Symbolic Differentiation Research Articles

It Is Not the Whole Story: Toward a Broader Understanding of Entrepreneurial Ventures’ Symbolic Differentiation

It Is Not the Whole Story: Toward a Broader Understanding of Entrepreneurial Ventures’ Symbolic Differentiation

IMPLEMENTATION OF THE ALGEBRA OF HYPERDUAL NUMBERS IN NEURAL NETWORKS

For the numerical solution of problems arising in various fields of mathematics and mechanics, it is often necessary to determine the values of derivatives included in the model. Currently, numerical values of derivatives can be obtained using automatic differentiation libraries in many programming languages. This paper discusses the use of the Python programming language, which is widely used in the scientific community. It should be noted that the principles of automatic differentiation are not related to numerical or symbolic differentiation methods. The work consists of three parts. The introduction reviews the historical development of the general theory of complex numbers and the use of simple complex, double and dual numbers, which are a subset of the set of general complex numbers, in various fields of mathematics. The second part is devoted to the algebra of dual and hyperdual numbers and their properties. This section presents tables of the basis element of elementary functions with dual and hyperdual arguments, based on multiplication rules. Two important formulas for finding the numerical values of a complex function's first and second derivatives by expanding functions with dual and hyperdual arguments in the Taylor series are also obtained. A simple test function was used to verify the correctness of these formulas, the results of which were checked analytically as well as through implementation in a programming language. The third part of the paper focuses on practical applications and the implementation of these methods in Python. It includes detailed examples of case studies demonstrating the effectiveness of using hyperdual numbers in automatic differentiation. The results highlight the accuracy and computational efficiency of these methods, making them valuable tools for researchers and engineers. This comprehensive approach not only validates the theoretical aspects but also showcases the practical utility of dual and hyperdual numbers in solving complex mathematical and mechanical problems.

Bifurcation tracking on moving meshes and with consideration of azimuthal symmetry breaking instabilities

We present a black-box method to numerically investigate the linear stability of arbitrary multi-physics problems. While the user just has to enter the system's residual in weak formulation, e.g. by a finite element method, all required discretized matrices are automatically assembled based on just-in-time generated and compiled C codes. Based on this method, entire phase diagrams in the parameter space can be obtained by bifurcation tracking and continuation at low computational costs. Particular focus is put on problems with moving domains, e.g. free surface problems in fluid dynamics, since a moving mesh introduces a plethora of complicated nonlinearities to the system. By symbolic differentiation before the code generation, however, these moving mesh problems are made accessible to bifurcation tracking methods. In a second step, our method is generalized to investigate symmetry-breaking instabilities of axisymmetric stationary solutions by effectively utilizing the symmetry of the base state. Each bifurcation type is validated on the basis of results reported in the literature on versatile fluid dynamics problems, for which we subsequently present novel results as well.

Tinned: A symbolic library for response theory and high-order derivatives.

A symbolic C++ library-Tinned-has been developed for symbolic differentiation and manipulation in response theory. By recognizing different key building blocks in the density matrix-based (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) and coupled-cluster response theories, we have implemented their corresponding C++ symbolic classes, including but not limited to one- and two-electron operators, exchange-correlation energy and potential, and coupled-cluster operator. Formulas of response theory can be well expressed in terms of the symbolic classes in the library Tinned. Their high-order perturbation-strength derivatives can be straightforwardly computed and extracted afterwards for numerical evaluation. The library Tinned will greatly facilitate the development work of response theory and may lead to a unified framework for response theory at different levels of electronic structure theory.

편의점 브랜드 캐릭터의 개발과 활용 연구

The purpose of this study is to find out how brand characters help to form a positive brand image and brand equity through the case of CU. To this end, Heyroo Friends and CU Friends were analyzed based on previous studies on brands and brand characters. Heyroo Friends was only a PB product brand character based on CU's function, but it developed into CU brand character CU Friends by expanding the media from webtoons to web dramas, and metaverse. CU Friends is being actively used as a means to enhance the brand image as the basis for characters in flagship stores and short-form web drama series. Contents based on brand characters is contributing to deepening the relationship between consumers and brands. Through this study, it was confirmed that brand characters can be an effective means of enhancing brand identity and forming familiarity, as well as building brand assets through symbolic differentiation.

Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation.

