Accurate ROMs for Industrial Applications at Virginia Tech (ARIA@VT)
Department of Biomedical Engineering and Mechanics & Department of Mathematics, Virginia Tech, July 6-8, 2022
ParticipantsThis page will be frequently updated as new confirmations are received.
AbstractsReduced-Order Modeling using Time-Dependent Bases with Applications to Fluid Dynamics and Combustion Hessam Babaee, University of Pittsburgh Many important problems in fluid mechanics are described by high-dimensional partial differential equations (PDEs). The computational cost of solving these problems using classical discretization techniques increases exponentially with respect to the number of dimensions --- a fundamental challenge that is dubbed the curse of dimensionality. On the other hand, many of these high-dimensional problems have a much lower intrinsic dimensionality, that if discovered, can mitigate the curse of dimensionality. This calls for techniques that extract and exploit correlated structures directly from the PDE. This approach is in direct contrast to classical discretization techniques that disregard multidimensional correlations and result in inefficient solutions for high-dimensional problems. While there are numerous data-driven dimension reduction techniques that can extract these correlated structures by solving the full-dimensional PDE, these techniques are only feasible for lower-dimensional PDEs (e.g., 2D/3D). This same workflow is impracticable for many high-dimensional PDEs as computing the solution of the full-dimensional PDE is the very problem we cannot afford to solve. To this end, we present a reduced-order modeling framework, in which the correlated structures are extracted directly from the PDE --- bypassing the need to generate data. These structures are exploited by building on-the-fly reduced-order models (ROM). The correlated structures are represented by a set of time-dependent orthonormal bases and their evolution is prescribed by the physics of the problem. We present several demonstration cases including reduced-order modeling of reactive species transport equation in turbulent combustion as well as sensitivity analysis and uncertainty quantification in fluid dynamics problems. Driving Bifurcating Parametrized Nonlinear PDEs by Optimal Control Strategies: Application to Navier-Stokes Equations Francesco Ballarin, Catholic University of the Sacred Heart, Italy This talk deals with optimal control problems as a strategy to drive bifurcating solutions of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows the solution to change profile and the stability of state solution branches. We present a general framework for nonlinear optimal control problems to reconstruct each branch of optimal solutions, discussing the stability properties of the obtained controlled solutions. Then, we apply the proposed framework to several optimal control problems governed by bifurcating Navier-Stokes equations, and analyze the qualitative and quantitative effect of the control over a pitchfork bifurcation, as well as the stability eigenvalue analysis of the controlled state. Finally, we propose reduced order modeling as a tool to efficiently and reliably solve parametric stability analysis of such optimal control systems, which would otherwise be unbearable to perform with standard discretization techniques. This is a joint work with Dr. Federico Pichi, Dr. Maria Strazzullo and Prof. Gianluigi Rozza, based on manuscript [1]. References: A Computational Approach to Nonlinear Balanced Truncation Jeff Borggaard, Virginia Tech For linear control systems, balanced truncation is well established as a strategy for simultaneous model reduction and control design. The theory was extended to nonlinear systems in the 1990's, but there are limited computational tools for computing these reduced-order models. In this talk, we outline an approach to compute the nonlinear transformations and reduced-order models for balanced truncation of polynomial systems. We begin by describing the nonlinear generalizations of the controllability and observability gramians used in balancing: their corresponding energy functions. We then present an approximation methodology for problems with polynomial nonlinearity. Finally, we use these approximations to develop nonlinear transformations that balance the input and output energies of the system leading to reduced-order models. One advantage of this approach is we simultaneously compute feedback controllers as well. This is joint work with Boris Kramer and Serkan Gugercin. A Physics-Informed Data-Driven Statistical Reduced Order Model for Ensemble Forecast of Complex Turbulent Systems Nan Chen, University of Wisconsin-Madison A new method of building effective statistical reduced order model, named as the physics-informed data-driven algorithm with conditional Gaussian statistics (PIDD-CG), is developed to predict the probability density functions (PDFs) of complex turbulent systems with partial observations. The PIDD-CG algorithm integrates a unique multiscale statistical closure modeling strategy with an extremely efficient nonlinear data assimilation scheme to create a mixture of conditional statistics. These conditional statistics serve as the forecast ensemble members to mitigate the curse of dimensionality in recovering high-dimensional PDF. The multiscale features in the time evolution of these conditional statistics ensembles are effectively predicted by an appropriate combination of physics-informed analytic formulae and recurrent neural networks. An information metric is adopted as the loss function for the latter to capture the desirable turbulent features more accurately. The proposed algorithm succeeds in efficiently forecasting both the transient and statistical equilibrium