# A Neural Network-Inspired Matrix Formulation of Chemical Kinetics for Acceleration on GPUs
## 1 Introduction  
Computational modeling of reacting flows has become an integral part of the design and analysis of complex propulsion concepts [1, 2]. While a wide variety of tools constitute the modeling pathways, simulations for reacting flows require detailed models for chemical reactions. In most applications, the multi-physics behavior stemming from chemical reactions induces a highly complex Arrhenius-based expression for the species source terms, the nonlinearity of which produces an extremely stiff set of equations for the time evolution of the reacting chemical species [1, 2, 3]. The wide range of timescales in this setting must be resolved to accurately represent the characteristic turbulence-chemistry interactions. As a result, efforts to accelerate the computation of turbulent reacting flows with a focus on the chemical reaction (kinetics) terms have been, and still is, a major research effort in fields related to turbulent combustion. The efforts can be categorized into two classes: The first, more traditional class is model-oriented, i.e., an approximation to the exact formulation for the source terms and the species evolution is constructed with thegoal of decreasing the time-to-solution. The second class, which is the central focus of this work, is hardware-oriented, i.e., the computation of the source terms and time-evolution of the species is accelerated by porting existing exact algorithms to state-of-the-art hardware. The former is driven by innovations in physical understanding, whereas the latter is driven by advances in emerging heterogeneous computing architecture. To motivate the need for the latter, some previous work concerning the former class (traditional modeling approaches) is first briefly reviewed. Note that both classes are not mutually exclusive--the tools presented by machine learning (namely neural networks), to be touched on further below, very much blur the line between the two. While there has been considerable growth in the physical understanding and the development of reliable yet computationally efficient models [1; 2], representation of complex chemical kinetics remains a challenge. In particular, the use of detailed mechanisms that involve a hundred or more species and an even larger number of reactions in a turbulent flow configuration remains out of reach [4]. Although progress towards their use in canonical flows has been reported [5], their use in the simulation of complex geometries is still limited. When modeling turbulent combustion, manifold methods have overcome this computational issue by representing multi-step kinetics using a reduced-set of tracking variables such as mixture fraction and progress variable [6]. However, other combustion models such as the transported probability density function (PDF) approach [7; 8] or the linear-eddy model [9] require detailed chemistry to be directly evolved. In this regard, methods and algorithms that allow detailed chemical processes to be included in such approaches are a critical requirement. Models that attempt to tackle this issue from a reduction point of view include tabulation methods such as in-situ adaptive tabulation (ISAT) [10] and the PRISM [11] approach. In these methods, the computationally expensive numerical integration of chemical source terms, which can be cast as a set of ordinary differential equations (ODEs), is replaced by a look-up table. In particular, ISAT builds a trust region in thermochemical composition space using a set of ellipsoids determined by the Jacobian of the source terms. The cost of building and accessing such tables, however, can become expensive for large mechanisms, especially on modern high-performance computers (HPCs) that are memory-limited and use extensive concurrency in computations to reach high throughput efficiency.