same end. The first approach involves constructing and factorizing a rectangular tall and skinny matrix of size proportional to the number of unknowns of the FOM. The LSPG method avoids the use of the large matrix by directly constructing the product by its transpose. Unfortunately, constructing such a product element by element is not feasible and requires the use of a complementary mesh, which adds an extra layer of complexity to the hyper-reduction process when performing mesh sampling. The main idea of this work is to propose an alternative technique based on the idea of Petrov–Galerkin minimization. Essentially, we choose a left basis so that a least-squares minimization procedure can be carried out on a reduced problem while guaranteeing that the discrete full order residual is minimized. The resulting procedure is applicable to problems with both SPD and non-SPD Jacobians. Additionally, the resulting minimization problem can be assembled element by element, avoiding the use of the complementary mesh and simplifying implementation in the context of finite elements. The resulting technique is amenable to hyper-reduction by the use of the Empirical Cubature Method and can be readily applied in the context of nonlinear reduction procedures.

Scalable Projection-Based Reduced-Order Models for Large Multiscale Fluid Systems

Although projection-based reduced-order models (PROMs) have existed for decades, they have rarely been applied to large, nonlinear, multiscale, and multi-physics systems due to the complexity of effectively implementing such methods. Advances in hyper-reduction have enabled the scalable computation of PROMs for general nonlinear dynamical systems. Further, the recent model-form-preserving least squares with variable transformation method has proven capable of generating stable PROMs for extremely stiff multiphysics problems. In this work, we formulate a PROM framework combining these methodologies and demonstrate that robust, accurate, and cost-effective PROMs can be realized for complex nonreacting and reacting compressible flows. Along with an open-source toolchain for hyper-reduction sample mesh generation from extremely large data sets, this represents an end-to-end effort to assess the applicability of PROMs to large-scale, multiphysics problems of engineering interest. We examine practical considerations for implementing hyper-reduction methods and their effect on memory consumption, load balancing, and interprocessor communications. These considerations produce accurate PROMs that are three to four orders of magnitude more computationally efficient than the full-order model in recreating transonic flow over a cavity and reacting flow in a rocket combustor. Guidelines for data preparation, sample mesh construction, and online PROM solution which promote robust simulations are also provided.

A multifidelity deep operator network approach to closure for multiscale systems

Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed “in-the-loop” training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier–Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: Application to shape optimization

We present a numerical method to efficiently solve optimization problems governed by large-scale nonlinear systems of equations, including discretized partial differential equations, using projection-based reduced-order models accelerated with hyperreduction (empirical quadrature) and embedded in a trust-region framework that guarantees global convergence. The proposed framework constructs a hyperreduced model on-the-fly during the solution of the optimization problem, which completely avoids an offline training phase. This ensures all snapshot information is collected along the optimization trajectory, which avoids wasting samples in remote regions of the parameters space that are never visited, and inherently avoids the curse of dimensionality of sampling in a high-dimensional parameter space. At each iteration of the proposed algorithm, a reduced basis and empirical quadrature weights are constructed precisely to ensure the global convergence criteria of the trust-region method are satisfied, ensuring global convergence to a local minimum of the original (unreduced) problem. Numerical experiments are performed on two fluid shape optimization problems to verify the global convergence of the method and demonstrate its computational efficiency; speedups over 18× (accounting for all computational cost, even cost that is traditionally considered “offline” such as snapshot collection and data compression) relative to standard optimization approaches that do not leverage model reduction are shown.

Evaluation of dual-weighted residual and machine learning error estimation for projection-based reduced-order models of steady partial differential equations

Projection-based reduced-order models (pROMs) show great promise as a means to accelerate many-query applications such as forward error propagation, solving inverse problems, and design optimization. In order to deploy pROMs in the context of high-consequence decision making, accurate error estimates are required to determine the region(s) of applicability in the parameter space. The following paper considers the dual-weighted residual (DWR) error estimate for pROMs and compares it to another promising pROM error estimate, machine learned error models (MLEM). In this paper, we show how DWR can be applied to ROMs and then evaluate DWR on two partial differential equations (PDEs): a two-dimensional linear convection–reaction–diffusion equation, and a three-dimensional static hyper-elastic beam. It is shown that DWR is able to estimate errors for pROMs extrapolating outside of their training set while MLEM is best suited for pROMs used to interpolate within the pROM training set.

