(388e) Using Co-Optimized Machine Learned Manifolds for Modeling Chemically Reacting Flows

Background: Chemically reacting flows play a key role in a wide range of engineered systems, from chemical and polymer processing to combustion-based energy conversion technologies. Simulations of these flows involve solving a coupled set of partial differential equations for mass, momentum, energy, and all relevant chemical species in the system. Chemical reaction pathways may be extremely complex and involve hundreds or more intermediate species, with reactions that occur over timescales varying by several orders of magnitude – presenting a significant numerical stiffness challenge. The combination of these factors makes simulation of chemically reacting flows vastly more expensive than nonreactive simulations, and often makes direct solution of the governing equations intractable. It is necessary to apply lower-fidelity models in place of the detailed governing equations in order to reduce computational cost to enable reacting flow simulation tools to be used in the engineering design process. Many of the models employed for this purpose are based on reducing the dimension of the thermochemical state, motivated by the observation that the observed thermochemical states in a system lie on a low-dimensional manifold in thermochemical state space. This behavior occurs due to the fast equilibration of certain reactive and transport processes, and physics-based manifold models rely on idealized assumptions about the balance of timescales and the way in which chemistry and transport are coupled. In this work, we apply a novel method for data-driven manifold-based modeling that can leverage data from high-fidelity reacting flow simulations to improve model accuracy in cases where the physics-based modeling assumptions break down. The approach is designed to be broadly applicable across chemically reacting flow systems but is applied here to turbulent combustion modeling.

Methods: This work is based on the recently proposed co-optimized machine-learned manifolds (CMLM) modeling approach, which uses a bespoke neural network structure to simultaneously define quantities that can parameterize the low-dimensional manifold and a nonlinear mapping to reaction rates and other thermochemical quantities necessary the close the flow simulation, so the network training algorithm co-optimizes these aspects of the model. The manifold parameters are constrained to being sparse linear combinations of species mass fractions, allowing transport equations to be easily defined. When the model is applied, equations are solved for the small number of manifold parameters and the nonlinear mapping is applied to compute all thermochemical quantities during simulation. Additionally, the CMLM network can also learn the necessary closures of the nonlinear source terms necessary in large eddy simulation (LES) or Reynolds-averaged Navier Stokes (RANS) simulation, in which case output quantities are predicted based on low-order moments of the manifold parameters.

The CMLM approach is a physically-inspired data-driven approach in that the model form mimics physics-based turbulent combustion models such as Flamelet Generated Manifolds (FGM), where thermochemical quantities are tabulated based on a reaction progress variable and mixture fraction, both linear combinations of species mass fractions. Neural networks have been used to replace traditional tabulation in these models, making the adoption of neural network based strategies that also include manifold identification a logical extension of this class of model. Furthermore, the co-optimization procedure more fully accounts for nonlinearity in the chemical reaction system than the current-state-of-the-art data-driven approach of using principal component analysis (PCA), a linear method, to identify the manifold parameters coupled with a simple fully connected neural network for the nonlinear mapping. Another advantage of CMLM is that the sparsity in the definition of the manifold parameters allows for identification of the key chemical species for computing reaction rates, making it helpful in physically interpreting complex chemical systems.

The Pele suite of reacting flow solvers is chosen for the implementation of CMLM. These codes are being developed as part of the Department of Energy’s Exascale Computing Project to run large-scale, high-fidelity simulations of practical combustion systems. Implementation in Pele will allow the data generated in these large-scale simulations to be used for model development, and then lower cost simulations can be run over wider range of conditions, all within the same software framework. The included figure demonstrates the full intended workflow and demonstrates progress made thus far in training and evaluation of the model. Initially, implementation is targeted toward PeleLM, the low Mach number solver, to avoid the added modeling challenge of compressibility.

Results: A priori evaluation of the model decoupled from the flow solver is a key first step in demonstrating the model that has been completed. In this phase, we compare CMLM against state-of-the-art physics-based models (FGM) and data-driven dimension reduction approaches (PCA). The first data set considered is a set of one-dimensional flames, which exist on a two-dimensional manifold in state space by construction. When trained on this data, the two-dimensional manifold model “learned” by CMLM is equivalent to FGM and slightly more favorable numerically. This demonstrates that CMLM recovers the physical behavior in the appropriate limit. In contrast, PCA cannot accurately model the data set with only two manifold dimensions, and therefore does not recover the physically expected behavior in this simple case. To demonstrate the ability of CMLM to be applied where physics-based models fail, data from direct numerical simulations (DNS) of a turbulent ignition kernel (A. Krisman et al., Combustion & Flame, 2021) is considered. In this case, CMLM allows for accurate computation of reaction rates while reducing the thermochemical state space from 30 dimensions to 4 dimensions. For comparison, PCA requires 7 dimensions for similar accuracy.

Manifold-based modeling capability has been added to the PeleLM and the physical property evaluation library PelePhysics. This will allow for a posteriori evaluation of the models, initially targeting similar turbulent combustion cases as were used for the a priori training. These tests will quantify the benefits of the manifold-based approach in terms of computational cost and stiffness reduction relative to the full high-dimensional chemical reaction network, as well as a comparison of the accuracy relative to other manifold-based models, again using FGM and PCA as a baseline to show improvement over state-of-the-art models.

Implications: In the study of reacting flows, there has been a trend toward larger, more computationally expensive simulations that generate ever more data. In order to realize the benefits of these simulations, it is necessary to develop data-driven methodologies that take advantage of these data sets to improve upon physics-based models. The CMLM methodology that is demonstrated in this work provides an avenue to do that for the purpose of dimension reduction and manifold-based modeling, which is useful both for reducing simulation cost and interpreting complex chemical systems. Having demonstrated the model successfully for a turbulent ignition kernel problem, we will seek to demonstrate its broader applicability for other problems in combustion modeling and other reacting flow applications.