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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

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

Abstract 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 $$10^3{\times }$$ 10 3 × 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).

42 ENGINEERING↗
DOE OSTI2021

Model Reduction by Generalized Falk Method for Efficient Field-Circuit Simulations

The Generalized Falk Method (GFM) for coordinate transformation, together with two model-reduction strategies based on this method, are presented for efficient coupled field-circuit simulations. Each model-reduction strategy is based on a decision to retain specific linearly-independent vectors, called trial vectors, to construct a vector basis for coordinate transformation. The reduced-order models are guaranteed to be stable and passive since the GFM is a congruence transformation of originally symmetric positive definite systems. We also show that, unlike the Pad´e-via-Lanczos (PVL) method, the GFM does not generate unstable positive poles while reducing the order of circuit problems. Further, the proposed GFM is also faster when compared to methods of the type Lanczos (or Krylov) that are already widely used in circuit simulations for electrothermal and electromagnetic problems. The concept of response participation factors is introduced for the selection of the trial vectors in the proposed model-reduction methods. Further, we present methods to develop simple equivalent circuit networks for the field component of the overall field-circuit system. The implementation of these equivalent circuit networks in circuit simulators is discussed. With the proposed model-reduction strategies, significant improvement on the efficiency of the generalized Falk method is illustrated for coupled field-circuit problems.

42 ENGINEERING↗
NASA NTRSDate not supplied

Aeroservoelastic Control Law Development for the Integrated Adaptive Wing Technology Maturation Wind-Tunnel Test

The NASA Advanced Air Transport Technology (AATT) Project seeks to “explore and develop technologies and concepts for improved energy efficiency and environmental compatibility for fixed wing subsonic transports” [1]. The IAWTM project is a subproject within AATT and is a joint Boeing/NASA effort to, in part, demonstrate the active control systems necessary to employ high-aspect ratio wings for improved aerodynamic efficiency. The aircraft configuration under study for the IAWTM project is a generic jet transport based on the NASA Common Research Model (CRM) whose wing aspect ratio has been increased to 13.5 (as opposed to nominal CRM aspect ratio of 9) and whose development is given in Ref. [2]. A semi-span model of the high-aspect-ratio CRM with 10 trailing edge control surfaces is currently being fabricated and will be delivered to the NASA Langley Transonic Dynamics Tunnel (TDT). There, several tests will be conducted to demonstrate maneuver load alleviation (MLA), gust load alleviation (GLA), active flutter suppression (AFS), and drag reduction, all at transonic conditions. A layout of the wind tunnel model is shown in Fig. 1. The wind-tunnel model is a spar/pod wing design with a relatively rigid fuselage attached to a mounting structure (seen in Fig. 1), which in turn is attached to a load balance behind the tunnel wall. Shims exist that can be attached to force the load to go through the fuselage rather than through the balance so as not to damage the balance in the event of flutter. The spar/pod design is similar to that used in the Boeing truss-braced wing aeroelastic model tested in the TDT in 2013 [3]. There is a central spar running the span of the wing to which the pods are attached that form the aerodynamic shape of the wing. The pods are constructed of ribs for attachment to the spar and skins to create the outer mold line. There are ten active control surfaces that will be used for the test objectives discussed above. Three of these (two outboard ailerons and one inboard aileron) are high-speed electrohydraulic actuators, and the remaining seven are low-speed mini-plane flaps, which have slower servo motors that will be used primarily for drag and maneuver load reductions. Control laws must be developed for each test (MLA, GLA, AFS, and drag reduction) and must be able to work in tandem when two or more control strategies are being tested simultaneously (e.g., when control laws to mitigate maneuver loads and gust loads operate simultaneously). The development of these control laws requires accurate mathematical representations of the aerodynamics, structural dynamics, and control surface dynamics of the system. Taken together, this mathematical representation—which is often in state-space form— is the ASE model. Three low-order ASE models of differing fidelities were developed for the IAWTM test and will be presented here. The differences in fidelity appeared in the aerodynamic modeling—one model assumed purely linear aerodynamics, while the other two models used different techniques to linearize the aerodynamics about a nonlinear mean. Furthermore, simple observer-based full-state feedback controllers for each test (MLA, GLA, AFS, and drag reduction) were designed from these low-order ASE models and implemented in a higher-order, nonlinear computational ASE simulation to assess the effectiveness of each controller. Observers were designed from each of the three lower-order ASE models, and each (linear) observer was then used as an estimate of the nonlinear dynamics of the system. Using the estimated states and the full-state feedback controller, control surface commands were fed back to the higher-order, nonlinear ASE system. The insight gained during this development will be applied to control law development for the physical IAWTM wind-tunnel test.