Robust and Scalable Projection-based Reduced-order Models for Simulations of Reacting Flows

Robust and Scalable Projection-based Reduced-order Models for Simulations of Reacting Flows

Linear Reduced-Order Model Predictive Control

Model predictive controllers use dynamics models to solve <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constrained</i> optimal control problems. However, computational requirements for real-time control have limited their use to systems with low-dimensional models. Nevertheless, high-dimensional models arise in many settings, for example, discretization methods for generating finite-dimensional approximations to partial differential equations can result in models with thousands to millions of dimensions. In such cases, reduced-order models (ROMs) can significantly reduce computational requirements, but model approximation error must be considered to guarantee controller performance. In this article, a reduced-order model predictive control (ROMPC) scheme is proposed to solve robust, output feedback, constrained optimal control problems for high-dimensional linear systems. Computational efficiency is obtained by using projection-based ROMs, and guarantees on robust constraint satisfaction and stability are provided. The performance of the approach is demonstrated in simulation for several examples, including an aircraft control problem leveraging an inviscid computational fluid dynamics model with dimension 998 930.

Model order reduction for compressible flows solved using the discontinuous Galerkin methods

Projection-based reduced order models (ROM) based on the weak form and the strong form of the discontinuous Galerkin (DG) method are proposed and compared for shock-dominated problems. The incorporation of dissipation components of DG in a consistent manner, including the upwinding flux and the localized artificial viscosity model, is employed to enhance stability of the ROM. To ensure efficiency, the discrete empirical interpolation method (DEIM) is adopted to enable hyper-reduction, for which the upwinding flux is decomposed into the central part and the dissipation part. The maximum local wave speed in the upwinding dissipation part is compressed and approximated using the DEIM approach, and the same strategy is applied to the artificial viscosity. Energy stability is proved with the strong-form-based ROM prior to hyper-reduction for the one-dimensional scalar cases. Eigenvalue spectrum is analyzed to verify and compare the stability properties of the two proposed ROMs. Several benchmark cases are conducted to test the performance of the proposed models. Results show that stable computations with reasonable acceleration for shock-dominated cases can be achieved with the ROM built on the strong form.

Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction

A quadratic approximation manifold is presented for performing nonlinear, projection-based, model order reduction (PMOR). It constitutes a departure from the traditional affine subspace approximation that is aimed at mitigating the Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated transport problems. It builds on the data-driven approach underlying the traditional construction of projection-based reduced-order models (PROMs); is application-independent; is linearization-free; and therefore is robust for highly nonlinear problems. Most importantly, this approximation leads to quadratic PROMs that deliver the same accuracy as their traditional counterparts using however a much smaller dimension – typically, n2∼n1, where n2 and n1 denote the dimensions of the quadratic and traditional PROMs, respectively. The computational advantages of the proposed high-order approach to nonlinear PMOR over the traditional approach are highlighted for the detached-eddy simulation-based prediction of the Ahmed body turbulent wake flow, which is a popular CFD benchmark problem in the automotive industry. For a fixed accuracy level, these advantages include: a reduction of the total offline computational cost by a factor greater than five; a reduction of its online wall clock time by a factor greater than 32; and a reduction of the wall clock time of the underlying high-dimensional model by a factor greater than two orders of magnitude.

