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Physical models often contain unknown functions and relations. In order to gain more insights into the nature of physical processes, these unknown functions have to be identified or reconstructed. Mathematically, we can formulate this…
The statistical essence of the Transformer architecture has long remained elusive: Is it a universal approximator, or a neural network version of known computational algorithms? Through rigorous algebraic proof, we show that the latter…
The operator fidelity is a measure of the information-theoretic distinguishability between perturbed and unperturbed evolutions. The response of this measure to the perturbation may be formulated in terms of the operator fidelity…
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $\omega$, coupled by means of springs. Such systems have been studied before, and appear in various models. In…
The metabolic process in a cell is modeled with the use of the Fourier transformation. The histograms of the invariant measures of chaotic attractors are constructed. In particular, a scenario of adaptation of the metabolic process under a…
Large language models exhibit sophisticated capabilities, yet understanding how they work internally remains a central challenge. A fundamental obstacle is that training selects for behavior, not circuitry, so many weight configurations can…
The one-dimensional quantum harmonic oscillator problem is examined via the Laplace transform method. The stationary states are determined by requiring definite parity and good behaviour of the eigenfunction at the origin and at infinity.
Fourier Transforms is a first in a series of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such…
Transformers have dominated empirical machine learning models of natural language processing. In this paper, we introduce basic concepts of Transformers and present key techniques that form the recent advances of these models. This includes…
Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet…
Teaching by direct models in science has been weakening the learning process of the students, because the real problems in engineering are not solved by direct models instead commonly they are solved by inverse models. On the other hand,…
When a Simple Harmonic Oscillator (SHO) wave function is used as an effective wave function, a very important parameter in the SHO wave function is the effective $\beta$ value. We obtain the analytical expression of $\beta_{eff}$…
Can general-purpose AI architectures go beyond prediction to discover the physical laws governing the universe? True intelligence relies on "world models" -- causal abstractions that allow an agent to not only predict future states but…
Neutrino oscillations are studied in the general framework of open quantum systems by means of extended dynamics that take into account possible dissipative effects. These new phenomena induce modifications in the neutrino oscillation…
The asymmetrically forced, damped Duffing oscillator is introduced as a prototype model for analyzing the homoclinic tangle of symmetric dissipative systems with \textit{symmetry breaking} disturbances. Even a slight fixed asymmetry in the…
Transformer models underpin many recent advances in practical machine learning applications, yet understanding their internal behavior continues to elude researchers. Given the size and complexity of these models, forming a comprehensive…
Previous work on the learnability of transformers \textemdash\ focused primarily on examining their ability to approximate specific algorithmic patterns through training \textemdash\ has largely been data-driven, offering only probabilistic…
Through a set of generators that preserves the hermiticity and trace of density matrices, we analyze the damping of harmonic oscillator in open quantum systems into four modes, distinguished by their specific effects on the covariance…
This paper studies hidden oscillations appearing in electromechanical systems with and without equilibria. Three different systems with such effects are considered: translational oscillator-rotational actuator, drilling system actuated by a…
Coupled quantum harmonic oscillators, studied by many authors using many different techniques over the decades, are frequently used toy-models to study open quantum systems. In this manuscript, we explicitly study the simplest oscillator…