Publications

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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.

Computational simulations of high-speed flow play an important role in the design of hypersonic vehicles, for which experimental data are scarce; however, high-fidelity simulations of hypersonic flow are computationally expensive. Reduced order models (ROMs) have the potential to make many-query problems, such as design optimization and uncertainty quantification, tractable for this domain. Residual minimization-based ROMs, which formulate the projection onto a reduced basis as an optimization problem, are one promising candidate for model reduction of large-scale fluid problems. This work analyzes whether specific choices of norms and objective functions can improve the performance of ROMs of hypersonic flow. Specifically, we investigate the use of dimensionally consistent inner products and modifications designed for convective problems, including ℓ1 minimization and constrained optimization statements to enforce conservation laws. Particular attention is paid to accuracy for problems with strong shocks, which are common in hypersonic flow and challenging for projection-based ROMs. We demonstrate that these modifications can improve the predictability and efficiency of a ROM, though the impact of such formulations depends on the quantity of interest and problem considered.

Computational simulations of high-speed flow play an important role in the design of hypersonic vehicles, for which experimental data are scarce; however, high-fidelity simulations of hypersonic flow are computationally expensive. Reduced order models (ROMs) have the potential to make many-query problems, such as design optimization and uncertainty quantification, tractable for this domain. Residual minimization-based ROMs, which formulate the projection onto a reduced basis as an optimization problem, are one promising candidate for model reduction of large-scale fluid problems. This work analyzes whether specific choices of norms and objective functions can improve the performance of ROMs of hypersonic flow. Specifically, we investigate the use of dimensionally consistent inner products and modifications designed for convective problems, including ℓ1 minimization and constrained optimization statements to enforce conservation laws. Particular attention is paid to accuracy for problems with strong shocks, which are common in hypersonic flow and challenging for projection-based ROMs. We demonstrate that these modifications can improve the predictability and efficiency of a ROM, though the impact of such formulations depends on the quantity of interest and problem considered.

The Reynolds-averaged Navier–Stokes (RANS) equations remain a workhorse technology for simulating compressible fluid flows of practical interest. Due to model-form errors, however, RANS models can yield erroneous predictions that preclude their use on mission-critical problems. This work presents a data-driven turbulence modeling strategy aimed at improving RANS models for compressible fluid flows. The strategy outlined has three core aspects: (1) prediction for the discrepancy in the Reynolds stress tensor and turbulent heat flux via machine learning (ML), (2) estimating uncertainties in ML model outputs via out-of-distribution detection, and (3) multi-step training strategies to improve feature-response consistency. Results are presented across a range of cases publicly available on NASA’s turbulence modeling resource involving wall-bounded flows, jet flows, and hypersonic boundary layer flows with cold walls. We find that one ML turbulence model is able to provide consistent improvements for numerous quantities-of-interest across all cases.

This paper presents the uncertainty propagation of turbulent coefficients for the Spalart– Allmaras (SA) turbulence model using projection-based reduced-order models (ROMs). ROMs are used instead of Reynolds-averaged Navier–Stokes (RANS) solvers and stochastic collocation/ Galerkin and Monte Carlo methods because they are computationally inexpensive and tend to offer more accuracy than a polynomial surrogate. The uncertainty propagation is performed on two benchmark RANS cases documented on NASA’s turbulence modeling resource. Uncertainty propagation of the SA turbulent coefficients using a ROMis shown to compare well against uncertainty propagation performed using only RANS and using a Gaussian process regression (GP) model. The ROM is shown to be more robust to the size and spread of the training data compared to a GP model.

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A projection-based reduced order model (pROM) methodology has been developed for transient heat transfer problems involving coupled conduction and enclosure radiation. The approach was demonstrated on two test problems of varying complexity. The reduced order models demonstrated substantial speedups (up to 185×) relative to the full order model with good accuracy (less than 3% L∞ error). An attractive feature of pROMs is that there is a natural error indicator for the ROM solution: the final residual norm at each time-step of the converged ROM solution. Using example test cases, we discuss how to interpret this error indicator to assess the accuracy of the ROM solution. The approach shows promise for many-query applications, such as uncertainty quantification and optimization. The reduced computational cost of the ROM relative to the full-order model (FOM) can enable the analysis of larger and more complex systems as well as the exploration of larger parameter spaces.

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This work focuses on the space-time reduced-order modeling (ROM) method for solving large-scale uncertainty quantification (UQ) problems with multiple random coefficients. In contrast with the traditional space ROM approach, which performs dimension reduction in the spatial dimension, the space-time ROM approach performs dimension reduction on both the spatial and temporal domains, and thus enables accurate approximate solutions at a low cost. We incorporate the space-time ROM strategy with various classical stochastic UQ propagation methods such as stochastic Galerkin and Monte Carlo. Numerical results demonstrate that our methodology has significant computational advantages compared to state-of-the-art ROM approaches. By testing the approximation errors, we show that there is no obvious loss of simulation accuracy for space-time ROM given its high computational efficiency.

This work aims to advance computational methods for projection-based reduced-order models (ROMs) of linear time-invariant (LTI) dynamical systems. For such systems, current practice relies on ROM formulations expressing the state as a rank-1 tensor (i.e., a vector), leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern architectures. This weakness can be particularly limiting when tackling many-query studies, where one needs to run a large number of simulations. This work introduces a reformulation, called rank-2 Galerkin, of the Galerkin ROM for LTI dynamical systems which converts the nature of the ROM problem from memory bandwidth to compute bound. We present the details of the formulation and its implementation, and demonstrate its utility through numerical experiments using, as a test case, the simulation of elastic seismic shear waves in an axisymmetric domain. We quantify and analyze performance and scaling results for varying numbers of threads and problem sizes. Finally, we present an end-to-end demonstration of using the rank-2 Galerkin ROM for a Monte Carlo sampling study. We show that the rank-2 Galerkin ROM is one order of magnitude more efficient than the rank-1 Galerkin ROM (the current practice) and about 970 times more efficient than the full-order model, while maintaining accuracy in both the mean and statistics of the field.

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This work proposes a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space–time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov–Galerkin (LSPG), and space–time LSPG projection. In addition, the approach addresses key deficiencies in existing model reduction techniques, e.g., the dependence of LSPG and space–time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space–time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space–time residuals than Galerkin and LSPG projection.

This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter instances. Our extension is inspired by the concept of parameterized ODEs, which are widely investigated in computational science and engineering contexts, where characteristics of the governing equations vary over the input parameters. We apply the proposed parameterized NODEs (PNODEs) for learning latent dynamics of complex dynamical processes that arise in computational physics, which is an essential component for enabling rapid numerical simulations for time-critical physics applications. For this, we propose an encoder-decoder-type framework, which models latent dynamics as PNODEs. We demonstrate the effectiveness of PNODEs on benchmark problems from computational physics.