A divide and conquer method for symbolic regression

Symbolic regression aims to find a function that best explains the relationship between independent variables and the objective value based on a given set of sample data. Genetic programming (GP) is usually considered as an appropriate method for the problem since it can optimize functional structure and coefficients simultaneously. However, the convergence speed of GP might be too slow for large scale problems that involve a large number of variables. Fortunately, in many applications, the target function is separable or partially separable. This feature motivated us to develop a new method, divide and conquer (D&C), for symbolic regression, in which the target function is divided into a number of sub-functions and the sub-functions are then determined by any of a GP algorithm. The separability is probed by a new proposed technique, Bi-Correlation test (BiCT). D&C powered GP has been tested on some real-world applications, and the study shows that D&C can help GP to get the target function much more rapidly

Elite Bases Regression: A Real-time Algorithm for Symbolic Regression

Symbolic regression is an important but challenging research topic in data mining. It can detect the underlying mathematical models. Genetic programming (GP) is one of the most popular methods for symbolic regression. However, its convergence speed might be too slow for large scale problems with a large number of variables. This drawback has become a bottleneck in practical applications. In this paper, a new non-evolutionary real-time algorithm for symbolic regression, Elite Bases Regression (EBR), is proposed. EBR generates a set of candidate basis functions coded with parse-matrix in specific mapping rules. Meanwhile, a certain number of elite bases are preserved and updated iteratively according to the correlation coefficients with respect to the target model. The regression model is then spanned by the elite bases. A comparative study between EBR and a recent proposed machine learning method for symbolic regression, Fast Function eXtraction (FFX), are conducted. Numerical results indicate that EBR can solve symbolic regression problems more effectively.Comment: The 2017 13th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD 2017

Initiation characteristics of wedge-induced oblique detonation wave in a stoichiometric hydrogen-air mixture

The initiation features of two-dimensional, oblique detonations from a wedge in a stoichiometric hydrogen-air mixture are investigated via numerical simulations using the reactive Euler equations with detailed chemistry. A parametric study is performed to analyze the effect of inflow pressure P0, and Mach number M0 on the initiation structure and length. The present numerical results demonstrate that the two transition patterns, i.e., an abrupt transition from a multi-wave point connecting the oblique shock and the detonation surface and a smooth transition via a curved shock, depend strongly on the inflow Mach number, while the inflow pressure is found to have little effect on the oblique shock-to-detonation transition type. The present results also reveal a slightly more complex structure of abrupt transition type in the case of M0 = 7.0, consisting of various chemical and gasdynamic processes in the shocked gas mixtures. The present results show quantitatively that the initiation length decreases with increasing M0, primarily due to the increase of post-shock temperature. Furthermore, the effect of M0 on initiation length is independent of P0, but given the same M0, the initiation length is found to be inversely proportional to P0. Theoretical analysis based on the constant volume combustion (CVC) theory is also performed, and the results are close to the numerical simulations in the case of high M0 regardless of P0, demonstrating that the post-oblique-shock condition, i.e., post-shock temperature, is the key parameter affecting the initiation. At decreasing M0, the CVC theory breaks down, suggesting a switch from chemical kinetics-controlled to a wave-controlled gasdynamic process. For high inflow pressure P0 at decreasing M0, the CVC theoretical estimations depart from numerical results faster than those of low P0, due to the presence of the non-monotonic effects of chemical kinetic limits in hydrogen oxidation at high pressure

Eau

Modeling aluminum (Al) particle-air detonation is extremely difficult because the combustion is shock-induced, and there are multi-phase heat release and transfer in supersonic flows. Existing models typically use simplified combustion to reproduce the detonation velocity, which introduces many unresolved problems. The hybrid combustion model, coupling both the diffused- and kinetics-controlled combustion, is proposed recently, and then improved to include the effects of realistic heat capacities dependent on the particle temperature. In the present study, 2D cellular Al particle-air detonations are simulated with the realistic heat capacity model and its effects on the detonation featured parameters, such as the detonation velocity and cell width, are analyzed. Numerical results show that cell width increases as particle diameter increases, similarly to the trend observed with the original model, but the cell width is underestimated without using the realistic heat capacities. Further analysis is performed by averaging the 2D cellular detonations to quasi-1D, demonstrating that the length scale of quasi-1D detonation is larger than that of truly 1D model, similar to gaseous detonations

