1 Introduction
The emerging field of scientific machine learning brings together the perspectives of physicsbased modeling and datadriven learning. In the field of fluid dynamics, physicsbased modeling and simulation have played a critical role in advancing scientific discovery and driving engineering innovation in domains as diverse as biomedical engineering (Yin et al. 2010; Nordsletten et al. 2011), geothermal modeling (O’Sullivan et al. 2001; Cui et al. 2011), and aerospace (Spalart and Venkatakrishnan 2016). These advances are based on decades of mathematical and algorithmic developments in computational fluid dynamics (CFD). Scientific machine learning builds upon these rigorous physicsbased foundations while seeking to exploit the flexibility and expressive modeling capabilities of machine learning (Baker et al. 2019). This paper presents a scientific machine learning approach that blends datadriven learning with the theoretical foundations of physicsbased model reduction. This creates the capability to learn predictive reducedorder models (ROMs) that provide approximate predictions of complex physical phenomena while exhibiting several orders of magnitude computational speedup over CFD.
Projectionbased model reduction considers the class of problems for which the governing equations are known and for which we have a highfidelity (e.g., CFD) model (Antoulas 2005; Benner et al. 2015). The goal is to derive a ROM that has lower complexity and yields accurate solutions with reduced computation time. Projectionbased approaches define a lowdimensional manifold on which the dominant dynamics evolve. This manifold may be defined as a function of the operators of the highfidelity model, as in interpolatory methods that employ a Krylov subspace (Bai 2002; Freund 2003), or it may be determined empirically from representative highfidelity simulation data, as in the proper orthogonal decomposition (POD) (Lumley 1967; Sirovich 1987; Berkooz et al. 1993). The POD has been particularly successful in fluid dynamics, dating back to the early applications in unsteady flows and turbulence modeling (Sirovich 1987; Deane et al. 1991; Gatski and Glauser 1992), and in unsteady fluidstructure interaction (Dowell and Hall 2001).
Model reduction methods have advanced to include error estimation
(Veroy et al. 2003; Veroy and Patera 2005; Grepl and Patera 2005; Rozza et al. 2008) and to address parametric and nonlinear problems (Barrault et al. 2004; Astrid et al. 2008; Chaturantabut and Sorensen 2010; Carlberg et al. 2013), yet the intrusive nature of the methods has limited their impact in practical applications. When legacy or commercial codes are used, as is often the case for CFD applications, it can be difficult or impossible to implement classical projectionbased model reduction. Blackbox surrogate modeling instead derives the ROM by fitting to simulation data; such methods include response surfaces and Gaussian process models, long used in engineering, as well as machine learning surrogate models. These methods are powerful and often yield good results, but since the approximations are based on generic datafit representations, they are not equipped with the guarantees (e.g., stability guarantees, error estimators) that accompany projectionbased ROMs. Nonlinear system identification techniques seek to illuminate the black box by discovering the underlying physics of a system from data (Brunton et al. 2016). However, when the governing dynamics are known and simulation data are available, reduced models may be directly tailored to the specific dynamics without access to the details of the largescale CFD code.Building on the work in Swischuk et al. (2020), this paper presents a nonintrusive alternative to blackbox surrogate modeling. We use the Operator Inference method of Peherstorfer and Willcox (2016)
to learn a ROM from simulation data; the structure of the ROM is defined by the known governing equations combined with the theory of projectionbased model reduction. Our approach may be termed ‘glassbox modeling’, as the targeted dynamics are known via the governing partial differential equations that define the problem of interest but the inner workings of the CFD code are not accessed. This paper extends our prior work by formally introducing regularization to the approach, which is critical to avoid overfitting for problems with complex dynamics, as is the case for the combustion example considered here. A second contribution of this paper is a scalable implementation of the approach, which is available via an opensource implementation. Section
2 presents the methodology and regularization approach and describes the scalable implementation. Section 3 presents numerical results for a singleinjector combustion problem and Section 4 concludes the paper.2 Methodology
This section begins with an overview the Operator Inference approach in Section 2.1. Section 2.2 augments Operator Inference with a new regularization formulation, posed as an optimization problem, and presents a complete algorithm for regularization selection and model learning. In Section 2.3, we discuss a scalable implementation of the algorithm, which can then be applied to CFD problems of high dimension.