Acquiring reliable knowledge amidst uncertainty is a topical issue of modern science. Interval mathematics has proved to be of central importance in coping with uncertainty and imprecision. Algorithmic differentiation, being superior to both numeric and symbolic differentiation, is nowadays one of the most celebrated techniques in the field of computational mathematics. In this connexion, laying out a concrete theory of interval differentiation arithmetic, combining subtlety of ordinary algorithmic differentiation with power and reliability of interval mathematics, can extend real differentiation arithmetic so markedly both in method and objective, and can so far surpass it in power as well as applicability. This article is intended to lay out a systematic theory of dyadic interval differentiation numbers that wholly addresses first and higher order automatic derivatives under uncertainty. We begin by axiomatizing a differential interval algebra and then we present the notion of an interval extension of a family of real functions, together with some analytic notions of interval functions. Next, we put forward an axiomatic theory of interval differentiation arithmetic, as a two-sorted extension of the theory of a differential interval algebra, and provide the proofs for its categoricity and consistency. Thereupon, we investigate the ensuing structure and show that it constitutes a multiplicatively non-associative S-semiring in which multiplication is subalternative and flexible. Finally, we show how to computationally realize interval automatic differentiation. Many examples are given, illustrating automatic differentiation of interval functions and families of real functions.

Research into the Dynamics of a System of Two Connected Bodies Moving in the Plane of a Circular Orbit by Applying Computer Algebra Methods

Computer algebra methods are used to determine the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves in a central Newtonian force field on a circular orbit under the action of gravitational torque. Primary attention is given to the study of equilibrium orientations of the two-body system in the plane of the circular orbit. By applying symbolic differentiation, differential equations of motion are derived in the form of Lagrange equations of the second kind. A method is proposed for transforming the system of trigonometric equations determining the equilibria into a system of algebraic equations, which in turn are reduced by calculating the resultant to a single algebraic equation of degree 12 in one unknown. The roots of the resulting algebraic equation determine the equilibrium orientations of the two-body system in the circular orbit plane. By applying symbolic factorization, the algebraic equation is decomposed into three polynomial factors, each specifying a certain class of equilibrium configurations. The domains with an identical number of equilibrium positions are classified using algebraic methods for constructing a discriminant hypersurface. The equations for the discriminant hypersurface determining the boundaries of domains with an identical number of equilibrium positions in the parameter space of the problem are obtained via symbolic computations of the determinant of the resultant matrix. By numerical analysis of the real roots of the resulting algebraic equations, the number of equilibrium positions of the two-body system is determined depending on the parameters.

Combinatory Adjoints and Differentiation

We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars, vectors, matrices or tensors represented as multi-dimensional arrays. We show that both symbolic and automatic differentiation can be performed using a differential calculus for generating linear functions representing Fr\'echet derivatives based on rules for primitive, constant, linear and bilinear functions as well as their sequential and parallel composition. Linear functions are represented in a combinatory domain-specific language. Finally, we provide a calculus for symbolically computing the adjoint of a derivative without using matrices, which are too inefficient to use on high-dimensional spaces. The resulting symbolic representation of a derivative retains the data-parallel operations from the input program. The combination of combinatory differentiation and computing formal adjoints turns out to be behaviorally equivalent to reverse-mode automatic differentiation. In particular, it provides opportunities for optimizations where matrices are too inefficient to represent linear functions.

The German social space and its homologies: National variation on a basic structure

This article constructs a comprehensive new model of the contemporary class structure of Germany. More specifically, inspired by Pierre Bourdieu’s geometric conception of class relations and drawing on original survey data, it adopts multiple correspondence analysis paired with cluster analysis to chart the German ‘social space’, that is, the relational configuration of key forms of capital. It then explores correspondences with occupational groups, ethnic groups, other demographic features, lifestyle practices and tastes. The results disclose specific structuring effects of German peculiarities on the distribution of social power, including East–West reunification and the long-running guestworker programme. More fundamentally, though, in its basic structure, the space resembles that mapped by Bourdieu in France and those documented by others elsewhere, suggesting common principles of social and symbolic differentiation among Western capitalist societies.

Recent progress on equation-oriented optimization of complex chemical processes

Process optimization in equation-oriented (EO) modeling environments favors the gradient-based optimization algorithms by their abilities to provide accurate Jacobian matrices via automatic or symbolic differentiation. However, computational inefficiencies including that in initial-point-finding for Newton type methods have significantly limited its application. Recently, progress has been made in using a pseudo-transient (PT) modeling method to address these difficulties, providing a fresh way forward in EO-based optimization. Nevertheless, research in this area remains open, and challenges need to be addressed. Therefore, understanding the state-of-the-art research on the PT method, its principle, and the strategies in composing effective methodologies using the PT modeling method is necessary for further developing EO-based methods for process optimization. For this purpose, the basic concepts for the PT modeling and the optimization framework based on the PT model are reviewed in this paper. Several typical applications, e.g., complex distillation processes, cryogenic processes, and optimizations under uncertainty, are presented as well. Finally, we identify several main challenges and give prospects for the development of the PT based optimization methods.

