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Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties. This results in models that often do not capture any underlying continuous dynamics—either of the system of interest, or indeed of any related system. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates an underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications. Many challenging problems in science and engineering rely on the study of dynamical systems that evolve continuously in time, and yet this feature proves difficult to be captured reliably using modern machine learning (ML) models. This paper develops a convergence test based on numerical analysis and illustrates how this methodology can be combined with existing ML techniques to validate models for science and engineering applications. [ABSTRACT FROM AUTHOR]

In terms of the first- and second-order sensitivities of the complex mode for a symmetric damped linear discrete dynamic system with respect to design parameters, the algebraic method, direct method, full-mode method and truncated modal method corresponding to full-mode method are proposed. These four algorithms insist on using the expression forms in N-dimensional vector space, which provide the convenience in engineering application. In the numerical examples, the correctness and validity of all the proposed algorithms are demonstrated for the single-frequency, closed-frequency and multiple-frequency systems. The algebraic method, direct method and full-mode method are accurate algorithms. No statement is made that one method is absolutely superior to the other, but this paper provides the chance to compare their efficiency and accuracies of all the three accurate algorithms. Truncated modal method proposed here is an approximate algorithm. The truncation criterion is simple and efficient, which is a more competitive alternative to the full-mode method, and it is shown that the truncated modal method has high precision and good convergence speed for sensitivity analysis by numerical examples. [ABSTRACT FROM AUTHOR]

The present paper addresses the dynamical motion of two degrees-of-freedom (DOF) auto-parametric system consisting of a connected rolling cylinder with a damped spring. This motion has been considered under the action of an excitation force. Lagrange's equations from second kind are utilized to obtain the governing system of motion. The uniform approximate solutions of this system are acquired up to higher order of approximation using the technique of multiple scales in view of the abolition of emerging secular terms. All resonance cases are characterized, and the primary and internal resonances are examined simultaneously to set up the corresponding modulation equations and the solvability conditions. The time histories of the amplitudes, modified phases, and the obtained solutions are graphed to illustrate the system's motion at any given time. The nonlinear stability approach of Routh–Hurwitz is used to examine the stability of the system, and the different zones of stability and instability are drawn and discussed. The characteristics of the nonlinear amplitude for the modulation equations are investigated and described, as well as their stabilities. The gained results can be considered novel and original, where the methodology was applied to a specific dynamical system. [ABSTRACT FROM AUTHOR]

In this paper, the Rayleigh beam system with two dynamical boundary controls is treated. Theoretically, the well-posedness of the weak solution is obtained. Later, we discretize the system by using the Implicit Euler scheme in time and the P 3 Hermite finite element in space. In addition, we show the decay of the discrete energy and we establish some a priori error estimates. Finally, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]