Phish: A Novel Hyper-Optimizable Activation Function

Deep-learning models estimate values using backpropagation. The activation function within hidden layers is a critical component to minimizing loss in deep neural-networks. Rectified Linear (ReLU) has been the dominant activation function for the past decade. Swish and Mish are newer activation functions that have shown to yield better results than ReLU given specific circumstances. Phish is a novel activation function proposed here. It is a composite function defined as f(x) = xTanH(GELU(x)), where no discontinuities are apparent in the differentiated graph on the domain observed. Four generalized networks were constructed using Phish, Swish, Sigmoid, and TanH. SoftMax was the output function. Using images from MNIST and CIFAR-10 databanks, these networks were trained to minimize sparse categorical crossentropy. A large scale cross-validation was simulated using stochastic Markov chains to account for the law of large numbers for the probability values. Statistical tests support the research hypothesis stating Phish could outperform other activation functions in classification. Future experiments would involve testing Phish in unsupervised learning algorithms and comparing it to more activation functions.

Phish: A Novel Hyper-Optimizable Activation Function

Deep-learning models estimate values using backpropagation. The activation function within hidden layers is a critical component to minimizing loss in deep neural-networks. Rectified Linear (ReLU) has been the dominant activation function for the past decade. Swish and Mish are newer activation functions that have shown to yield better results than ReLU given specific circumstances. Phish is a novel activation function proposed here. It is a composite function defined as f(x) = xTanH(GELU(x)), where no discontinuities are apparent in the differentiated graph on the domain observed. Four generalized networks were constructed using Phish, Swish, Sigmoid, and TanH. SoftMax was the output function. Using images from MNIST and CIFAR-10 databanks, these networks were trained to minimize sparse categorical crossentropy. A large scale cross-validation was simulated using stochastic Markov chains to account for the law of large numbers for the probability values. Statistical tests support the research hypothesis stating Phish could outperform other activation functions in classification. Future experiments would involve testing Phish in unsupervised learning algorithms and comparing it to more activation functions.

Schur-Convexity for Elementary Symmetric Composite Functions and Their Inverse Problems and Applications

This paper investigates the Schur-convexity, Schur-geometric convexity, and Schur-harmonic convexity for the elementary symmetric composite function and its dual form. The inverse problems are also considered. New inequalities on special means are established by using the theory of majorization.

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.

MULTIVARIATE COMPOSITE COPULAS

In this paper, we present a method for generating a copula by composing two arbitrary n-dimensional copulas via a vector of bivariate functions, where the resulting copula is named as the multivariate composite copula. A necessary and sufficient condition on the vector guaranteeing the composite function to be a copula is given, and a general approach to construct the vector satisfying this necessary and sufficient condition via bivariate copulas is provided. The multivariate composite copula proposes a new framework for the construction of flexible multivariate copula from existing ones, and it also includes some known classes of copulas. It is shown that the multivariate composite copula has a clear probability structure, and it satisfies the characteristic of uniform convergence as well as the reproduction property for its component copulas. Some properties of multivariate composite copulas are discussed. Finally, numerical illustrations and an empirical example on financial data are provided to show the advantages of the multivariate composite copula, especially in capturing the tail dependence.

On the Use of Composite Functions in the Simple Equations Method to Obtain Exact Solutions of Nonlinear Differential Equations

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of a class of nonlinear differential equations containing polynomial nonlinearities. We present an amended version of the methodology, which is based on the use of composite functions. The number of steps of the SEsM was reduced from seven to four in the amended version of the methodology. For the case of nonlinear differential equations with polynomial nonlinearities, SEsM can reduce the solved equations to a system of nonlinear algebraic equations. Each nontrivial solution of this algebraic system leads to an exact solution of the solved nonlinear differential equations. We prove the theorems and present examples for the use of composite functions in the methodology of the SEsM for the following three kinds of composite functions: (i) a composite function of one function of one independent variable; (ii) a composite function of two functions of two independent variables; (iii) a composite function of three functions of two independent variables.

Path Tracking Control of Autonomous Vehicle Based on Nonlinear Tire Model

The tire forces of vehicles will fall into the non-linear region under extreme handling conditions, which cause poor path tracking performance. In this paper, a model predictive controller based on a nonlinear tire model is designed. The tire forces are characterized with nonlinear composite functions of the magic formula instead of a simple linear relation model. Taylor expansion is used to linearize the controller, the first-order difference quotient method is used for discretization, and the partial derivative of the composite function is used for matrix transformation. Constant velocity and variable velocity conditions are selected to compare the designed controller with the conventional controller in Carsim/Simulink. The results show that when the tire forces fall in the nonlinear region, two controllers have good stability, and the tracking accuracy of the controller designed in this paper is slightly better. However, after the tire forces become nonlinear, the controller with linear tire force becomes worse, the tracking accuracy is far worse than the controller with the nonlinear tire model, and the vehicle stability is also degraded. In addition, an active steering test platform based on LabVIEW-RT is established, and hardware-in-the-loop tests are carried out. The effectiveness of the designed controller is verified.

Formulation and evaluation of a composite acoustic objective incorporating air-borne and structure-borne transmission loss for optimization of building components

Technological advancements in computational building modeling have enabled designers to conduct many simulations at both the building and component levels. With the evolution of parametric modeling at the early stage of building design, designers can evaluate multiple design options and identify the best performing solutions. However, to conduct design space exploration or optimization, an objective function is needed to evaluate a design's performance. While defined objectives exist for building design considerations such as sustainability, energy usage, and structural performance there is not a single, encompassing objective that can accurately assess acoustic performance for optimization. This paper proposes the development of a novel acoustic objective function that encompasses sound transmission when designing floors, walls, or other acoustic barriers. The composite function will incorporate both air-borne and structure-borne sound simultaneously to determine the appropriate percentages for the formulation of the composite function. The results of the composite acoustic function for multiple floor constructions will be compared for the determination of a final acoustic transmission composite function. This study will detail why the implementation of a composite acoustic function is valuable for design optimization for sound transmission, what the limitations of this method are, and future applications of a composite acoustic function.