Application of Sparse Identification of Nonlinear Dynamics for Physics-Informed Learning

Advances in machine learning and deep neural networks has enabled complex engineering tasks like image recognition, anomaly detection, regression, and multi-objective optimization, to name but a few. The complexity of the algorithm architecture, e.g., the number of hidden layers in a deep neural network, typically grows with the complexity of the problems they are required to solve, leaving little room for interpreting (or explaining) the path that results in a specific solution. This drawback is particularly relevant for autonomous aerospace and aviation systems, where certifications require a complete understanding of the algorithm behavior in all possible scenarios. Including physics knowledge in such data-driven tools may improve the interpretability of the algorithms, thus enhancing model validation against events with low probability but relevant for system certification. Such events include, for example, spacecraft or aircraft sub-system failures, for which data may not be available in the training phase. This paper investigates a recent physics-informed learning algorithm for identification of system dynamics, and shows how the governing equations of a system can be extracted from data using sparse regression. The learned relationships can be utilized as a surrogate model which, unlike typical data-driven surrogate models, relies on the learned underlying dynamics of the system rather than large number of fitting parameters. The work shows that the algorithm can reconstruct the differential equations underlying the observed dynamics using a single trajectory when no uncertainty is involved. However, the training set size must increase when dealing with stochastic systems, e.g., nonlinear dynamics with random initial conditions

Investigation of Fuzzy Inductive Modeling Method in Forecasting Problems

This paper is devoted to the investigation and application of fuzzy inductive modeling method group method of data handling (GMDH) in problems of forecasting in the financial sphere. GMDH method belongs to self-organizing methods and allows to discover internal hidden laws in the appropriate object area. The advantage of GMDH algorithms is the possibility of constructing optimal models. In the generalization of GMDH in case of uncertainty, new method fuzzy GMDH is described which enables to construct fuzzy models almost automatically. The algorithms of fuzzy GMDH for different membership functions are considered. The extensions of fuzzy GMDH for different partial descriptions—orthogonal polynomials of Chebyshev and trigonometric polynomials of Fourier—are considered. The problem of adaptation of fuzzy models obtained by FGMDH is considered, and the corresponding adaptation algorithm is described. The experimental investigations of the suggested FGMDH in the problem of forecasting macroeconomic indicators of Ukraine are carried out, and comparison with classic GMDH and neural network BP is performed

The generalization complexity measure for continuous input data

We introduce in this work an extension for the generalization complexity measure to continuous input data. The measure, originally defined in Boolean space, quantifies the complexity of data in relationship to the prediction accuracy that can be expected when using a supervised classifier like a neural network, SVM, and so forth. We first extend the original measure for its use with continuous functions to later on, using an approach based on the use of the set of Walsh functions, consider the case of having a finite number of data points (inputs/outputs pairs), that is, usually the practical case. Using a set of trigonometric functions a model that gives a relationship between the size of the hidden layer of a neural network and the complexity is constructed. Finally, we demonstrate the application of the introduced complexity measure, by using the generated model, to the problem of estimating an adequate neural network architecture for real-world data sets.http://dx.doi.org/10.1155/2014/815156publishedVersionFil: Gómez, Iván. Universidad de Málaga. Departamento de Lenguajes y Ciencias de la Computación; España.Fil: Franco, Leonardo. Universidad de Málaga. Departamento de Lenguajes y Ciencias de la Computación; España.Fil: Jerez, José M. Universidad de Málaga. Departamento de Lenguajes y Ciencias de la Computación; España.Fil: Cannas, Sergio Alejandro. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Osenda, Omar. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Otras Ciencias de la Computación e Informació