Assessment of machine learning methods for state-to-state approaches

TL;DR

This work addresses the high computational cost of state-to-state (STS) models in high-speed reacting flows by evaluating machine-learning strategies. It first benchmarks regression surrogates to predict STS relaxation terms, finding Kernel Ridge regression offers the best accuracy while Decision Trees provide fastest predictions, with PCA enabling effective dimensionality reduction. The authors then explore coupling pre-trained ML models with ODE solvers in both MATLAB-Python and Fortran-Python interfaces, achieving substantial speed-ups but highlighting stability and integration challenges that depend on the IVP/BVP nature of the problem. Additionally, a deep neural network is trained to directly infer the full 1D STS Euler solution behind a shock, achieving satisfactory agreement for several variables, though generalization and interpretability remain open questions. Overall, the results point to a promising pathway for accelerating STS simulations, with implications for larger-scale and transport-coefficient regression in future work.

Abstract

It is well known that numerical simulations of high-speed reacting flows, in the framework of state-to-state formulations, are the most detailed but also often prohibitively computationally expensive. In this work, we start to investigate the possibilities offered by the use of machine learning methods for state-to-state approaches to alleviate such burden. In this regard, several tasks have been identified. Firstly, we assessed the potential of state-of-the-art data-driven regression models based on machine learning to predict the relaxation source terms which appear in the right-hand side of the state-to-state Euler system of equations for a one-dimensional reacting flow of a N$_2$/N binary mixture behind a plane shock wave. It is found that, by appropriately choosing the regressor and opportunely tuning its hyperparameters, it is possible to achieve accurate predictions compared to the full-scale state-to-state simulation in significantly shorter times. Secondly, we investigated different strategies to speed-up our in-house state-to-state solver by coupling it with the best-performing pre-trained machine learning algorithm. The embedding of machine learning methods into ordinary differential equations solvers may offer a speed-up of several orders of magnitude but some care should be paid for how and where such coupling is realized. Performances are found to be strongly dependent on the mutual nature of the interfaced codes. Finally, we aimed at inferring the full solution of the state-to-state Euler system of equations by means of a deep neural network completely by-passing the use of the state-to-state solver while relying only on data. Promising results suggest that deep neural networks appear to be a viable technology also for these tasks.

Figures (6)

• Figure 1: Machine learning for state-to-state: conceptual map.
• Figure 2: Parity plot of cross-validated predictions against "ground truth" values for the Decision Tree algorithm (left) and plot of cumulative variance as a function of the PCA number of components (right).
• Figure 3: Comparison of Matlab and ML solution for the one-dimensional reacting shock flow in STS approach for binary N$_2$/N mixture.
• Figure 4: Comparison of Matlab and ML solution for the one-dimensional reacting shock flow in STS approach for Air 5 mixture.
• Figure 5: Comparison of Matlab and ML time-to-solution for the one-dimensional reacting shock flow in STS approach for binary N$_2$/N mixture. The ML call is performed within the ODE system integration before the matrix inversion.