The symmetrical cell eigenfrequency method for periodic structure stop-band definition

The Floquet theory that is used in order to find the stop-bands is not defined for non-linear cases. The eigenfrequencies of a symmetrical periodicity cell can serve as alternative indicators of stop-band boundaries. In the linear cases, eigenfrequencies of a structure with symmetrical boundary conditions are exactly placed on stop-bands boundaries. In the non-linear cases, however, the notion of the Floquet zones are not clearly defined. However, the eigenfrequencies method show the stop-band positions that agree with the energy flow analysis. In the cases are considered, the axial rod vibration, the radial-periodic membrane case, the axial rods connected with spring with non-linear stiffness

Comparison of Single- and Multi- Objective Optimization Quality for Evolutionary Equation Discovery

Evolutionary differential equation discovery proved to be a tool to obtain equations with less a priori assumptions than conventional approaches, such as sparse symbolic regression over the complete possible terms library. The equation discovery field contains two independent directions. The first one is purely mathematical and concerns differentiation, the object of optimization and its relation to the functional spaces and others. The second one is dedicated purely to the optimizational problem statement. Both topics are worth investigating to improve the algorithm's ability to handle experimental data a more artificial intelligence way, without significant pre-processing and a priori knowledge of their nature. In the paper, we consider the prevalence of either single-objective optimization, which considers only the discrepancy between selected terms in the equation, or multi-objective optimization, which additionally takes into account the complexity of the obtained equation. The proposed comparison approach is shown on classical model examples -- Burgers equation, wave equation, and Korteweg - de Vries equation

Adaptation of NEMO-LIM3 model for multigrid high resolution Arctic simulation

High-resolution regional hindcasting of ocean and sea ice plays an important role in the assessment of shipping and operational risks in the Arctic Ocean. The ice-ocean model NEMO-LIM3 was modified to improve its simulation quality for appropriate spatio-temporal resolutions. A multigrid model setup with connected coarse- (14 km) and fine-resolution (5 km) model configurations was devised. These two configurations were implemented and run separately. The resulting computational cost was lower when compared to that of the built-in AGRIF nesting system. Ice and tracer boundary-condition schemes were modified to achieve the correct interaction between coarse- and fine grids through a long ice-covered open boundary. An ice-restoring scheme was implemented to reduce spin-up time. The NEMO-LIM3 configuration described in this article provides more flexible and customisable tools for high-resolution regional Arctic simulations

Towards stable real-world equation discovery with assessing differentiating quality influence

This paper explores the critical role of differentiation approaches for data-driven differential equation discovery. Accurate derivatives of the input data are essential for reliable algorithmic operation, particularly in real-world scenarios where measurement quality is inevitably compromised. We propose alternatives to the commonly used finite differences-based method, notorious for its instability in the presence of noise, which can exacerbate random errors in the data. Our analysis covers four distinct methods: Savitzky-Golay filtering, spectral differentiation, smoothing based on artificial neural networks, and the regularization of derivative variation. We evaluate these methods in terms of applicability to problems, similar to the real ones, and their ability to ensure the convergence of equation discovery algorithms, providing valuable insights for robust modeling of real-world processes

Automated Differential Equation Solver Based on the Parametric Approximation Optimization

The classical numerical methods for differential equations are a well-studied field. Nevertheless, these numerical methods are limited in their scope to certain classes of equations. Modern machine learning applications, such as equation discovery, may benefit from having the solution to the discovered equations. The solution to an arbitrary equation typically requires either an expert system that chooses the proper method for a given equation, or a method with a wide range of equation types. Machine learning methods may provide the needed versatility. This article presents a method that uses an optimization algorithm for a parameterized approximation to find a solution to a given problem. We take an agnostic approach without dividing equations by their type or boundary conditions, which allows for fewer restrictions on the algorithm. The results may not be as precise as those of an expert; however, our method enables automated solutions for a wide range of equations without the algorithm’s parameters changing. In this paper, we provide examples of the Legendre equation, Painlevé transcendents, wave equation, heat equation, and Korteweg–de Vries equation, which are solved in a unified manner without significant changes to the algorithm’s parameters

Automated differential equation solver based on the parametric approximation optimization

The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of the equations, on which the convergence with a given parameter set or range is proved. Only a few "cheap and dirty" numerical methods converge on a wide class of equations without parameter tuning with the lower approximation order price. The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation. The result may not be as precise as an expert one. However, it allows solving the wide class of equations in an automated manner without the algorithm's parameters change