2.1 Operator Inference
We target problems governed by systems of nonlinear partial differential equations. Consider the governing equations of the system of interest written, after spatial discretization, in semidiscrete form
(1) 
where
is the state vector discretized over
points in space at time , denotes the inputs at time , typically related to boundary conditions or forcing terms, and are respectively the initial and final time, and is the given initial condition. We refer to Eq. (1) as the fullorder model (FOM) and note that it has been written to have a polynomial structure: are constant terms; are the terms that are linear in the state , with the discretized operator ; are the terms that are quadratic in , with ; and are the terms that are linear in the input , with . This polynomial structure arises in three ways: (1) it may be an attribute of the governing equations; (2) it may be exposed via variable transformations; or (3) it may be derived by introducing auxiliary variables through the process of lifting. As examples of each: (1) the incompressible NavierStokes equations have a quadratic form; (2) the Euler equations can be transformed to quadratic form by using pressure, velocity, and specific volume as state variables; (3) a nonlinear tubular reactor model with Arrhenius reaction terms can be written in quadratic form via the lifting transformation shown in Kramer and Willcox (2019) that introduces six auxiliary variables.A projectionbased reducedorder model (ROM) of Eq. (1) preserves the polynomial structure (Benner et al. 2015). Approximating the highdimensional state in a lowdimensional basis , with , we write , where is the reduced state. Using a Galerkin projection, this yields the intrusive ROM of Eq. (1):
where , , , and are the ROM operators corresponding to the FOM operators , , , and , respectively. The ROM is intrusive because computing these ROM operators requires access to the discretized FOM operators, which typically entails intrusive queries and/or access to source code.
The nonintrusive Operator Inference (OpInf) approach proposed by Peherstorfer and Willcox (2016) parallels the intrusive projectionbased ROM setting, but learns ROMs from simulation data without direct access to the FOM operators. Recognizing that the intrusive ROM has the same polynomial form as Eq. (1), OpInf uses a datadriven regression approach to derive a ROM of Eq. (1) as
(2) 
where , , , and are determined by solving a datadriven regression problem, and is the state of the OpInf ROM.^{1}^{1}1From here on we use to indicate a compact Kronecker product with only the unique quadratic terms (, , , ); for matrices, the product is applied columnwise.
OpInf solves a regression problem to find reduced operators that yield the ROM that best matches projected snapshot data in a minimumresidual sense. Mathematically, OpInf solves the leastsquares problem
(3) 
where is the dataset used to drive the learning with denoting a reducedstate snapshot at timestep , are the associated time derivatives, and is the collection of inputs corresponding to the data with . To generate the dataset, we employ the following steps: (1) Collect a set of highfidelity state snapshots by solving the original highfidelity model at times with inputs . (2) Apply a variable transformation to obtain snapshots of the transformed variables. Here is the map representing an invertible transformation (e.g., from density to specific volume) or a lifting transformation (Qian et al. 2020). (3) Compute the proper orthogonal decomposition (POD) basis of the transformed snapshots.^{2}^{2}2Or any other lowdimensional basis as desired. (4) Project the transformed snapshots onto the POD subspace as . (5) Estimate projected time derivative information . The training period for which we have data is a subset of the full time domain of interest ; results from the ROM over will be entirely predictive.
Eq. (3) can also be written in matrix form as
(4) 
where
(unknown operators)  
(known data)  
(snapshots)  
(time derivatives)  
(inputs) 
and where and is the length column vector with all entries set to unity. The OpInf leastsquares problem Eq. (3) is therefore linear in the coefficients of the unknown ROM operators , , , and .
Note that the OpInf approach permits us to compute the ROM operators , , , and without explicit access to the original highdimensional operators , , , and . This point is key since we apply variable transformations only to the snapshot data, not to the operators or the underlying model. Thus, even in a setting where deriving a classical intrusive ROM might be possible, the OpInf approach enables us to work with variables other than those used for the original highfidelity discretization. In Section 3 we will see the importance of this for a reacting flow application. We also note that under some conditions, OpInf recovers the intrusive POD ROM (Peherstorfer and Willcox 2016; Peherstorfer 2019).