Josiah M Waite↗
DOE OSTI2025

First‐Order Empirical Interpolation Method for Real‐Time Solution of Parametric Time‐Dependent Nonlinear PDEs

ABSTRACT We present a model reduction approach for the real‐time solution of time‐dependent nonlinear partial differential equations (PDEs) with parametric dependencies. A major challenge in constructing efficient and accurate reduced‐order models for nonlinear PDEs is the efficient treatment of nonlinear terms. We address this by unifying the implementation of hyperreduction methods to deal with nonlinear terms. Furthermore, we introduce a first‐order empirical interpolation method (EIM) to provide an efficient approximation of the nonlinear terms in time‐dependent PDEs. We demonstrate the effectiveness of our approach on the Allen–Cahn equation, which models phase separation, and the Buckley–Leverett equation, which describes two‐phase fluid flow in porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed method compared with both the Galerkin–Newton approach and hyper‐reduced models using the standard EIM.

Nguyen, Ngoc Cuong [Center for Computational Engin↗
DOE OSTI2020

Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.

Xie, Xuping↗
DOE OSTI2020

Gradient-based constrained optimization using a database of linear reduced-order models

A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Physics-constrained, low-dimensional models for magnetohydrodynamics: First-principles and data-driven approaches

Plasmas are highly nonlinear and multiscale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD), although these reduced models hold promise for understanding key physical mechanisms, efficient computation, and real-time optimization and control. Galerkin models, obtained by projection of the MHD equations onto a truncated modal basis, and data-driven models, obtained by modern machine learning and system identification, can furnish this gap in the lower levels of the model hierarchy. This work develops a reduced-order modeling framework for compressible plasmas, leveraging decades of progress in projection-based and data-driven modeling of fluids. We begin by formalizing projection-based model reduction for nonlinear MHD systems. To avoid separate modal decompositions for the magnetic, velocity, and pressure fields, we introduce an energy inner product to synthesize all of the fields into a dimensionally consistent, reduced-order basis. Next, we obtain an analytic model by Galerkin projection of the Hall-MHD equations onto these modes. We illustrate how global conservation laws constrain the model parameters, revealing symmetries that can be enforced in data-driven models, directly connecting these models to the underlying physics. We demonstrate the effectiveness of this approach on data from high-fidelity numerical simulations of a three-dimensional spheromak experiment. Finally, this manuscript builds a bridge to the extensive Galerkin literature in fluid mechanics and facilitates future principled development of projection-based and data-driven models for plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Koopman-Based Approach to Nonintrusive Reduced Order Modeling: Application to Aerodynamic Shape Optimization and Uncertainty Propagation