Projection-based reduced order models for parameterized nonlinear time-dependent problems arising in cardiac mechanics

&lt;abstract&gt;&lt;p&gt;The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.&lt;/p&gt;&lt;/abstract&gt;

Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to 10^3{times } in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

Data-driven closure of projection-based reduced order models for unsteady compressible flows

A data-driven closure modeling based on proper orthogonal decomposition (POD) temporal modes is used to obtain stable and accurate reduced order models (ROMs) of unsteady compressible flows. Model reduction is obtained via Galerkin and Petrov–Galerkin projection of the non-conservative compressible Navier–Stokes equations. The latter approach is implemented using the least-squares Petrov–Galerkin (LSPG) technique and the present methodology allows pre-computation (i.e., not requiring hyper-reduction) of both Galerkin and LSPG coefficients. Closure is performed by adding linear and non-linear coefficients to the original ROMs and minimizing the error with respect to the POD temporal modes. In order to further reduce the computational cost of the ROMs, an accelerated greedy missing point estimation (MPE) hyper-reduction method is employed. A canonical compressible cylinder flow is first analyzed and serves as a benchmark. The second problem studied consists of the turbulent flow over a plunging airfoil undergoing deep dynamic stall. For both cases, regularization is required and an iterative Tikhonov methodology is proposed. For the first case, linear and non-linear closure coefficients are both low in intrusiveness, capable of providing results in excellent agreement with the full order model. Regularization of calibrated models is also straightforward for this case. On the other hand, the dynamic stall flow is significantly more challenging, specially when only linear coefficients are used. Results show that non-linear calibration coefficients outperform their linear counterparts when a POD basis with fewer modes is used in the reconstruction. However, determining a correct level of regularization is more complicated with non-linear coefficients. Hyper-reduced models show good results when combined with non-linear calibration and an appropriate sized POD basis.

Windowed space–time least-squares Petrov–Galerkin model order reduction for nonlinear dynamical systems

Projection-based reduced-order models (PROMs) restrict the full-order model (FOM) to a low-dimensional subspace. Space–time PROMs in particular restrict the FOM to a temporally space–time trial subspace, and compute approximate solutions through a residual orthogonalization or minimization process. One such technique of interest is the space–time least-squares Petrov–Galerkin method (ST-LSPG), which reduces both the spatial and temporal dimensions. However, ST-LSPG is computationally expensive, because it requires solving a dense space–time system with a space–time basis that is calculated over the entire global time domain, which can be unfeasible for large-scale applications. To address these challenges, this paper presents the windowed space–time least-squares Petrov–Galerkin method (WST-LSPG) for model reduction of nonlinear parameterized dynamical systems. The proposed WST-LSPG approach addresses the aforementioned challenges by (1) dividing the time simulation into windows, (2) devising a unique low-dimensional high-fidelity space–time trial subspace for each window, and (3) minimizing the discrete-in-time space–time residual of the dynamical system over each window. In this formulation, the problem confines coupling within each window, but solves space–time residual minimization problems sequentially across the windows. WST-LSPG is equipped with hyper-reduction techniques to further reduce the computational cost. Numerical experiments for the one-dimensional Burgers’ equation and the two-dimensional compressible Navier–Stokes equations for flow over a NACA 0012 airfoil demonstrate that WST-LSPG is superior to ST-LSPG in terms of accuracy and computational gain by as much as 99%.

Active Manifold and Model-Order Reduction to Accelerate Multidisciplinary Analysis and Optimization

A computational framework is proposed for efficiently solving multidisciplinary analysis and optimization (MDAO) problems in a relatively high-dimensional design parameter space. It relies on projection-based reduced-order models (PROMs) and a new concept of an active manifold (AM) to mitigate the curse of dimensionality during the training of the PROMs. The AM is discovered using a deep convolutional autoencoder for dimensionality reduction: it is proposed as a superior alternative to the concept of active subspace whose capabilities are limited by the associated affine approximation. The computational framework also blends the concept of a global PROM as a surrogate model of a nonlinear partial differential equation (PDE)-based behavior function with that of a database of local, linear PROMs for approximating a linear PDE-based behavior function. It is demonstrated for the solution of a flexible instance of NASA’s Common Research Model for transport aircraft, where the objective function pertains to aerodynamics, the constraint relates to flutter, and the design space contains 58 structural and shape parameters. The obtained results illustrate the potential of the AM concept for mitigating the aforementioned curse of dimensionality. They also demonstrate the feasibility of the proposed computational framework for realistic MDAO problems and its ability to reduce solution time.