2.2 Regularization
Eq. (4) decouples into independent linear leastsquares problems, one for each of the rows of (Peherstorfer and Willcox 2016). Each problem is generally overdetermined, but is also noisy due to error in the numerically estimated time derivatives , model misspecification (e.g., if the system is not truly quadratic), and truncated POD modes that leave some system dynamics unresolved. The ROMs resulting from Eq. (4) can thus suffer from overfitting the operators to the data and therefore exhibit poor predictive performance over the time domain of interest .
To avoid overfitting, we introduce a Tikhonov regularization to the subproblems in Eq. (4), which then become
(5) 
where is the th row of (the th column of ), is the th row of , and is a fullrank regularizer. Each regularized subproblem in Eq. (5) admits a unique solution since, by construction, the augmented data matrix is taller than it is wide and has full column rank.
An regularizer , and
the identity matrix, penalizes each entry of the inferred ROM operators
, , , and , thereby driving the ROM toward the globally stable system. An appropriate value for the scalar hyperparameter
, which balances the minimization between the data fit and the regularization, must be chosen with care. The ideal produces a ROM that minimizes some error metric over the full time domain ; however, since data are only available for the smaller training domain , we choose so that the resulting ROM minimizes error over while maintaining a bound on the integrated POD coefficients over . That is, we require that any produced by the ROM trained with parameter satisfy for some . This in turn ensures a bound on the magnitude of the entries of the highdimensional state :where is the th element of the th POD basis vector. Note that could be chosen with the intent of imposing a particular bound on the since the sums can be precomputed.
Algorithm 1 details our regularized OpInf procedure, in which we choose as a multiple of the maximum absolute entry of the projected training data . This particular strategy for selecting could be replaced with a crossvalidation or resampling grid search technique, but our experiments in this vein did not produce robust results. The training error in step 14 may compare and directly in the reduced space (e.g., with a matrix norm or an norm), or it may be replaced with a more targeted comparison of some quantity of interest.
2.3 Scalable Implementation
The steps of Algorithm 1 are highly modular and amenable to large problems. The variable transformation of step 2 consists of elementary computations on the original snapshot matrix . To compute the rank POD basis in step 3
, we use a randomized singularvalue decomposition (SVD) algorithm requiring
operations (Halko et al. 2011); since , the leading order behavior is . The projection in step 4 costs operations. Note that is small compared to , as typically – and –. The time derivatives in step 5 may be provided by the fullorder solver or estimated, e.g., with finite differences of the columns of . In the latter case, the cost is . The computational cost of steps 2–6 is therefore .Solving the regularized problems Eq. (5) in steps 9 and 16 requires computing the SVD of the augmented data matrix — equivalently, the eigendecomposition of — which costs operations (Demmel 1997). This SVD is then applied to each of the augmented vectors to produce the unique solution. The ROM integration in step 10 can be carried out with any timestepping scheme; for explicit methods, evaluating the ROM at a single point, i.e., computing the righthand side of Eq. (2), costs operations. The total cost of evaluating the subroutine TrainError is therefore which, importantly, is independent of . The minimization in step 15 is carried out with a bisectiontype search method, which enables fewer total evaluations of the subroutine than a blind grid search. However, an initial coarse grid search is useful for identifying an appropriate search interval .
3 Results
This section applies regularized OpInf to a singleinjector combustion problem, studied previously by Swischuk et al. (2020), on the twodimensional computational domain shown in Figure 1. Section 3.1 describes the governing dynamics, a set of highfidelity data obtained from a CFD code, and the variable transformations used to produce training data for learning reduced models with Algorithm 1. The resulting OpInf ROM performance is analyzed in Section 3.2 and compared to a stateoftheart intrusive model reduction method in Section 3.3.
3.1 Problem setup
The combustion dynamics for this problem are governed by conservation laws
(6) 
where are the conservative variables, are the inviscid flux terms, are the viscous flux terms, and are the source terms. Here is the density , and are the and velocity , is the total energy , and is the th species mass fraction with . The chemical species are CH, O, HO, and CO, which follow a global onestep chemical reaction (Westbrook and Dryer 1981). See Harvazinski et al. (2015) for more details on the governing equations.
At the downstream end of the combustor, we impose a nonreflecting boundary condition while maintaining the chamber pressure via
(7) 
where Pa and Hz. The top and bottom wall boundary conditions are noslip conditions, and for the upstream boundary we impose a constant mass flow at the inlets.
