Karen Willcox

Boris Kramer, Karen Willcox

This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model---it uses a change of variables, but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential algebraic equations, depending on the problem and lifting transformation. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be pre-computed. Thus, a key benefit of the approach is that there is no need for additional approximations of nonlinear terms, in contrast with existing nonlinear model reduction methods requiring sparse sampling or hyper-reduction. Application of the lifting and POD model reduction to the FitzHugh-Nagumo benchmark problem and to a tubular reactor model with Arrhenius reaction terms shows that the approach is competitive in terms of reduced model accuracy with state-of-the-art model reduction via POD and discrete empirical interpolation, while having the added benefits of opening new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.

Nicole Aretz, Karen Willcox

This paper introduces a multifidelity formulation that reduces the computational cost of the proper orthogonal decomposition (POD) of a high-fidelity model by leveraging data from cheaper, lower-fidelity models. POD is a prevalent technique for extracting a low-dimensional basis from training data to achieve subsequent dimension reduction or reduced-order modeling. In scientific and engineering applications, the training data are typically numerical snapshot solutions of a high-fidelity model, and computation of a sufficiently rich snapshot set can be prohibitively expensive, especially when sampling over a high-dimensional parameter space. Insufficient snapshot training data risks overfitting and poor generalizability of the POD basis to outside the training regime. Our multifidelity POD (MFPOD) formulation reallocates computational budget to cheaper, low-fidelity models that can be sampled more extensively. MFPOD then weights high- and low-fidelity snapshot data via a control-variate formulation to guarantee an unbiased estimate of the expected high-fidelity least-squares projection error. The MFPOD subspace is chosen to minimize the estimate of this projection error, and converges in probability to the same subspace as single-fidelity POD in the limit of an arbitrarily large budget. For restrictive computational budgets, the MFPOD cost function has (under some assumptions) lower variance than the POD cost function, which makes the MFPOD subspace more robust against variations in the training data and thus less prone to overfitting. For a numerical example modeling the velocity of the Pine Island glacier, MFPOD achieves the same accuracy as single-fidelity POD with an order of magnitude reduction in the offline computational cost of snapshot generation.

Alessio Spantini, Tiangang Cui, Karen Willcox et al.

We propose optimal dimensionality reduction techniques for the solution of goal-oriented linear-Gaussian inverse problems, where the quantity of interest (QoI) is a function of the inversion parameters. These approximations are suitable for large-scale applications. In particular, we study the approximation of the posterior covariance of the QoI as a low-rank negative update of its prior covariance, and prove optimality of this update with respect to the natural geodesic distance on the manifold of symmetric positive definite matrices. Assuming exact knowledge of the posterior mean of the QoI, the optimality results extend to optimality in distribution with respect to the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose approximation of the posterior mean of the QoI as a low-rank linear function of the data, and prove optimality of this approximation with respect to a weighted Bayes risk. Both of these optimal approximations avoid the explicit computation of the full posterior distribution of the parameters and instead focus on directions that are well informed by the data and relevant to the QoI. These directions stem from a balance among all the components of the goal-oriented inverse problem: prior information, forward model, measurement noise, and ultimate goals. We illustrate the theory using a high-dimensional inverse problem in heat transfer.

Rudy Geelen, Laura Balzano, Stephen Wright et al.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Rudy Geelen, Laura Balzano, Karen Willcox

We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem's intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques.

Nicole Aretz, Karen Willcox

This paper presents a data-driven, nested Operator Inference (OpInf) approach for learning physics-informed reduced-order models (ROMs) from snapshot data of high-dimensional dynamical systems. The approach exploits the inherent hierarchy within the reduced space to iteratively construct initial guesses for the OpInf learning problem that prioritize the interactions of the dominant modes. The initial guess computed for any target reduced dimension corresponds to a ROM with provably smaller or equal snapshot reconstruction error than with standard OpInf. Moreover, our nested OpInf algorithm can be warm-started from previously learned models, enabling versatile application scenarios involving dynamic basis and model form updates. We demonstrate the performance of our algorithm on a cubic heat conduction problem, with nested OpInf achieving a four times smaller error than standard OpInf at a comparable offline time. Further, we apply nested OpInf to a large-scale, parameterized model of the Greenland ice sheet where, despite model form approximation errors, it learns a ROM with, on average, 3% error and computational speed-up factor above 19,000.

Vignesh Sella, Julie Pham, Karen Willcox et al.