A methodology for nonintrusive projection-based nonlinear model reduction originally presented by Renganathan et al. is further extended here toward parametric systems with a focus on application to aerospace design. Specifically, the method is extended to address static systems with parametric geometry (that deforms the mesh) in addition to parametric freestream boundary conditions. The main idea is to first perform a transformation on the governing equations such that it is lifted to a higher-dimensional but linear underdetermined system. This enables one to extract the system matrices easily as compared to that of the original nonlinear system. The underdetermined system is closed with a set of model-dependent nonlinear constraints upon which the model reduction is finally performed. The methodology is validated on the subsonic and transonic inviscid flows past the NACA0012 and the RAE2822 airfoils with parametrized shapes. The utility of the approach is further demonstrated by applying it to two common problems in aerospace design, namely, derivative-free global optimization and parametric uncertainty quantification with Monte Carlo sampling. Overall, the methodology is shown to achieve accuracy up to 5% and computational speedup of two to three orders of magnitude relative to the full order model. Comparison against another nonintrusive model reduction method revealed that the proposed approach is more robust, accurate, and retains the consistency between the state variables.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders

A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Furthermore, our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD-Galerkin technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

VpROM: a novel variational autoencoder-boosted reduced order model for the treatment of parametric dependencies in nonlinear systems

Reduced Order Models (ROMs) are of considerable importance in many areas of engineering in which computational time presents difficulties. Established approaches employ projection-based reduction, such as Proper Orthogonal Decomposition. The limitation of the linear nature of such operators is typically tackled via a library of local reduction subspaces, which requires the assembly of numerous local ROMs to address parametric dependencies. Our work attempts to define a more generalisable mapping between parametric inputs and reduced bases for the purpose of generative modeling. We propose the use of Variational Autoencoders (VAEs) in place of the typically utilised clustering or interpolation operations, for inferring the fundamental vectors, termed as modes, which approximate the manifold of the model response for any and each parametric input state. The derived ROM still relies on projection bases, built on the basis of full-order model simulations, thus retaining the imprinted physical connotation. However, it additionally exploits a matrix of coefficients that relates each local sample response and dynamics to the global phenomena across the parametric input domain. The VAE scheme is utilised for approximating these coefficients for any input state. This coupling leads to a high-precision low-order representation, which is particularly suited for problems where model dependencies or excitation traits cause the dynamic behavior to span multiple response regimes. Moreover, the probabilistic treatment of the VAE representation allows for uncertainty quantification on the reduction bases, which may then be propagated to the ROM response. The performance of the proposed approach is validated on an open-source simulation benchmark featuring hysteresis and multi-parametric dependencies, and on a large-scale wind turbine tower characterised by nonlinear material behavior and model uncertainty.

Conditional VAEs↗
DOE OSTI2022

A comparison of neural network architectures for data-driven reduced-order modeling

The popularity of deep convolutional autoencoders (CAEs) has engendered new and effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. Despite this, it is still unknown whether deep CAEs provide superior performance over established linear techniques or other network-based methods in all modeling scenarios. To elucidate this, the effect of autoencoder architecture on its associated ROM is studied through the comparison of deep CAEs against two alternatives: a simple fully connected autoencoder, and a novel graph convolutional autoencoder. Through benchmark experiments, it is shown that the superior autoencoder architecture for a given ROM application is highly dependent on the size of the latent space and the structure of the snapshot data, with the proposed architecture demonstrating benefits on data with irregular connectivity when the latent space is sufficiently large.

42 ENGINEERING↗
NASA NTRS1978

Procedures for generation and reduction of linear models of a turbofan engine

A real time hybrid simulation of the Pratt & Whitney F100-PW-F100 turbofan engine was used for linear-model generation. The linear models were used to analyze the effect of disturbances about an operating point on the dynamic performance of the engine. A procedure that disturbs, samples, and records the state and control variables was developed. For large systems, such as the F100 engine, the state vector is large and may contain high-frequency information not required for control. This, reducing the full-state to a reduced-order model may be a practicable approach to simplifying the control design. A reduction technique was developed to generate reduced-order models. Selected linear and nonlinear output responses to exhaust-nozzle area and main-burner fuel flow disturbances are presented for comparison.

