The universal approximation Theorem 3.1 immediately implies another universal approximation Thus y (t) solves the ODE (2.6), with initial condition y (0) = y (0) = 0 . Reconstruction of a continuous signal from its sine transform. Step 0: (Equicontinuity) We recall the following fact from topology. F (τ):= null f (τ), for τ 0, f ( τ), for τ 0. Since F is odd, the Fourier transform of F is given by We provide the details below. The next step in the proof of the fundamental Lemma 3.5 needs the following preliminary result in By (B.3), this implies that It follows from Lemma 3.4 that for any input By the sine transform reconstruction Lemma B.1, there exists It follows from Lemma 3.6, that there exists Indeed, Lemma 3.7 shows that time-delays of any given input signal can be approximated with any Step 1: By the Fundamental Lemma 3.5, there exist It follows from Lemma 3.6, that there exists an oscillator Step 3: Finally, by Lemma 3.8, there exists an oscillator network,

Given this wide prevalence of (networks of) oscillators in nature and man-made devices, it is not surprising that oscillators have inspired various machine learning architectures in recent years.

Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of that encompasses these architectures and prove that neural oscillators are universal, i.e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.

We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_\varepsilon$ of excursions of magnitude at least $\varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.

Rhythmic fluctuations in acoustic energy and accompanying neuronal excitations in cortical oscillations are characteristic of human speech, yet whether a corresponding rhythmicity inheres in the articulatory movements that generate speech remains unclear. The received understanding of speech movements as discrete, goal-oriented actions struggles to make contact with the rhythmicity findings. In this work, we demonstrate that an unintuitive -- but no less principled than the conventional -- representation for discrete movements reveals a pervasive limit cycle organization and unlocks the recovery of previously inaccessible rhythmic structure underlying the motor activity of speech. These results help resolve a time-honored tension between the ubiquity of biological rhythmicity and discreteness in speech, the quintessential human higher function, by revealing a rhythmic organization at the most fundamental level of individual articulatory actions.

The question whether algorithmic trading systems (ATS) can improve human trading in terms of effectiveness is eliciting an increasingly relevant debate among traders and investors, as well as quantitative studies that address this issue through numerical testing [[9]]. In recent years, the discussion regarding whether algorithmic trading systems (ATS) can surpass human traders in terms of efficiency, consistency, and adaptability has gained significant traction in both academic and professional circles. Empirical evidence indicates that algorithmic strategies tend to exhibit superior performance in volatile or declining markets, whereas human-managed funds may retain a relative advantage during upward market trends due to behavioral and intuitive factors [[2]]. Moreover, large-scale behavioral studies reveal that algorithms largely eliminate well-known cognitive biases such as the disposition effect that continue to affect human traders [[23]]. Complementary research has also emphasized the growing integration of artificial intelligence and machine learning methods in modern ATS, which enhances predictive accuracy and execution speed [[7]]. Nonetheless, experimental findings suggest that algorithmic trading may still be constrained by design limitations, challenging the notion of its absolute superiority over human decision-making [[16]]. These findings collectively indicate that algorithmic and human trading approaches might be best viewed as complementary, each offering unique strengths under different market conditions.

Discrete methodologies use the discrete language of finite state machines while continuous methodologies use the continuous modelling language of differential equations. The resulting need to interface discrete automata and continuous differential equations has made the theory of hybrid and cyberphysical systems a key component of control design. The advantages of separating the discrete and the continuous design are many. Yet, this separation has become a bottleneck in the design of scalable architectures that can control and regulate across a range of spatial and temporal scales [1, 2]. The divide between automation and regulation does not exist in animal machines. Nervous systems make decisions and regulate muscle activation with one and the same architecture that is event-based rather than continuous or discrete.

Recent advances in artificial intelligence allow researchers to recover laws of physics and predict dynamics of physical systems from observed data by utilizing machine learning techniques, e.g.,

In this work, we ask what advantages biologically inspired ANN architecture can provide in the domain of motor control.

Here, we propose a new family of "dynamical" priors