Search

This work formulates a new approach to reduced modeling of parameterized, timedependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model. [ABSTRACT FROM AUTHOR]

This paper presents data-driven learning of localized reduced models. Instead of a global reduced basis, the approach employs multiple local approximation subspaces. This localization permits adaptation of a reduced model to local dynamics, thereby keeping the reduced dimension small. This is particularly important for reduced models of nonlinear systems of partial differential equations, where the solution may be characterized by different physical regimes or exhibit high sensitivity to parameter variations. The contribution of this paper is a non-intrusive approach that learns the localized reduced model from snapshot data using operator inference. In the offline phase, the approach partitions the state space into subregions and solves a regression problem to determine localized reduced operators. During the online phase, a local basis is chosen adaptively based on the current system state. The non-intrusive nature of localized operator inference makes the method accessible, portable and applicable to a broad range of scientific problems, including those that use proprietary or legacy high-fidelity codes. We demonstrate the potential for achieving large computational speedups while maintaining good accuracy for a Burgers' equation governing shock propagation in a one-dimensional domain and a phase-field problem governed by the Cahn–Hilliard equation. This article is part of the theme issue 'Data-driven prediction in dynamical systems'. [ABSTRACT FROM AUTHOR]

We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an enabling technology for many computational algorithms used in engineering settings. Our formulation generalizes to the function space PDE setting the Operator Inference method previously developed in [B. Peherstorfer and K. Willcox, Comput. Methods Appl. Mech. Engrg., 306 (2016), pp. 196--215] for systems governed by ordinary differential equations. The method brings together two main elements. First, ideas from projection-based model reduction are used to explicitly parametrize the learned model by low-dimensional polynomial operators which reflect the known form of the governing PDE. Second, supervised machine learning tools are used to infer from data the reduced operators of this physics-informed parametrization. For systems whose governing PDEs contain more general (nonpolynomial) nonlinearities, the learned model performance can be improved through the use of lifting variable transformations, which expose polynomial structure in the PDE. The proposed method is demonstrated on two examples: a heat equation model problem that demonstrates the benefits of the function space formulation in terms of consistency with the underlying continuous truth, and a three-dimensional combustion simulation with over 18 million degrees of freedom, for which the learned reduced models achieve accurate predictions with a dimension reduction of five orders of magnitude and model runtime reduction of up to nine orders of magnitude. [ABSTRACT FROM AUTHOR]

This work presents a data-driven nonintrusive model reduction approach for large-scale time-dependent systems with linear state dependence. Traditionally, model reduction is performed in an intrusive projection-based framework, where the operators of the full model are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a vector. Our nonintrusive approach constructs reduced models directly from trajectories of the inputs and outputs of the full model, without requiring the full-model operators. These trajectories are generated by running a simulation of the full model; our method then infers frequency-response data from these simulated time-domain trajectories and uses the data-driven Loewner framework to derive a reduced model. Only a single time-domain simulation is required to derive a reduced model with the new data-driven nonintrusive approach. We demonstrate our model reduction method on several benchmark examples and a finite element model of a cantilever beam; our approach recovers the classical Loewner reduced models and, for these problems, yields high-quality reduced models despite treating the full model as a black box. [ABSTRACT FROM AUTHOR]

We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive and accurately inferring parameters can be expensive, too. We address both problems by proposing an offline-online strategy for controlling systems with time- varying parameters. During the offline phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reflects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider two numerical test problems in the form of partial differential equations: a convection-diffusion system and a model for flow through a porous medium. We demonstrate on those models that the proposed method successfully stabilizes the system model in the presence of process noise. [ABSTRACT FROM AUTHOR]

This work presents a nonintrusive projection-based model reduction approach for full models based on time-dependent partial differential equations. Projection-based model reduction constructs the operators of a reduced model by projecting the equations of the full model onto a reduced space. Traditionally, this projection is intrusive, which means that the full-model operators are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a given vector; however, in many situations the full model is given as a black box that computes trajectories of the full-model states and outputs for given initial conditions and inputs, but does not provide the full-model operators. Our nonintrusive operator inference approach infers approximations of the reduced operators from the initial conditions, inputs, trajectories of the states, and outputs of the full model, without requiring the full-model operators. Our operator inference is applicable to full models that are linear in the state or have a low-order polynomial nonlinear term. The inferred operators are the solution of a least-squares problem and converge, with sufficient state trajectory data, in the Frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full-model operators. Our numerical results demonstrate operator inference on a linear climate model and on a tubular reactor model with a polynomial nonlinear term of third order. [ABSTRACT FROM AUTHOR]

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification--settings that require repeated model evaluations over different parameter values. [ABSTRACT FROM AUTHOR]