Swischuk et al. (2020) show that if the governing equations (6) are transformed to be written in the specific volume variables, many of the terms take a quadratic form. Following that idea, we choose as learning variables the transformed and augmented state where is the pressure , is the temperature , is the specific volume , and are the species molar concentrations given by with the molar mass of the th species . As shown in Swischuk et al. (2020), the equations for , , and are exactly quadratic in , while the remaining equations are quadratic with some nonpolynomial terms in . Note that, differently from Swischuk et al. (2020), here is chosen to contain specific volume, pressure, and temperature, even though only two of the three quantities are needed to fully define the highfidelity model (and the equation of state then defines the third). We augment the learning variables in this way because doing so exposes the quadratic form while also directly targeting the variables that are of primary interest for assessing ROM performance. In particular, the resulting ROMs provide more accurate predictions of temperature when temperature is included explicitly as a learning variable. We can do this since the transformations are applied only to the snapshot data, not to the CFD model itself. This flexibility is a major advantage of the nonintrusive OpInf approach in comparison to traditional intrusive projectionbased model reduction methods.
To generate highfidelity training data, we use the finitevolume based General Equation and Mesh Solver (GEMS) (Harvazinski et al. 2015) to solve for the variables over cells, resulting in snapshots with entries each. The snapshots are computed for 60,000 time steps beyond the initial condition with a temporal discretization of s, from s to s. The computational cost of computing this dataset is approximately 1,200 CPU hours on two computing nodes, each of which contains two Haswell CPUs at GHz and cores per node.
Scaling is an essential aspect of successful model reduction and is particularly critical for this problem due to the wide range of scales across variables. After transforming the GEMS snapshot data to the learning variables , the species molar concentrations are scaled to , and all other variables are scaled to . This scaling ensures that null velocities and null molar concentrations are preserved. For example, some upstream regions of the injector have zero methane concentration at all times. By construction, the POD basis vectors and thus the ROM predictions will preserve those zero concentration values.
We implement Algorithm 1 via the rom_operator_inference Python package,^{3}^{3}3See https://github.com/WillcoxResearchGroup/romoperatorinferencePython3. which is built on NumPy, SciPy, and scikitlearn (Walt et al. 2011; Virtanen et al. 2020; Buitinck et al. 2013). The time derivatives in step 5 of Algorithm 1 are estimated with fourthorder finite differences, and the leastsquares problems in step 9 are solved by the divideandconquer LAPACK routine DGELSD. The learned ROMs are integrated in step 10 with the explicit, adaptive, fourthorder RungeKutta scheme RK45, and the error evaluation of step 14 uses the norm in the reduced space. To minimize the scalar function TrainError in step 15, we use Brent’s method (Brent 2002) with enforced bounds . The code and details are publicly available at https://github.com/WillcoxResearchGroup/ROMOpInfCombustion2D.
3.2 Sensitivity to Training Data
We study the sensitivity of our approach to the training data by varying the number of snapshots used to compute the POD basis and learn the OpInf ROM. Specifically, we consider the three cases where we use the first , , and snapshots from GEMS as training data sets. In each case we compute the POD basis, and to select an appropriate reduced dimension , we compute the cumulative energy based on the POD singular values: , where are the singular values of the learning variable snapshot matrix (see Figure 2). Specifically, we choose the minimal such that is greater than a fixed threshold energy. Table 3.2 shows that is increasing linearly with the number of snapshots, indicating that the basis is not being saturated as additional information is incorporated. This is an indication of the challenging nature of this application, due to the rich and complex dynamics. It is also an indication of the importance of having sufficient training data.
The basis size required to reach a given cumulative energy level increases linearly with the number of snapshots in the training set, .