Technology computer aided design based deep level transient spectra: simulation of high-purity germanium crystals

Very often deep level transient spectroscopy (DLTS) specimens deviate from ideal textbook examples, making the interpretation of spectra a huge challenge. This challenge introduces inaccurate estimates of emission signatures and the lack of appropriate estimates for the concentrations of the observed trap levels. In this work it is shown with the example of high-purity germanium that technology computer aided design including symbolic differentiation provides the necessary numerical stability over a wide temperature range to model DLTS spectra. Moreover high-purity germanium is a quasi intrinsic semiconductor for which it is well-known that original small signal theory can introduce strong errors. It is furthermore shown that the parasitic impact of fractional filling and high resistivity material can be modelled and that these modelled spectra can in the future assist the interpretation of experimental results.

Coarsening optimization for differentiable programming

This paper presents a novel optimization for differentiable programming named coarsening optimization. It offers a systematic way to synergize symbolic differentiation and algorithmic differentiation (AD). Through it, the granularity of the computations differentiated by each step in AD can become much larger than a single operation, and hence lead to much reduced runtime computations and data allocations in AD. To circumvent the difficulties that control flow creates to symbolic differentiation in coarsening, this work introduces phi-calculus, a novel method to allow symbolic reasoning and differentiation of computations that involve branches and loops. It further avoids "expression swell" in symbolic differentiation and balance reuse and coarsening through the design of reuse-centric segment of interest identification. Experiments on a collection of real-world applications show that coarsening optimization is effective in speeding up AD, producing several times to two orders of magnitude speedups.

Symbolic and automatic differentiation of languages

Formal languages are usually defined in terms of set theory. Choosing type theory instead gives us languages as type-level predicates over strings. Applying a language to a string yields a type whose elements are language membership proofs describing how a string parses in the language. The usual building blocks of languages (including union, concatenation, and Kleene closure) have precise and compelling specifications uncomplicated by operational strategies and are easily generalized to a few general domain-transforming and codomain-transforming operations on predicates. A simple characterization of languages (and indeed functions from lists to any type) captures the essential idea behind language “differentiation” as used for recognizing languages, leading to a collection of lemmas about type-level predicates. These lemmas are the heart of two dual parsing implementations—using (inductive) regular expressions and (coinductive) tries—each containing the same code but in dual arrangements (with representation and primitive operations trading places). The regular expression version corresponds to symbolic differentiation, while the trie version corresponds to automatic differentiation. The relatively easy-to-prove properties of type-level languages transfer almost effortlessly to the decidable implementations. In particular, despite the inductive and coinductive nature of regular expressions and tries respectively, we need neither inductive nor coinductive/bisimulation arguments to prove algebraic properties.

Symbolic Differentiation Algorithm for Inverse Dynamics of Serial Robots With Flexible Joints

AbstractA new symbolic differentiation algorithm is proposed in this paper to automatically generate the inverse dynamics of flexible-joint robots in symbolic form, and results obtained can be used in real-time applications. The proposed method with O(n) computational complexity is developed based on the recursive Newton–Euler algorithm, the chain rule of differentiation, and the computer algebra system. The input of the proposed algorithm consists of symbolic matrices describing the kinematic and dynamic parameters of the robot. The output is the inverse dynamics solution written in portable and optimized code (C-code/Matlab-code). An exemplary, numerical simulation for inverse dynamics of the Kuka LWR4 robot with seven flexible joints is conducted using matlab, in which the computational time per cycle of inverse dynamics is about 0.02 ms. The numerical example provides very good matching results versus existing methods, while requiring much less computation time and complexity.

Automatic Differentiation in MetaPhysicL and Its Applications in MOOSE

Efficient solution via Newton’s method of nonlinear systems of equations requires an accurate representation of the Jacobian, corresponding to the derivatives of the component residual equations with respect to the degrees of freedom. In practice these systems of equations often arise from spatial discretization of partial differential equations used to model physical phenomena. These equations may involve domain motion or material equations that are complex functions of the systems’ degrees of freedom. Computing the Jacobian by hand in these situations is arduous and prone to error. Finite difference approximations of the Jacobian or its action are prone to truncation error, especially in multiphysics settings. Symbolic differentiation packages may be used, but often result in an excessive number of terms in realistic model scenarios. An alternative to symbolic and numerical differentiation is automatic differentiation (AD), which propagates derivatives with every elementary operation of a computer program, corresponding to continual application of the chain rule. Automatic differentiation offers the guarantee of an exact Jacobian at a relatively small overhead cost. In this work, we outline the adoption of AD in the Multiphysics Object Oriented Simulation Environment (MOOSE) via the MetaPhysicL package. We describe the application of MOOSE’s AD capability to several sets of physics that were previously infeasible to model via hand-coded or Jacobian-free simulation techniques, including arbitrary Lagrangian-Eulerian and level-set simulations of laser melt pools, phase-field simulations with free energies provided through neural networks, and metallic nuclear fuel simulations that require inner Newton loop calculation of nonlinear material properties.