non-Gaussian PDFs of strongly turbulent systems with intermittency, regime switching and extreme events. Neural-Network-Augmented Projection-based Reduced-Order Models for Mitigating the Kolmogorov Barrier Joshua L. Barnett and Charbel Farhat, Stanford University and Yvon Maday, Sorbonne Université This lecture will expand upon our recent work on a projection-based reduced-order model (PROM) grounded in a data-driven quadratic approximation manifold by proposing a more general, neural-network (NN)-augmented PROM, referred to as a PROM-NN. Given a desired level of accuracy, the main objective of a PROM-NN is to dramatically reduce the dimensionality of the online approximation beyond what was achieved using a quadratic approximation manifold; and therefore far beyond what is possible using a traditional affine approximation. Compared to previous works that attempted to combine projection-based model order reduction (PMOR) and machine learning, the training of the NN in a PROM-NN does not scale with the large dimension of the high-dimensional model (HDM). This makes a PROM-NN significantly more practical for large-scale, industrial-grade problems with hundreds of millions of degrees of freedom. This computational efficiency comes at the price of a limitation of the low-dimensional approximation to the span of all generated basis vectors. However, with a sufficiently rich nontruncated basis, the affine approximation is little affected by this limitation. Equipping a PROM-NN with hyperreduction is relatively straight forward, particularly using the ECSW method. The proposed PMOR-NN technology will be demonstrated in the parametric setting for a couple academic problems that exhibit the Kolmogorov barrier and offer the benefit of being easily reproducible by the interested reader. Its superior performance will be demonstrated through comparisons with the performances of the PMOR methods based on a data-driven quadratic approximation manifold and on the traditional affine subspace approximation. Model Reduction of Navier-Stokes Equations using the Loewner Framework Matthias Heinkenschloss, Rice University The Loewner framework is a data-driven extension of interpolation based model reduction. Traditional interpolation based model reduction computes a projection based reduced order model (ROM) so that transfer function components of the ROM interpolate the corresponding transfer function components of the original full order model at points specified by the user. In contrast, the Loewner framework computes a ROM directly from measurements of transfer function components. Given these measurements, no explicit access to the system is required to construct the ROM. For systems governed by quadratic ordinary differential equations (ODEs) the transfer function components are derived from an expansion of the system's input-to-output map. To derive transfer functions for systems governed by incompressible Navier-Stokes (NS) equations, the NS equations are transformed into ODEs by projecting onto the subspace defined by the incompressibility condition. This projection is used theoretically, but avoided computationally. This talk presents the overall approach. Numerical results show the potential of the Loewner framework, but also reveal a possible lack of stability of the ROM. A possible remedy, which currently requires access to the NS system, to deal with these instabilities is outlined. Sketching for Nonsmooth PDE Constrained Optimization Problems Evelyn Herberg, George Mason University This work applies matrix sketching type techniques to nonsmooth optimization problems constrained by dynamical systems. Such problems are notoriously difficult with respect to storage and computational time. Even the second order methods may struggle to converge. Sketching is shown to be a promising approach to overcome these challenges. Convergence of inexact methods for nonsmooth problems, such as semi-smooth Newton is studied. Theoretical findings are illustrated with the help of a numerical example, where we consider an initial measure control. This is a joint work with Harbir Antil, Mohammed Alshehri (George Mason University) and Drew P. Kouri (Sandia National Labs). Discussion of Some Examples of Linear or Nonlinear, Intrusive or Non-intrusive Reduced Models based on Convex Displacement Interpolation Angelo Iollo, University of Bordeaux & Inria, France The presentation focuses on examples of parametric problems governed by partial differential equations in which the linear representation of the reduced space fails. We introduce a nonlinear approximation technique based on a solution mapping as a function of the parameters via optimal transport. We conclude by discussing the advantages and disadvantages in this framework of linear or nonlinear, intrusive or non-intrusive model reduction approaches. The presented work is in collaboration with S. Cucchiara, T. Taddei, and H. Telib. Progress and Challenges in Projection-based Model Order Reduction of Turbulent Combustion Cheng Huang, University of Kansas Even with exascale computing capabilities, high-fidelity, full-scale simulations of turbulent combustion in realistic applications remain computationally expensive and inaccessible for many-query applications such as engineering design, optimization, and control. Projection-based model order reduction methods have shown promise in greatly improving computational efficiency. However, classical model order reduction methods that seek reduced solutions in low-dimensional subspaces fail for realistic turbulent combustion problems because reacting flows feature extreme stiffness, sharp gradients and multi scale transport, posing great challenges in deriving low order representations. In this talk, recent advances are discussed in