Surrogate modeling for systems with high-dimensional quantities of interest remains challenging, particularly when training data are costly to acquire. This work develops multifidelity methods for multiple-input multiple-output linear regression targeting data-limited applications with high-dimensional outputs. Multifidelity methods integrate many inexpensive low-fidelity model evaluations with limited, costly high-fidelity evaluations. We introduce two projection-based multifidelity linear regression approaches that leverage principal component basis vectors for dimensionality reduction and combine multifidelity data through: (i) a direct data augmentation using low-fidelity data, and (ii) a data augmentation incorporating explicit linear corrections between low-fidelity and high-fidelity data. The data augmentation approaches combine high-fidelity and low-fidelity data into a unified training set and train the linear regression model through weighted least squares with fidelity-specific weights. Various weighting schemes and their impact on regression accuracy are explored. The proposed multifidelity linear regression methods are demonstrated on approximating the surface pressure field of a hypersonic vehicle in flight. In a low-data regime of no more than ten high-fidelity samples, multifidelity linear regression achieves approximately 3% - 12% improvement in median accuracy compared to single-fidelity methods with comparable computational cost.

Thomas O'Leary-Roseberry, Xiaosong Du, Anirban Chaudhuri et al.

We propose a scalable framework for the learning of high-dimensional parametric maps via adaptively constructed residual network (ResNet) maps between reduced bases of the inputs and outputs. When just few training data are available, it is beneficial to have a compact parametrization in order to ameliorate the ill-posedness of the neural network training problem. By linearly restricting high-dimensional maps to informed reduced bases of the inputs, one can compress high-dimensional maps in a constructive way that can be used to detect appropriate basis ranks, equipped with rigorous error estimates. A scalable neural network learning framework is thus to learn the nonlinear compressed reduced basis mapping. Unlike the reduced basis construction, however, neural network constructions are not guaranteed to reduce errors by adding representation power, making it difficult to achieve good practical performance. Inspired by recent approximation theory that connects ResNets to sequential minimizing flows, we present an adaptive ResNet construction algorithm. This algorithm allows for depth-wise enrichment of the neural network approximation, in a manner that can achieve good practical performance by first training a shallow network and then adapting. We prove universal approximation of the associated neural network class for $L^2_ν$ functions on compact sets. Our overall framework allows for constructive means to detect appropriate breadth and depth, and related compact parametrizations of neural networks, significantly reducing the need for architectural hyperparameter tuning. Numerical experiments for parametric PDE problems and a 3D CFD wing design optimization parametric map demonstrate that the proposed methodology can achieve remarkably high accuracy for limited training data, and outperformed other neural network strategies we compared against.

Elizabeth Qian, Ionut-Gabriel Farcas, Karen Willcox

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, Data-driven operator inference for non-intrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering, 306 (2016)] 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 (non-polynomial) 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.

Peter Benner, Pawan Goyal, Boris Kramer et al.

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 in 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.

Elizabeth Qian, Boris Kramer, Benjamin Peherstorfer et al.

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.

Renee Swischuk, Boris Kramer, Cheng Huang et al.

This paper presents a physics-based data-driven method to learn predictive reduced-order models (ROMs) from high-fidelity simulations, and illustrates it in the challenging context of a single-injector combustion process. The method combines the perspectives of model reduction and machine learning. Model reduction brings in the physics of the problem, constraining the ROM predictions to lie on a subspace defined by the governing equations. This is achieved by defining the ROM in proper orthogonal decomposition (POD) coordinates, which embed the rich physics information contained in solution snapshots of a high-fidelity computational fluid dynamics (CFD) model. The machine learning perspective brings the flexibility to use transformed physical variables to define the POD basis. This is in contrast to traditional model reduction approaches that are constrained to use the physical variables of the high-fidelity code. Combining the two perspectives, the approach identifies a set of transformed physical variables that expose quadratic structure in the combustion governing equations and learns a quadratic ROM from transformed snapshot data. This learning does not require access to the high-fidelity model implementation. Numerical experiments show that the ROM accurately predicts temperature, pressure, velocity, species concentrations, and the limit-cycle amplitude, with speedups of more than five orders of magnitude over high-fidelity models. Our ROM simulation is shown to be predictive 200% past the training interval. Moreover, ROM-predicted pressure traces accurately match the phase of the pressure signal and yield good approximations of the limit-cycle amplitude.

Boris Kramer, Alexandre Noll Marques, Benjamin Peherstorfer et al.

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 $10^5$ 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.