Cumulative energy  

Figure 3 plots the GEMS and OpInf ROM results for pressure, temperature and velocity predictions over time at the monitor locations in Figure 1.^{4}^{4}4See https://github.com/WillcoxResearchGroup/ROMOpInfCombustion2D for additional results. While it can be misleading to assess accuracy based on predictions at a single spatial point, these plots reveal several representative insights. First, we see the importance of the training data—as the amount of training data increases, the ROM predictions change significantly and generally (but not always) improve. This is yet another indication of the complexity of the dynamics we are aiming to approximate. Second, we see that good ROM performance over the training region is not sufficient to produce good predictive behavior. In particular, the ROM with only training snapshots can accurately repredict the training dynamics, but shows large errors as it predicts further beyond the training horizon. Third, we see that pressure and velocity are better approximated by the ROM than is temperature. The effects of the 5,000 Hz downstream pressure forcing are clearly visible in the pressure and velocity traces. The temperature dynamics, on the other hand, are more irregular, and exhibit shortterm swings exceeding 1,000 K. This is because the temperature profile is influenced by both the flow dynamics and the local chemical reactions, which in combination lead to a highly nonlinear and multiscale behavior. The ROM cannot accurately predict the detailed variations, but with training snapshots, the ROM does an adequate job of predicting the general trends of temperature evolution beyond the training horizon.
Figure 4 plots integrated quantities as a function of time: the temperature averaged over the spatial domain, and the CH and O concentrations integrated over the spatial domain. These measures give a more global sense of the ROM predictive accuracy. In each case, the ROMs are able to accurately repredict the training data and capture much of the overall system behavior in the prediction phase.
3.3 Comparison to PODDEIM
We now compare regularized OpInf to a stateoftheart nonlinear model reduction method that uses a leastsquares PetrovGalerkin POD projection coupled with the discrete empirical interpolation method (DEIM)
(Chaturantabut and Sorensen 2010), as implemented for the same combustion problem in Huang et al. (2019, 2018). This PODDEIM method is intrusive—it requires nonlinear residual evaluations of the GEMS code at sparse discrete interpolation points. This also increases the computational cost of solving the PODDEIM ROM in comparison to the OpInf ROM: integrating a PODDEIM ROM with for 6,000 time steps of size s takes approximately minutes on two nodes, each with two Haswell CPUs processors at GHz and cores per node; for OpInf, using Python and a single CPU on an AMD EPYC 7,702 64core processor at GHz with TB RAM, we solve Eq. (5) with training snapshots and POD modes in approximately s and integrate the resulting OpInf ROM for 60,000 time steps of size s in approximately s. While these measurements are made on different hardware, and though the execution time for PODDEIM can be improved with optimal load balancing, the difference in execution times (30 minutes versus 0.5 seconds) is representative and illustrates one of the advantages over PODDEIM of the polynomial form employed in the OpInf approach.Figure 5 compares select GEMS outputs to PODDEIM and OpInf ROM outputs, with each ROM trained on training snapshots and the ROM dimension chosen to achieve a cumulative energy of . Both approaches reconstruct the training data well and maintain appropriate pressure oscillation frequencies, but they struggle to predict the erratic temperature dynamics beyond the training horizon. The figure shows that, for this monitor location, the temperature signal over the training region is not particularly representative of the dynamics observed at later times.
Figures 6, 7 and 8 show, respectively, fulldomain results for the pressure, temperature, and molar concentration of CH. The figures show the solution at time instants within the training regime, at the end of the training regime, and into the prediction regime. As with the point traces shown earlier, we see that the ROMs have impressive accuracy over the training region, but lose accuracy as they attempt to predict dynamics beyond the training horizon. Pressure is again well predicted, but the temperature and CH concentration fields predicted at s are not pointwise accurate. However, many of the coherent features are reasonably predicted, especially the recirculation zone dynamics near the dump plane ( in Figure 1) shown in the temperature fields.
4 Conclusions
The presented scientific machine learning approach is broadly applicable to problems where the governing equations are known but access to the highfidelity simulation code is limited. The approach is computationally as accessible as blackbox surrogate modeling while achieving the accuracy of intrusive projectionbased model reduction. While the conclusions drawn from the numerical studies apply to the singleinjector combustion example, they are relevant and likely apply to other problems. First, the quality and quantity of the training data are critical to the success of the method. Second, regularization is essential to avoid overfitting. Third, achieving a low error over the training regime is not necessarily indicative of a reduced model with good predictive capability. This emphasizes the importance of the training data. Fourth, physical quantities that exhibit largescale coherent structures (e.g., pressure) are more accurately predicted by a reducedorder model than quantities that exhibit multiscale behavior (e.g., temperature, species concentrations). Fifth, a significant advantage of the datadriven learning aspects of the approach is that the reduced model may be derived in any variables. This includes the possibility to include redundancy in the learning variables (e.g., to include both pressure and temperature). Overall, this paper illustrates the power and effectiveness of learning from data through the lens of physicsbased models as a physicsgrounded alternative to blackbox machine learning.
