Tool For Symbolic Regression Research Articles

Learning transport processes with machine intelligence

Transport processes ruled by complex micro-physics and impractical to theoretical investigation may exhibit emergent behavior describable by mathematical expressions. Such information, while implicitly contained in the results of microscopic-scale numerical simulations close to first principles or experiments is not in a form suitable for macroscopic modelling. Here we present a machine learning approach that leverages such information to deploy micro-physics informed transport flux representations applicable to a continuum mechanics description. One issue with deep neural networks, arguably providing the most generic of such representations, is their noisiness which is shown to break the performance of numerical schemes. The matter is addressed and a methodology suitable for schemes characterised by second order convergence rate is presented. The capability of the methodology is demonstrated through an idealized study of the long standing problem of heat flux suppression relevant to fusion and cosmic plasmas. Symbolic representations, although potentially less generic, are straightforward to use in numerical schemes and theoretical analysis, and can be even more accurate as shown by the application to the same problem of an advanced symbolic regression tool. These results are a promising initial step to filling the gap between micro and macro in this important area of modeling.

Reynolds-averaged stress and scalar-flux closures via symbolic regression for vertical natural convection

Turbulent natural convection is characterised by mutual coupling effects between velocity and thermal fields. This inter-coupling induces simulation errors with linear and non-coupled turbulence closures commonly used in many Reynolds–averaged Navier–Stokes (RANS) approaches. In this paper, we re-examine the buoyancy-accounting algebraic scalar-flux model proposed by Kenjereš et al., Int. J. Heat Fluid Flow, Vol. 26, pp. 569–586 (2005). Using a term-by-term analysis on the model with the aid of high-fidelity datasets for natural convection flow between two differentially heated vertical plates, it is demonstrated that there are significant discrepancies in the predicted turbulent heat fluxes once the model is combined with the existing algebraic Reynolds stress models. As a result, we suggest including the quadratic terms in buoyancy-extended explicit algebraic Reynolds stress models, and have developed non-linear Reynolds-stress and heat-flux closure models via an in–house symbolic regression tool based on gene expression programming (GEP). The evaluation of these GEP-developed models shows significant improvements in the prediction of mean quantities and second-moments in an a priori stage and in an a posteriori stage, with the latter being realized by embedding the new models into the elliptic relaxation v2‾/k-f-k-ε-θ2‾ equations, for vertical natural convection with Rayleigh numbers in the range of 106 to 109.

The Train Delay Model Developed by the Genetic Programming Algorithm

The paper discusses the problem of probability distribution category identification of train delay data by a genetic programming algorithm. This train delay frequency function and the probability distribution simply derived from it are significant to train traffic modelling and management. The genetic programming algorithm was used as an uninformed tool to prevent the influence of a priori information, which should be biased. The real traffic data were aggregated into predefined bins and then the frequencies of the individual delays were computed. The genetic programming algorithm was used in the next step as a symbolic regression tool to discover their frequency function in the form of an algebraic expression. The results concluded that although data has no known distribution, their distributions are similar to exponential ones.

Analytic Programming – a Novel Tool for Synthesis of Controller for Chaotic Lozi Map

In this paper, it is presented a utilization of a novel tool for symbolic regression, which is analytic programming, for the purpose of the synthesis of a new feedback control law. This new synthesized chaotic controller secures the fully stabilization of selected discrete chaotic systems, which is the two-dimensional Lozi map. The paper consists of the descriptions of analytic programming as well as selected chaotic system, used heuristic and cost function design. For experimentation, Self-Organizing Migrating Algorithm (SOMA) and Differential evolution (DE) were used. Two selected experiments are detailed described.