One of the major challenges in the Bayesian solution of inverse problems governed by partial differential equations (PDEs) is the computational cost of repeatedly evaluating numerical PDE models, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. This paper proposes a data-driven projection-based model reduction technique to reduce this computational cost. The proposed technique has two distinctive features. First, the model reduction strategy is tailored to inverse problems: the snapshots used to construct the reduced-order model are computed adaptively from the posterior distribution. Posterior exploration and model reduction are thus pursued simultaneously. Second, to avoid repeated evaluations of the full-scale numerical model as in a standard MCMC method, we couple the full-scale model and the reduced-order model together in the MCMC algorithm. This maintains accurate inference while reducing its overall computational cost. In numerical experiments considering steady-state flow in a porous medium, the data-driven reduced-order model achieves better accuracy than a reduced-order model constructed using the classical approach. It also improves posterior sampling efficiency by several orders of magnitude compared with a standard MCMC method. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In many engineering problems, unknown parameters of a model are inferred in order to make predictions, to design controllers, or to optimize the model. When parameters are distributed (continuous) or very high-dimensional (discrete) and quantities of interest are low-dimensional, parameters need not be fully resolved to make accurate estimates of quantities of interest. In this work, we extend goal-oriented inference--the process of estimating predictions from observed data without resolving the parameter, previously justified theoretically in the linear setting--to Bayesian statistical inference problem formulations with nonlinear experimental and prediction processes. We propose to learn the joint density of data and predictions offline using Gaussian mixture models. When data are observed online, we condition the representation to arrive at a probabilistic description of predictions given observed data. Our approach enables real-time estimation of uncertainty in quantities of interest and renders tractable high-dimensional PDE-constrained Bayesian inference when there exist low-dimensional output quantities of interest. We demonstrate the method on a realistic problem in carbon capture and storage for which existing methods of Bayesian parameter estimation are intractable. [ABSTRACT FROM AUTHOR]

An inverse problem seeks to infer unknown model parameters using observed data. We consider a goal-oriented inverse problem , where the goal of inferring parameters is to use them in predicting a quantity of interest (QoI). Recognizing that multiple models of varying fidelity and computational cost may be available to describe the physical system, we formulate a goal-oriented model adaptivity approach that leverages multiple models while controlling the error in the QoI prediction. In particular, we adaptively form a mixed-fidelity model by using models of different levels of fidelity in different subregions of the domain. Taking the solution of the inverse problem with the highest-fidelity model as our reference QoI prediction, we derive an adjoint-based third-order estimate for the QoI error from using a lower-fidelity model. Localization of this error then guides the formation of mixed-fidelity models. We demonstrate the method for example problems described by convection–diffusion–reaction models. For these examples, our mixed-fidelity models use the high-fidelity model over only a small portion of the domain, but result in QoI estimates with small relative errors. We also demonstrate that the mixed-fidelity inverse problems can be cheaper to solve and less sensitive to the initial guess than the high-fidelity inverse problems. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace. [ABSTRACT FROM AUTHOR]

• Multifidelity method that enables estimation of failure probabilities for expensive-to- evaluate models. • For turbulent jet application, it reduces the CPU time to compute the biasing density by 65%. • The new approach draws from information fusion, importance sampling and multifidelity modeling. • The fused estimator is optimal in that it has minimal variance among all possible combinations of the estimators. This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which 105 samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65% while achieving a similar coefficient of variation. [ABSTRACT FROM AUTHOR]

This work presents a non-intrusive model reduction method to learn low-dimensional models of dynamical systems with non-polynomial nonlinear terms that are spatially local and that are given in analytic form. In contrast to state-of-the-art model reduction methods that are intrusive and thus require full knowledge of the governing equations and the operators of a full model of the discretized dynamical system, the proposed approach requires only the non-polynomial terms in analytic form and learns the rest of the dynamics from snapshots computed with a potentially black-box full-model solver. The proposed method learns operators for the linear and polynomially nonlinear dynamics via a least-squares problem, where the given non-polynomial terms are incorporated on the right-hand side. The least-squares problem is linear and thus can be solved efficiently in practice. The proposed method is demonstrated on three problems governed by partial differential equations, namely the diffusion–reaction Chafee–Infante model, a tubular reactor model for reactive flows, and a batch-chromatography model that describes a chemical separation process. The numerical results provide evidence that the proposed approach learns reduced models that achieve comparable accuracy as models constructed with state-of-the-art intrusive model reduction methods that require full knowledge of the governing equations. • Method learns low-dimensional models of dynamical systems with non-polynomial nonlinear terms. • Requires non-polynomial terms analytically; learns the other dynamics from snapshots. • Can achieve comparable accuracy as state-of-the-art intrusive model reduction methods on numerical examples. [ABSTRACT FROM AUTHOR]

We present Lift & Learn , a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system's governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh–Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model. • Lift & Learn: A new physics-informed machine learning method for nonlinear PDEs. • Lifting transformations expose quadratic structure that is exploited for learning. • Lift & Learn models generalize as well as intrusive projection-based reduced models. [ABSTRACT FROM AUTHOR]