Acknowledgements
This work has been supported in part by the Air Force Center of Excellence on MultiFidelity Modeling of Rocket Combustor Dynamics under award FA95501710195, and the US Department of Energy AEOLUS MMICC center under award DESC0019303.
References
 Antoulas (2005) Antoulas AC. 2005. Approximation of largescale dynamical systems. Philadelphia, PA: SIAM.
 Astrid et al. (2008) Astrid P, Weiland S, Willcox K, Backx T. 2008. Missing point estimation in models described by proper orthogonal decomposition. IEEE Transactions on Automatic Control. 53(10):2237–2251.
 Bai (2002) Bai Z. 2002. Krylov subspace techniques for reducedorder modeling of largescale dynamical systems. Applied Numerical Mathematics. 43(1–2):9–44.

Baker et al. (2019)
Baker N, Alexander F, Bremer T, Hagberg A, Kevrekidis Y, Najm H, Parashar M, Patra A, Sethian J, Wild S, et al. 2019. Workshop report on basic research needs for scientific machine learning: Core technologies for artificial intelligence. USDOE Office of Science (SC), Washington, DC (United States). Report No:.
https://www.osti.gov/servlets/purl/1478744.  Barrault et al. (2004) Barrault M, Maday Y, Nguyen NC, Patera AT. 2004. An ‘empirical interpolation’ method: application to efficient reducedbasis discretization of partial differential equations. Comptes Rendus Mathematique. 339(9):667–672.
 Benner et al. (2015) Benner P, Gugercin S, Willcox K. 2015. A survey of projectionbased model reduction methods for parametric dynamical systems. SIAM Review. 57(4):483–531.
 Berkooz et al. (1993) Berkooz G, Holmes P, Lumley J. 1993. The proper orthogonal decomposition in the analysis of turbulent flows. Annual Review of Fluid Mechanics. 25:539–575.
 Brent (2002) Brent RP. 2002. Algorithms for minimization without derivatives. New York, NY: Dover Publications.
 Brunton et al. (2016) Brunton SL, Proctor JL, Kutz JN. 2016. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences. 113(15):3932–3937.
 Buitinck et al. (2013) Buitinck L, Louppe G, Blondel M, Pedregosa F, Mueller A, Grisel O, Niculae V, Prettenhofer P, Gramfort A, Grobler J, et al. 2013. API design for machine learning software: Experiences from the scikitlearn project. In: ECML PKDD Workshop: Languages for Data Mining and Machine Learning. p. 108–122.
 Carlberg et al. (2013) Carlberg K, Farhat C, Cortial J, Amsallem D. 2013. The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows. Journal of Computational Physics. 242:623–647.
 Chaturantabut and Sorensen (2010) Chaturantabut S, Sorensen DC. 2010. Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing. 32(5):2737–2764.
 Cui et al. (2011) Cui T, Fox C, O’Sullivan MJ. 2011. Bayesian calibration of a largescale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm. Water Resources Research. 47(10).
 Deane et al. (1991) Deane A, Kevrekidis I, Karniadakis GE, Orszag S. 1991. Lowdimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Physics of Fluids A: Fluid Dynamics. 3(10):2337–2354.
 Demmel (1997) Demmel JW. 1997. Applied numerical linear algebra. Philadelphia, PA: SIAM.
 Dowell and Hall (2001) Dowell EH, Hall KC. 2001. Modeling of fluidstructure interaction. Annual Review of Fluid Mechanics. 33(1):445–490.
 Freund (2003) Freund R. 2003. Model reduction methods based on Krylov subspaces. Acta Numerica. 12:267–319.
 Gatski and Glauser (1992) Gatski T, Glauser M. 1992. Proper orthogonal decomposition based turbulence modeling. In: Instability, transition, and turbulence. Springer; p. 498–510.
 Grepl and Patera (2005) Grepl M, Patera A. 2005. A posteriori error bounds for reducedbasis approximations of parametrized parabolic partial differential equations. ESAIMMathematical Modelling and Numerical Analysis (M2AN). 39(1):157–181.
 Halko et al. (2011) Halko N, Martinsson PG, Tropp JA. 2011. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review. 53(2):217–288.