projection-based methods for reduced-order model (ROM) development of turbulent combustion problems. An adaptive reduced-order modeling technique is introduced which updates the low-dimensional space, thus circumventing representation barriers faced by static reduced dimensional spaces. The method leverages model-form preserving least-squares projections with variable transformation (MP-LSVT) for improved robustness of ROM and adapt the low-dimensional subspaces based on the evaluated dynamics during online calculations to greatly enhance predictive capabilities. In addition, applications of the adaptive ROM method are investigated towards modeling systems for which full order models are unaffordable (e.g. rocket engines). A component-based modeling framework is proposed, which only requires high-fidelity simulations of small components in the full system. These component ROMs are then integrated in a full system ROM. Empirical Basis Augmentation for Reduced-Order Models of Turbulent Flow Kento Kaneko, Massachusetts Institute of Technology Reduced-order models offer a promising approach for parametric analysis of engineering fluid dynamics applications. The standard procedure consists of using solution snapshots to produce a truncated POD basis, which is in turn used in a Galerkin projection of the governing Navier-Stokes equations. Unfortunately, the standard POD approach has well-known limitations for high Reynolds number flows. Even with stabilization, the required number of modes is often sufficiently large that these approaches are impaired by the cubic cost associated with evaluation of the third order advection tensor at each step of the ROM time-advancement. We present a novel non-intrusive stabilization technique in the form of basis augmentation that, in many cases, reduces the total number of modes required to produce a stable and accurate ROM reconstruction for turbulent flows at modest Reynolds numbers. The approach involves augmenting the standard POD modes with divergence-free projections of subsets of the advection term expansion in terms of POD basis. Bases that include interactions with lifting function and self-interactions have proven to be quite effective for several challenging flow problems with relatively low number of POD and augmentation modes. We demonstrate this proposed basis set on several challenging problems and compare its stability properties with alternative stabilization approaches for POD-based ROMs. This is a joint work with Paul Fischer. Nonlinear Projection-Based Model Order Reduction in the Presence of Adaptive Mesh Refinement Clayton Little, Stanford University Adaptive Mesh Refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models, but is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as Projection-based Reduced-Order Models (PROMs). This talk presents a complete framework for PROMs in the presence of AMR that builds on elements of existing methods while adding several critical new elements. In particular, an efficient algorithm is introduced for computing a pseudo-meshless inner product between AMR snapshots for the purpose of projection and clustering. Hyperreduction, specifically Energy-Conserving Sampling and Weighting (ECSW), is used to extend this approach to the solution of nonlinear problems. To achieve computational tractability in the presence of AMR, the concept of local ROBs in the state-space is also adopted. All presented ideas and their significance are illustrated and highlighted with the solution of three-dimensional, nonlinear, convection-dominated flow problems, for which significant speedups are achieved. Stochastic Data-driven ROM Closures Fei Lu, Johns Hopkins University Trajectory-wise data-driven reduced order models (ROMs) tend to be sensitive to training data, and thus lack robustness. We propose to construct a robust stochastic ROM closure (S-ROM) from data consisting of multiple trajectories from random initial conditions. The S-ROM is a low-dimensional time series model for the coefficients of the dominating proper orthogonal decomposition (POD) modes inferred from data. Thus, it achieves reduction both space and time, leading to simulations orders of magnitude faster than the full order model. We prove that both the estimated POD modes and parameters in the S-ROM converge when the number of trajectories increases. Thus, the S-ROM is not sensitive to training data. We demonstrate the S-ROM on a 1D Burgers equation with a viscosity $\nu= 0.002$ and with random initial conditions. The numerical results verify the convergence. Furthermore, the S-ROM makes accurate trajectory-wise predictions from new initial conditions and with a prediction time far beyond the training range, and it can quantify the spread of uncertainties due to the unresolved scales. An open question arises: what is an optimal space-time reduction? This is a joint work with Changhong Mou, Honghu Liu and Traian Iliescu. Learning Nonlinear Dynamical Systems from Data Using Scientific Machine Learning Romit Maulik, Argonne National Laboratory This talk will introduce recent work that aims to learn dynamical systems from data. Specifically, we will discuss novel neural architectures that identifying and leveraging physically relevant structures for preserving robustness and stability in the learned dynamical systems. In the first half of the talk, we will discuss a novel type of neural architecture that identifies an invariant manifold from high dimensional observations of a dynamical system. Furthermore, this architecture is also designed to forecast the evolution of the state while constrained on the identified manifold. Subsequently, we will discuss the development of a generative machine learning model that learns a dynamical system for a time-varying