Dynamic Response Analysis of a High Glide Ratio Parachute System

This paper is concerned with the dynamic stability study of a gliding parachute-payload system along its gliding path. To scrutinize the respective dynamic response characteristics after releasing from high altitude, a modified multi-body model is developed. In the stability analysis procedure, the yawing motion of the payload is considered in system dynamics, which in turn creates a state-dependent matrix in the stability analysis and makes the linearization algorithm more cumbersome. To solve the problem, a unified Jacobian-based symbolic differentiation algorithm is implemented and the dynamics is linearized about various operating points along gliding segment of a typical planned trajectory. Based on results, the system has short period and phugoid modes in longitudinal channel just like an aircraft. In addition to dutch roll mode, the system has a low frequency coupled roll-spiral mode in lateral-directional channel which is a result of effective canopy anhedral angle. It is shown, the coupled mode can be decomposed into two distinct roll and spiral modes for small anhedral angles. Based on results, as the parachute descends, both the period and damping ratio for the short period mode were increased by 18 and 30%, respectively. For the phugoid mode the period of oscillations is decreased by 20% and the damping ratio, almost remains constant. For the lateral-directional channel,. As the parachute descends, the dutch roll mode is destabilized whereas the other modes are stabilized. Furthermore, from a practical point of view, lengthening the suspension lines stabilizes the coupled roll-spiral mode whereas destabilizes the other modes.

Computation of Periodic Solutions to Pendulum Type Systems with a Small Parameter

Periodic solutions of pendulum-type ODE systems are considered. Finding such solutions is a classical problem in mechanics. Numerous methods are available for computing periodic solutions, and these methods have existed as long as the problems themselves. However, they were designed for manual calculation, and attempts to program them in computer algebra systems (CAS) are sometimes ineffective. For computing such solutions, we propose a method intended for CAS. The method is based on the use of high-order variational equations and symbolic differentiation. It is shown on a number of examples that all computations are reduced to operations with polynomials.

Computation of higher order Lie derivatives on the Infinity Computer

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the Infinity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique.

An Intersectional Exploration of Ethnic Identity, Class, and Inequality in a Multiethnic Context

In this article, we examine ethnicity, class/status, (ethnic) identity, and inequality from an intersectional perspective in two organizations in a multi-ethnic developing country. Inequalities and bounded/literal identity categories - such as gender, race/ethnicity, and class/status – and their interlinkages have been widely studied in organizations in the global North. However, how intersecting bounded and symbolic identity categories may (re)produce and legitimise inequalities in organizations in the global South, specifically in multi-ethnic contexts, remain under-researched. From analyzing vignette and interview data through an intersectionalist lens, we introduce a construct, symbolic identity differentiation, and advance two key findings. The suggests that how individuals construct identities in ethnoculturally diverse contexts, ad in effect the ethnocultural landscape, is changing. The second posits that managing diversity and inclusion in organizations in the global South is contingent on how individuals make meaning of inequalities through perceptions of status, and how they experience their identities, symbolically, as well as literally in these settings.

A machine learning based plasticity model using proper orthogonal decomposition

Data-driven material models have many advantages over classical numerical approaches, such as the direct utilization of experimental data and the possibility to improve performance of predictions when additional data is available. One approach to develop a data-driven material model is to use machine learning tools. These can be trained offline to fit an observed material behaviour and then be applied in online applications. However, learning and predicting history dependent material models, such as plasticity, is still challenging. In this work, a machine learning based material modelling framework is proposed for both elasticity and plasticity. The machine learning based hyperelasticity model is developed with the Feed forward Neural Network (FNN) directly whereas the machine learning based plasticity model is developed by using of a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN). In order to account for the loading history, the accumulated absolute strain is proposed to be the history variable of the plasticity model. Additionally, the strain–stress sequence data for plasticity is collected from different loading–unloading paths based on the concept of sequence for plasticity. By means of the POD, the multi-dimensional stress sequence is decoupled leading to independent one dimensional coefficient sequences. In this case, the neural network with multiple output is replaced by multiple independent neural networks each possessing a one-dimensional output, which leads to less training time and better training performance. To apply the machine learning based material model in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiation tool AceGen. The effectiveness and generalization of the presented models are investigated by a series of numerical examples using both 2D and 3D finite element analysis.