A local basis approximation approach for nonlinear parametric model order reduction
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Parametric dynamic mode decomposition for reduced order modeling
Dynamic Mode Decomposition (DMD) is a model-order reduction approach, whereby spatial modes of fixed temporal frequencies are extracted from numerical or experimental data sets. The DMD low-rank or reduced operator is typically obtained by singular value decomposition of the temporal data sets. For parameter-dependent models, as found in many multi-query applications such as uncertainty quantification or design optimization, the only parametric DMD technique developed was a stacked approach, with data sets at multiple parameter values were aggregated together, increasing the computational work needed to devise low-rank dynamical reduced-order models. Here in this paper, we present two novel approach to carry out parametric DMD: one based on the interpolation of the reduced-order DMD eigen-pair and the other based on the interpolation of the reduced DMD (Koopman) operator. Numerical results are presented for diffusion-dominated nonlinear dynamical problems, including a multiphysics radiative transfer example. All three parametric DMD approaches are compared.
97 MATHEMATICS AND COMPUTING↗
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↗
Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation
Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a low-dimensional emulation framework for systems that may be intrinsically high-dimensional. This is accomplished by utilizing a construction algorithm that is purely data-driven. It is no surprise, therefore, that the algorithmic advances of machine learning have led to non-intrusive ROMs with greater accuracy and computational gains. However, in bypassing the utilization of an equation-based evolution, it is often seen that the interpretability of the ROM framework suffers. This becomes more problematic when black-box deep learning methods are used which are notorious for lacking robustness outside the physical regime of the observed data. In this article, we propose the use of a novel latent-space interpolation algorithm based on Gaussian process regression. Notably, this reduced-order evolution of the system is parameterized by control parameters to allow for interpolation in space. The use of this procedure also allows for a continuous interpretation of time which allows for temporal interpolation. The latter aspect provides information, with quantified uncertainty, about full-state evolution at a finer resolution than that utilized for training the ROMs. This research assesses the viability of this algorithm for an advection-dominated system given by the inviscid shallow water equations.
97 MATHEMATICS AND COMPUTING↗
Data-driven models of nonautonomous systems
Nonautonomous dynamical systems are characterized by time-dependent inputs, which complicates the discovery of predictive models describing the spatiotemporal evolution of the state variables of quantities of interest from their temporal snapshots. When dynamic mode decomposition (DMD) is used to infer a linear model, this difficulty manifests itself in the need to approximate the time-dependent Koopman operators. Our approach is to approximate the original nonautonomous system with a modified system derived via a local parameterization of the time-dependent inputs. The modified system comprises a sequence of local parametric systems, which are subsequently approximated by a parametric surrogate model using the DRIPS (dimension reduction and interpolation in parameter space) framework. The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases for the observables mapped from training data. The online step interpolates on suitable manifolds to construct a sequence of iterative parametric surrogate models; the target/test parameter points on these manifolds are specified by a local parameterization of the test time-dependent inputs. Here, we use numerical experimentation to demonstrate the robustness of our method and compare its performance with that of deep neural networks.
97 MATHEMATICS AND COMPUTING↗
Crystal mechanics-based thermo-elastic constitutive modeling of orthorhombic uranium using generalized spherical harmonics and first-order bounding theories
In earlier works, a mathematical procedure for invertible microstructure-property linkages was developed using computationally efficient spectral methods for polycrystalline cubic and hexagonal metals. This paper formulates such invertible microstructure–property linkages for orthorhombic polycrystalline metals relying on the generalized spherical harmonics (GSH) spectral basis. The procedure is used to compute property closures of orthorhombic polycrystals. The closures represent the complete set of theoretically possible combinations of effective properties for a selected material. The procedure relies on the first-order bounding theories and considers orientation distribution functions (ODFs) as the main microstructural descriptor influencing homogenized properties. Numerous examples of these closures involving second-rank thermal expansion and fourth-rank elastic stiffness tensorial properties over a broad range of temperatures are presented for α-uranium (α-U). In doing so, certain key properties of these closures are exploited to facilitate their computation with drastically reduced computational effort. Along with the recently developed GSH-based interpolation procedure for ODFs from coarsely spaced experimental measurement grids to finely spaced finite element mesh resolution grids presented in Barrett et al., the developed computationally efficient ODF-effective property linkages are used to establish a crystal mechanics-based simulation framework coupled with the finite element method (FEM). The ODF dependent thermal expansion and elastic stiffness tensors are efficiently calculated at every integration point and used by the FEM to predict the overall distortion of a hemispherical part made of α-U during heating. In conclusion, it is shown that the developed framework can be used to simulate microstructurally heterogeneous components under thermo-mechanical loadings in a computationally efficient manner.