probability density function, given observations for samples from this density. Numerical studies for such architectures have been performed on various tasks ranging from the learning of canonical systems such as the Kuramoto-Sivashinsky and viscous Burgers equations to geophysical systems obtained from a combination of numerical simulations and satellite observations. Kolmogorov n-width, a Universal Barrier for Data-driven Models Rambod Mojgani, Rice University Maciej Balajewicz, Realtor.com and Pedram Hassanzadeh, Rice University Large Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts is known to be a barrier to the accuracy of projection-based reduced-order models (PROMs) on linear subspaces. In this talk, we demonstrate that the Kolmogorov barrier is a challenge in a wide range of data-driven approaches, including nonlinear neural network (NN) ROMs, compressed sensing, and physics-informed neural networks (PINNs). We explain how some developments in the training phase of NNs, such as weighted sampling of data, curriculum regularization, and sequence-to-sequence learning, inadvertently break this barrier. As a result, we conclude that many of the developments in PROMs can also be applied to other data-driven approaches. Therefore, and similar to ROMs with Lagrangian bases, we demonstrate that reformulation of nonlinear convection diffusion partial differential equations (PDEs) on the Lagrangian frame of reference improves the accuracy and convergence of PINNs. Moreover, in a fully data-driven (PDE-agnostic) formulation, a low-rank registration-based manifold is learned to decrease Kolmogorov n-width, and therefore to improve the accuracy and extrapolation capabilities of PROMs and NN-ROMs. This presentation is mainly based on [1-2]. References: An Efficient Data-Driven Multiscale Stochastic Reduced Order Modeling Framework for Complex Systems Changhong Mou, University of Wisconsin-Madison Suitable reduced order models (ROMs) are computationally efficient tools in characterizing key dynamical and statistical features of nature. In this paper, a systematic multiscale stochastic ROM framework is developed for complex systems with strong chaotic or turbulent behavior. The new ROMs are fundamentally different from the traditional Galerkin ROM (G-ROM) or those deterministic ROMs that aim at minimizing the path-wise errors and applying mainly to laminar systems. Here, the new ROM focuses on recovering the large-scale dynamics to the maximum extent while it also exploits cheap but effective conditional linear functions as the closure terms to capture the statistical features of the medium-scale variables and its feedback to the large scales. In addition, physics constraints are incorporated into the new ROM. One unique feature of the resulting ROM is that it facilitates an efficient and accurate scheme for nonlinear data assimilation, the solution of which is provided by closed analytic formulae. Such an analytic solvable data assimilation solution significantly accelerates the computational efficiency and allows the new ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. The overall model calibration is efficient and systematic via explicit mathematical formulae. The new ROM framework is applied to complex nonlinear systems, in which the intrinsic turbulent behavior is either triggered by external random forcing or deterministic nonlinearity. It is shown that the new ROM significantly outperforms the G-ROM in both scenarios in terms of reproducing the dynamical and statistical features as well as recovering unobserved states via the associated efficient data assimilation scheme. Dimensionally Consistent Inner Products for POD-Galerkin and Least-Squares Model Reduction of Compressible Flows Eric Parish, Sandia National Laboratories Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vector-valued L2({\Omega}) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this talk we investigate this challenge by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. We demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. We further demonstrate that physics-based inner products based on entropy principles result in significantly more accurate and robust ROM formulations than those based on non-dimensional L2({\Omega}) inner products. Numerical results are presented for the one-dimensional Sod shock tube, the two-dimensional Kelvin-Helmholtz instability, and two-dimensional homogeneous isotropic turbulence. NINNs: Nudging Induced Neural Networks Randy Price, George Mason University Training accurate neural networks is a challenging task requiring time consuming trial-and-error. To combat this we have developed NINNs which improve the accuracy of pre-existing neural networks. NINNs are based on the idea that neural networks can be nudged during forward propagation towards given data. In this talk we propose two method types that are based on the nudging algorithm for ODEs and PDEs. The two method types are then applied to two numerical studies. First, data assimilation results for the Lorenz 63 and 96 ODEs are presented. Second, NINNs are applied to a chemically reacting flow example. State of the Art and Perspectives for Reduced Order Methods in Computational Fluid Dynamics: Applications in Industry, Medicine and Environment Gianluigi Rozza, SISSA, Italy We provide the state of the art of Reduced Order Methods (ROM) for parametric Partial Differential Equations (PDEs), and we focus on some perspectives in their current trends and developments, with a special interest in parametric problems arising in offline-online Computational Fluid Dynamics (CFD) with application in industry, medicine and environment. Efficient parametrisations (random inputs, geometry, physics) are very important to be able to properly