Initiation structure of oblique detonation waves behind conical shocks

The understanding of oblique detonation dynamics has both inherent basic research value for high-speed compressible reacting flow and propulsion application in hypersonic aerospace systems. In this study, the oblique detonation structures formed by semi-infinite cones are investigated numerically by solving the unsteady, two-dimensional axisymmetric Euler equations with a one-step irreversible Arrhenius reaction model. The present simulation results show that a novel wave structure, featured by two distinct points where there is close-coupling between the shock and combustion front, is depicted when either the cone angle or incident Mach number is reduced. This structure is analyzed by examining the variation of the reaction length scale and comparing the flow field with that of planar, wedge-induced oblique detonations. Further simulations are performed to study the effects of chemical length scale and activation energy, which are both found to influence the formation of this novel structure. The initiation mechanism behind the conical shock is discussed to investigate the interplay between the effect of the Taylor-Maccoll flow, front curvature, and energy releases from the chemical reaction in conical oblique detonations. The observed flow fields are interpreted by means of the energetic limit as in the critical regime for initiation of detonation

Effects of inflow Mach number on oblique detonation initiation with a two-step induction-reaction kinetic model

Oblique detonations induced by two-dimensional, semi-infinite wedges are simulated by solving numerically the reactive Euler equations with a two-step induction-reaction kinetic model. Previous results obtained with other models have demonstrated that for the low inflow Mach number M0 regime past a critical value, the wave in the shocked gas changes from an oblique reactive wave front into a secondary oblique detonation wave (ODW). The present numerical results not only confirm the existence of such critical phenomenon, but also indicate that the structural shift is induced by the variation of the main ODW front which becomes sensitive to M0 near a critical value. Below the critical M0,cr, oscillations of the initiation structure are observed and become severe with further decrease of M0. For low M0 cases, the non-decaying oscillation of the initiation structure exists after a sufficiently long-time computation, suggesting the quasi-steady balance of initiation wave systems. By varying the heat release rate controlled by kR, the pre-exponential factor of the second reaction step, the morphology of initiation structures does not vary for M0 = 10 cases but varies for M0 = 9 cases, demonstrating that the effects of heat release rate become more prominent when M0 decreases. The instability parameter χ is introduced to quantify the numerical results. Although χ cannot reveal the detailed mechanism of the structural shift, a linear relation between χ and kR exists at the critical condition, providing an empirical criterion to predict the structural variation of the initiation structure

Numerical study of inflow equivalence ratio inhomogeneity on oblique detonation formation in hydrogen-air mixtures

In this study, numerical simulations using Euler equations with detailed chemistry are performed to investigate the effect of fuel-air composition inhomogeneity on the oblique detonation wave (ODW) initiation in hydrogen-air mixtures. This study aims for a better understanding of oblique detonation wave engine performance under practical operating conditions, among those is the inhomogeneous mixing of fuel and air giving rise to a variation of the equivalence ratio (ER) in the incoming combustible flow. This work focuses primarily on how a variable equivalence ratio in the inflow mixture affects both the formation and characteristic parameters of the oblique detonation wave. In this regard, the present simulation imposes initially a lateral linear distribution of the mixture equivalence ratio within the initiation region. The variation is either from fuel-lean or fuel-rich to the uniform stoichiometric mixture condition above the oblique shock wave. The obtained numerical results illustrate that the reaction surface is distorted in the cases of low mixture equivalence ratio. The so-called “V-shaped” flame is observed but differed from previous results that it is not coupled with any compression or shock wave. Analyzing the temperature and species density evolution also shows that the fuel-lean and fuel-rich inhomogeneity have different effects on the combustion features in the initiation region behind the oblique shock wave. Two characteristic quantities, namely the initiation length and the ODW surface position, are defined to describe quantitatively the effects of mixture equivalence ratio inhomogeneity. The results show that the initiation length is mainly determined by the mixture equivalence ratio in the initiation region. Additional computations are performed by reversing ER distribution, i.e., with the linear variation above the initiation region of uniform stoichiometric condition and results also demonstrate that the ODW position is effectively determined by the ER variation before the ODW, which has in turn only negligible effect on the initiation length