36 MATERIALS SCIENCE↗
$\text{GPLaSDI}$: Gaussian Process-based interpretable Latent Space Dynamics Identification through deep autoencoder
Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the latent space dynamics. By interpolating and solving the ODE system in the reduced latent space, fast and accurate ROM predictions can be made by feeding the predicted latent space dynamics into the decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework that relies on Gaussian process (GP) for latent space ODE interpolations. Using GPs offers two significant advantages. First, it enables the quantification of uncertainty over the ROM predictions. Second, leveraging this prediction uncertainty allows for efficient adaptive training through a greedy selection of additional training data points. This approach does not require prior knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently non-intrusive and can be applied to problems without a known PDE or its residual. Here we demonstrate the effectiveness of our approach on the Burgers equation, Vlasov equation for plasma physics, and a rising thermal bubble problem. Our proposed method achieves between 200 and 100,000 times speed-up, with up to 7% relative error.
97 MATHEMATICS AND COMPUTING↗
Islet: interpolation semi-Lagrangian element-based transport
Abstract. Advection of trace species, or tracers, also called tracer transport, in models of the atmosphere and other physical domains is an important and potentially computationally expensive part of a model's dynamical core. Semi-Lagrangian (SL) advection methods are efficient because they permit a time step much larger than the advective stability limit for explicit Eulerian methods without requiring the solution of a globally coupled system of equations as implicit Eulerian methods do. Thus, to reduce the computational expense of tracer transport, dynamical cores often use SL methods to advect tracers. The class of interpolation semi-Lagrangian (ISL) methods contains potentially extremely efficient SL methods. We describe a finite-element ISL transport method that we call the interpolation semi-Lagrangian element-based transport (Islet) method, such as for use with atmosphere models discretized using the spectral element method. The Islet method uses three grids that share an element grid: a dynamics grid supporting, for example, the Gauss–Legendre–Lobatto basis of degree three; a physics parameterizations grid with a configurable number of finite-volume subcells per element; and a tracer grid supporting use of Islet bases with particular basis again configurable. This method provides extremely accurate tracer transport and excellent diagnostic values in a number of verification problems.
58 GEOSCIENCES↗
Reducing Ground-based Astrometric Errors with Gaia and Gaussian Processes
Stochastic field distortions caused by atmospheric turbulence are a fundamental limitation to the astrometric accuracy of ground-based imaging. This distortion field is measurable at the locations of stars with accurate positions provided by the Gaia DR2 catalog; we develop the use of Gaussian process regression (GPR) to interpolate the distortion field to arbitrary locations in each exposure. We introduce an extension to standard GPR techniques that exploits the knowledge that the 2D distortion field is curl-free. Applied to several hundred 90 s exposures from the Dark Energy Survey as a test bed, we find that the GPR correction reduces the variance of the turbulent astrometric distortions ≈12× , on average, with better performance in denser regions of the Gaia catalog. The rms per-coordinate distortion in the riz bands is typically ≈7 mas before any correction and ≈2 mas after application of the GPR model. The GPR astrometric corrections are validated by the observation that their use reduces, from 10 to 5 mas rms, the residuals to an orbit fit to riz-band observations over 5 yr of the r = 18.5 trans-Neptunian object Eris. We also propose a GPR method, not yet implemented, for simultaneously estimating the turbulence fields and the 5D stellar solutions in a stack of overlapping exposures, which should yield further turbulence reductions in future deep surveys.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Discrete Empirical Interpolation Method Based Dynamic Load Model Reduction
Dynamic load models add significant complexity to bulk power system time-domain simulations. The complexity is due to the large number of ordinary differential equations (ODEs) introduced by the dynamic load components such as induction motors. It is challenging to derive reduced-order models (ROMs) for dynamic loads due to the nonlinear functions in their governing equations. This paper applies the discrete empirical interpolation method enhanced proper orthogonal decomposition (DEIM-POD) to approximate the full dynamic load model with the ROM that minimizes the projection error of the nonlinear functions in dynamic load ODEs onto their dominant modes. This approach only requires evaluation of nonlinear functions at selected observation points. The observation points selected by DEIM also provide information for screening critical load buses where dynamic load model parameters contribute the most to the accuracy of ROM across multiple contingencies. The proposed approach is validated on IEEE 9-bus, WECC 179-bus and 2384-bus Polish systems.