Improved Junction Body Flow Modeling Through Data-Driven Symbolic Regression

A novel data-driven turbulence modeling framework is presented and applied to the problem of junction body flow. In particular, a symbolic regression approach is used to find nonlinear analytical expressions of the turbulent stress-strain coupling that are ready for implementation in computational fluid dynamics (CFD) solvers using Reynolds-averaged Navier-Stokes (RANS) closures. Results from baseline linear RANS closure calculations of a finite square-mounted cylinder with a Reynolds number of 11; 000, based on diameter and freestream velocity, are shown to considerably overpredict the separated flow region downstream of the square cylinder, mainly because of the failure of the model to accurately represent the complex vortex structure generated by the junction flow. In the present study, a symbolic regression tool built on a gene expression programming technique is used to find a nonlinear constitutive stress-strain relationship. In short, the algorithm finds the most appropriate linear combination of basis functions and spatially varying coefficients that approximate the turbulent stress tensor from high-fidelity data. Here, the high-fidelity data, or the so-called training data, were obtained from a hybrid RANS/Large Eddy Simulation (LES) calculation also developed with symbolic regression that showed excellent agreement with direct numerical simulation data. The present study, therefore, also demonstrates that training data required for RANS closure development can be obtained using computationally more affordable approaches, such as hybrid RANS/LES. A procedure is presented to evaluate which of the individual basis functions that are available for model development are most likely to produce a successful nonlinear closure. A new model is built using those basis functions only. This new model is then tested, i.e., an actual CFD calculation is performed, on the well-known periodic hills case and produces significantly better results than the linear baseline model, despite this test case being fundamentally different from the training case. Finally, the new model is shown to also improve predictive accuracy for a surface-mounted cube placed in a channel at a cube height Reynolds number of 40; 000 over traditional linear RANS closures.

Data-Driven Predesign Tool for Small-Scale Centrifugal Compressor in Refrigeration

Domestic scale heat pumps and air conditioners are mainly driven by volumetric compressors. Yet the use of reduced scale centrifugal compressors is reconsidered due to their high efficiency and power density. The design procedure of centrifugal compressors starts with predesign tools based on the Cordier line. However, the optimality of the obtained predesign, which is the starting point of a complex and iterative process, is not guaranteed, especially for small-scale compressors operating with refrigerants. This paper proposes a data-driven predesign tool tailored for small-scale centrifugal compressors used in refrigeration applications. The predesign model is generated using an experimentally validated one-dimensional (1D) code which evaluates the compressor performance as a function of its detailed geometry and operating conditions. Using a symbolic regression tool, a reduced order model that predicts the performance of a given compressor geometry has been built. The proposed predesign model offers an alternative to the existing tools by providing a higher level of detail and flexibility. Particularly, the model includes the effect of the pressure ratio, the blade height ratio, and the shroud to tip radius ratio. The analysis of the centrifugal compressor losses allows identifying the underlying phenomena that shape the new isentropic efficiency contours. Compared to the validated 1D code, the new predesign model yields deviations below 4% on the isentropic efficiency, while running 1500 times faster. The new predesign model is, therefore, of significant interest when the compressor is part of an integrated system design process.

A multilevel block building algorithm for fast modeling generalized separable systems

Symbolic regression is an important application area of genetic programming (GP), aimed at finding an optimal mathematical model that can describe and predict a given system based on observed input-response data. However, GP convergence speed towards the target model can be prohibitively slow for large-scale problems containing many variables. With the development of artificial intelligence, convergence speed has become a bottleneck for practical applications. In this paper, based on observations of real-world engineering equations, generalized separability is defined to handle repeated variables that appear more than once in the target model. To identify the structure of a function with a possible generalized separability feature, a multilevel block building (MBB) algorithm is proposed in which the target model is decomposed into several blocks and then into minimal blocks and factors. The minimal factors are relatively easy to determine for most conventional GP or other non-evolutionary algorithms. The efficiency of the proposed MBB has been tested by comparing it with Eureqa, a state-of-the-art symbolic regression tool. Test results indicate MBB is more effective and efficient; it can recover all investigated cases quickly and reliably. MBB is thus a promising algorithm for modeling engineering systems with separability features.