address an offline-online decoupling of the computational procedures and to allow competitive computational performances. Current ROM developments in CFD include: a better use of stable high fidelity methods, considering also spectral element method, discontinuous Galerkin and finite volume discretisations, to enhance the quality of the reduced model too, and allowing to incorporate turbulent patterns, as well as increasing the Reynolds number and considering compressibility effects; more efficient sampling techniques to reduce the number of the basis functions, retained as snapshots, as well as the dimension of online systems; potential enhancements provided by automatic learning especially in turbulence modelling and in dealing with fast transient dynamics, as well as the guarantee of the stability of the approximation with proper space enrichments or ad hoc segregated solvers. Nonlinear Manifold ROM with Convolutional Autoencoders and Reduced Over-Collocation Method Giovanni Stabile, SISSA, Italy Non-affine parametric dependencies, nonlinearities, and advection-dominated regimes of the model of interest can result in a slow Kolmogorov n-width decay, which precludes the realization of efficient reduced-order models based on Proper Orthogonal Decomposition. Among the possible solutions, there are purely data-driven methods that leverage nonlinear approximation techniques such as autoencoders and their variants to learn a latent representation of the dynamical system, and then evolve it in time with another architecture. Despite their success in many applications where standard linear techniques fail, more has to be done to increase the interpretability of the results, especially outside the training range and not in regimes characterized by an abundance of data. Not to mention that none of the knowledge on the physics of the model is exploited during the predictive phase. In this talk, in order to overcome these weaknesses, we implement a variant of the nonlinear manifold method introduced in previous works with hyper-reduction achieved through reduced over-collocation and teacher-student training of a reduced decoder. We test the methodology on a 2d nonlinear conservation law model and compare the results we would obtain with a purely data-driven method for which the dynamics is evolved in time with a long-short term memory network. The presented work is in collaboration with F. Romor and G. Rozza. Registration-based Model Reduction of Parameterized Advection-Dominated PDEs with Spatio-Parameter Adaptivity Tommaso Taddei, University of Bordeaux & Inria, France We present a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent sharp gradient regions that cannot be adequately represented through linear approximation spaces. Our approach builds on the following ingredients: (i) a general (i.e., independent of the underlying equation) registration procedure for the computation of a parametric mapping that tracks moving features of the solution field; (ii) an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the solution field; (iii) a greedy procedure driven a residual-based error indicator for efficient exploration of the parameter domain; and (iv) an adaptive mesh refinement technique for the definition of an accurate discretization for all parameter values. We present results for a representative nonlinear problem in steady aerodynamics to demonstrate the effectiveness and the mathematical soundness of our proposal. Model Order Reduction in Clinical Applications: Examples and Perspectives Alessandro Veneziani, Emory University Model reduction is a key factor to bring model-driven analysis to the bedside and the operating rooms. As a matter of fact, the computational burden of traditional methods of numerical mechanics (together with not easy integration with available data) prevented a massive use of model-driven studies in clinical practice - beyond the pure proof-of-concept stage. Model reduction brings computational tools to the operating room, enabling rigorous optimization procedures for instance in surgical planning. Different model reduction techniques may fit different problems, with the common denominator of reducing computational costs while preserving accuracy in view of an aggressive integration of model-driven and data-driven numerical tools in healthcare. Efficient Model Reduction of Parametrized Aerodynamics Problems: Error Estimation and Adaptivity Masayuki Yano, University of Toronto, Canada We present goal-oriented model reduction of parametrized nonlinear PDEs, with an emphasis on aerodynamics problems that exhibit a wide range of scales, unsteadiness, and geometry changes. The key ingredients are as follows: an adaptive high-order discontinuous Galerkin (DG) method, which provides stability for convention-dominated flows and controls error in the snapshots; reduced basis (RB) spaces, which provide rapidly convergent approximations of the parametric manifold; the point-wise empirical quadrature procedure (EQP), which provides efficient and reliable hyperreduction of high-order DG methods; the dual-weighted residual (DWR) method, which provides effective error estimate for both the DG snapshots and reduced-order model (ROM); and an adaptive weak greedy algorithm, which simultaneously adapts the DG spaces, RB spaces, and EQP to meet the user-specified output error tolerance in an automated manner. We demonstrate the framework for parametrized aerodynamics problems modeled by the compressible Euler and Reynolds-averaged Navier-Stokes equations. In the offline stage, the adaptive greedy algorithm enables efficient and automated training of ROMs. In the online stage, the ROMs accelerate the computation by several orders of magnitude and also provide a posteriori error estimates. |