bulk power system↗
Eulerian finite element simulations of the drop weight impact test with a dislocation Density-based continuum model
During a drop weight impact test, the kinetic energy of the falling weight is transferred into the sample resting on the anvil. The plastic deformation in the sample is an important mechanism for the dissipation of the input kinetic energy. We use Eulerian finite element analysis to simulate the deformation and temperature evolution in a 1,3,5-trinitro-1,3,5-triazine (RDX) sample consisting of multiple crystals. In Eulerian finite element simulations, the mesh moves relative to the material. After every change of position between the mesh and the material, the state variables are interpolated to the new mesh position, i.e., advection. In an effort to reduce the advection errors, we use a rate form of a dislocation density-based continuum model by Luscher et al. Here, the simulations predict localization of plastic deformation, and plastic dissipation as a significant source of heat generation.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Subsetted model output on pressure levels for cloud-related variables from a LASSO-CACTI WRF simulation
Cloud-related model output interpolated to pressure levels from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m through the use of four nested domains, labeled d1 through d4. The simulations are of convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables interpolated to pressure levels.
54 ENVIRONMENTAL SCIENCES↗
Subsetted model output on pressure levels for meteorology-related variables from a LASSO-CACTI WRF simulation
Meteorology-related model output interpolated to pressure levels from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m through the use of four nested domains, labeled d1 through d4. The simulations are of convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables interpolated to pressure levels.
54 ENVIRONMENTAL SCIENCES↗
Subsetted model output on height-above-ground levels for cloud-related variables from a LASSO-CACTI WRF simulation
Cloud-related model output interpolated to height-above-ground levels from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m through the use of four nested domains, labeled d1 through d4. The simulations are of convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables interpolated to heights based on distance above ground level.
54 ENVIRONMENTAL SCIENCES↗
Subsetted model output on height-above-ground levels for meteorology-related variables from a LASSO-CACTI WRF simulation
Meteorology-related model output interpolated to height-above-ground levels from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m through the use of four nested domains, labeled d1 through d4. The simulations are of convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables interpolated to heights based on distance above ground level.
54 ENVIRONMENTAL SCIENCES↗
Subsetted model output on height-above-mean-sea-level levels for cloud-related variables from a LASSO-CACTI WRF simulation
Cloud-related model output interpolated to heights-above-mean-sea-level from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m through the use of four nested domains, labeled d1 through d4. The simulations are of convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables interpolated to heights based on distance above mean sea level.
54 ENVIRONMENTAL SCIENCES↗
Subsetted model output on height-above-mean-sea-level levels for meteorology-related variables from a LASSO-CACTI WRF simulation
Meteorology-related model output interpolated to heights-above-mean-sea-level from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m through the use of four nested domains, labeled d1 through d4. The simulations are of convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables interpolated to heights based on distance above mean sea level.
54 ENVIRONMENTAL SCIENCES↗
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↗