Utilization of analytic programming for the evolutionary synthesis of the robust multi-chaotic controller for selected sets of discrete chaotic systems

A novel tool for symbolic regression, analytical programming and its application for the synthesis of a new robust feedback control law are presented in this paper. This synthesized robust chaotic controller secures the fully stabilization of several selected sets containing one-dimensional, two-dimensional and evolutionary synthesized discrete chaotic systems. The paper consists of the descriptions of analytic programming as well as selected chaotic systems, used heuristic and cost function design. For experimentation, self-organizing migrating algorithm and differential evolution were used.

Analytic programming in the task of evolutionary synthesis of a controller for high order oscillations stabilization of discrete chaotic systems

This paper deals with the utilization of a symbolic regression tool, which is Analytic Programming (AP), together with two evolutionary algorithms, the Self-Organizing Migrating Algorithm (SOMA) and Differential Evolution (DE), for the synthesis of a new control law. This synthesized chaotic controller secures the stabilization of higher periodic orbits, which represent oscillations between several values of three selected discrete chaotic systems. Selected examples were: an artificially evolutionary synthesized system, logistic equation and Hénon map. The paper consists of the description of analytic programming as well as chaotic systems used, evolutionary techniques and the cost function.

Synthesis of feedback controller for three selected chaotic systems by means of evolutionary techniques: Analytic programming

This research deals with the utilization of analytic programming for a synthesis of control law for three selected discrete chaotic systems. The novelty of the approach is that a tool for symbolic regression–analytic programming–is used for such kinds of difficult problems. The paper consists of the descriptions of analytic programming as well as chaotic systems and used cost functions. For simulations, evolutionary algorithm SOMA (Self-Organizing Migrating Algorithm) and analytic programming were used. For each case study, repeated simulations were conducted to outline the effectiveness and robustness of used methods. All repeated simulations have given satisfactory results.

Investigation on Evolutionary Synthesis of Movement Commands

This paper deals with usage of an alternative tool for symbolic regression—analytic programming which is able to solve various problems from the symbolic domain, as well as genetic programming and grammatical evolution. This paper describes a setting of an optimal trajectory for a robot (originally designed as an artificial ant on Santa Fe trail) solved by means of analytic programming. Firstly, main principles of analytic programming are described and explained. The second part shows how analytic programming was used for the application of finding a suitable trajectory step by step. Because analytic programming needs evolutionary algorithms for its run, three evolutionary algorithms were used—self-organizing migrating algorithm, differential evolution, and simulated annealing—to show that anyone can be used. The total number of simulations was 150 and results show that the first two used algorithms were more successful than not so robust simulated annealing.

Parallel implementation of a genetic-programming based tool for symbolic regression

We report on a parallel implementation of a tool for symbolic regression, the algorithmic mechanism of which is based on genetic programming, and communication is handled using MPI. The implementation relies on a random islands model (RIM), which combines both the conventional islands model where migration of individuals between islands occurs periodically and niching where no migration takes place. The system was designed so that the algorithm is synergistic with parallel/distributed architectures, and works to make use of processor time and minimum use of network bandwidth without complicating the sequential algorithm significantly. Results on an IBM SP2 are included.

Critical heat flux prediction using genetic programming for water flow in vertical round tubes

The genetic programming method is used to develop critical heat flux (CHF) correlations for upward water flow in vertical round tubes under low pressure and low flow conditions. The genetic programming, as a symbolic regression tool, finds both the functional form and fitting coefficients of a correlation without any initial assumptions. Inlet and local condition type correlations are developed based on 414 and 314 CHF data from KAIST CHF data bank, respectively. The inlet condition type correlation shows the rms error of 15.2% and the local condition type one shows the rms errors of 32.7% and 13.2% by the direct substitution method and the heat balance method, respectively. Prediction errors are smaller than or comparable to those for other existing correlations.