Seldner, K.↗
NASA NTRS2020

Application of Quasi-Static Modal Analysis to an Orion Multi-Purpose Crew Vehicle Test

Bolted structural joints often exhibit load-dependent stiffness and energy dissipation that leads to nonlinear, amplitude dependent frequency and damping in the structure. As an alternative to direct integration of the nonlinear equations of motion, quasi-static modal analysis (QSMA) determines the dependence of frequency and damping on response amplitude using loading behavior from nonlinear static analyses. QSMA has previously been demonstrated to substantially reduce computational cost and maintain accuracy relative to full nonlinear dynamic simulation. This work explores the applicability of QSMA to a complex, large-scale aerospace structure. QSMA is employed to analyze a nonlinear model of test hardware developed to support the Orion Multi-Purpose Crew Vehicle program, which exhibited nonlinear behavior during dynamic testing at flight-like load levels. In addition to the extraction of amplitude-dependent frequency and damping curves, a Bouc-Wen hysteresis model was used in conjunction with the quasi-static results to develop nonlinear, uncoupled, time-domain modal equations of motion for the structure. Excellent agreement was observed between the reduced and full-order nonlinear models, encouraging future employment of QSMA to support accurate and efficient model reduction of structures with bolted joint nonlinearities.

QSMA↗
DOE OSTI2023

DRIPS: A framework for dimension reduction and interpolation in parameter space

Reduced-order models are often used to describe the behavior of complex systems, whose simulation with a full model is too expensive, or to extract salient features from the full model’s output. We introduce a new model-reduction framework DRIPS (dimension reduction and interpolation in parameter space) that combines the offline local model reduction with the online parameter interpolation of reduced-order bases (ROBs). The offline step of this framework relies on dynamic mode decomposition (DMD) to build a low-rank linear surrogate model, equipped with a local ROB, for quantities of interest derived from the training data generated by repeatedly solving the (nonlinear) high-fidelity model for multiple parameter points. The online step consists of the construction of a parametric reduced-order model for each target/test point in the parameter space, with the interpolation of ROBs done on a Grassman manifold and the interpolation of reduced-order operators done on a matrix manifold. The DMD component enables DRIPS to model (typically low-dimensional) quantities of interest directly, without having to access the (typically high-dimensional and possibly nonlinear) operators in a high-fidelity model that governs the dynamics of the underlying high-dimensional state variables, as required in projection-based reduced-order modeling. A series of numerical experiments suggests that DRIPS yields a model reduction, which is computationally more efficient than the commonly used projection-based proper orthogonal decomposition; it does so without requiring a prior knowledge of the governing equation for quantities of interest. Furthermore, for the nonlinear systems considered, DRIPS is more accurate than Gaussian-process interpolation (Kriging).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1990

Projection and assembly method for multibody component model reduction

The problem addressed is that of obtaining reduced-order component models for use in simulating the dynamics of a multibody system. In certain cases, nonlinear system models may be constructed using linear dynamic models for each component, but allowing large angle motion between components. Without some form of model reduction, system models constructed in this manner may be too large for use in control system design and simulation trades. This paper analyzes one method of component model reduction that allows systems level requirements (e.g., capturing the effect of body 1 reaction wheel noise on body 2 camera pointing) to aid in the selection of the reduced-order component models. Briefly stated, important modes are selected at the system level and projected onto the components, and reduced-order components are then assembled into a reduced-order system model that retains the projected modes.

Bernard, Douglas E.↗
DOE OSTI2021

Reduced order models for Lagrangian hydrodynamics

It is reported, as a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor-Green vortices, and triple-point problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Alternatives for jet engine control

Tensor model order reduction, recursive tensor model identification, input design for tensor model identification, software development for nonlinear feedback control laws based upon tensors, and development of the CATNAP software package for tensor modeling, identification and simulation were studied. The last of these are discussed.

Sain, M. K.↗
DOE OSTI2021

A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Lastly, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.

97 MATHEMATICS AND COMPUTING↗