 Harvazinski et al. (2015) Harvazinski ME, Huang C, Sankaran V, Feldman TW, Anderson WE, Merkle CL, Talley DG. 2015. Coupling between hydrodynamics, acoustics, and heat release in a selfexcited unstable combustor. Physics of Fluids. 27(4):045102.
 Huang et al. (2019) Huang C, Duraisamy K, Merkle CL. 2019. Investigations and improvement of robustness of reducedorder models of reacting flow. AIAA Journal. 57(12):5377–5389.
 Huang et al. (2018) Huang C, Xu J, Duraisamy K, Merkle C. 2018. Exploration of reducedorder models for rocket combustion applications. In: 2018 AIAA Aerospace Sciences Meeting; Orlando, FL. Paper AIAA20181183.
 Kramer and Willcox (2019) Kramer B, Willcox K. 2019. Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition. AIAA Journal. 57(6):2297–2307.
 Lumley (1967) Lumley J. 1967. The structures of inhomogeneous turbulent flow. Atmospheric Turbulence and Radio Wave Propagation:166–178.
 Nordsletten et al. (2011) Nordsletten D, McCormick M, Kilner P, Hunter P, Kay D, Smith N. 2011. Fluid–solid coupling for the investigation of diastolic and systolic human left ventricular function. International Journal for Numerical Methods in Biomedical Engineering. 27(7):1017–1039.
 O’Sullivan et al. (2001) O’Sullivan M, Pruess K, Lippmann M. 2001. State of the art of geothermal reservoir simulation. Geothermics. 30:395–429.
 Peherstorfer (2019) Peherstorfer B. 2019. Sampling lowdimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference. arXiv:190811233.
 Peherstorfer and Willcox (2016) Peherstorfer B, Willcox K. 2016. Datadriven operator inference for nonintrusive projectionbased model reduction. Computer Methods in Applied Mechanics and Engineering. 306:196–215.
 Qian et al. (2020) Qian E, Kramer B, Peherstorfer B, Willcox K. 2020. Lift & Learn: Physicsinformed machine learning for largescale nonlinear dynamical systems. Physica D: Nonlinear Phenomena. 406:132401.
 Rozza et al. (2008) Rozza G, Huynh D, Patera A. 2008. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Archives of Computational Methods in Engineering. 15(3):229–275. Available from: http://dx.doi.org/10.1007/s1183100890199.
 Sirovich (1987) Sirovich L. 1987. Turbulence and the dynamics of coherent structures. I. Coherent structures. Quarterly of Applied Mathematics. 45(3):561–571.
 Spalart and Venkatakrishnan (2016) Spalart P, Venkatakrishnan V. 2016. On the role and challenges of cfd in the aerospace industry. The Aeronautical Journal. 120(1223):209.
 Swischuk et al. (2020) Swischuk R, Kramer B, Huang C, Willcox K. 2020. Learning physicsbased reducedorder models for a singleinjector combustion process. AIAA Journal. 58(6):2658–2672.
 Veroy and Patera (2005) Veroy K, Patera A. 2005. Certified realtime solution of the parametrized steady incompressible NavierStokes equations: Rigorous reducedbasis a posteriori error bounds. International Journal for Numerical Methods in Fluids. 47:773–788.
 Veroy et al. (2003) Veroy K, Prud’homme C, Rovas D, Patera A. 2003. A posteriori error bounds for reducedbasis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference; Orlando, FL. Paper AIAA20033847.
 Virtanen et al. (2020) Virtanen P, Gommers R, Oliphant TE, Haberland M, Reddy T, Cournapeau D, Burovski E, Peterson P, Weckesser W, Bright J, et al. 2020. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods. 17:261–272.
 Walt et al. (2011) Walt Svd, Colbert SC, Varoquaux G. 2011. The NumPy array: A structure for efficient numerical computation. Computing in Science & Engineering. 13(2):22–30.
 Westbrook and Dryer (1981) Westbrook CK, Dryer FL. 1981. Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames. Combustion Science and Technology. 27(12):31–43.
 Yin et al. (2010) Yin Y, Choi J, Hoffman EA, Tawhai MH, Lin CL. 2010. Simulation of pulmonary air flow with a subjectspecific boundary condition. Journal of Biomechanics. 43(11):2159–2163.
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