相关论文: How Long Does Infinite Width Last? Signal Propagat…

Infinite width limits of deep neural networks often have tractable forms. They have been used to analyse the behaviour of finite networks, as well as being useful methods in their own right. When investigating infinitely wide convolutional…

Although overparameterized models have achieved remarkable practical success, their theoretical properties, particularly their generalization behavior, remain incompletely understood. The well known double descents phenomenon suggests that…

The widest spanwise scales in turbulent channel flows are studied through the use of three periodic channel-flow simulations at friction Reynolds number $\mathrm{Re}_{\tau}=550$. The length and height of the channels are the same in all…

A theoretical investigation is made of acoustic wave propagation in a periodically stubbed waveguide. In general the waveguide segments and stubs are made of different materials. The acoustic wave in such a system has two independent…

The energy transition towards high shares of renewable energy will affect the stability of electricity grids in many ways. Here, we aim to study its impact on propagation of disturbances by solving nonlinear swing equations describing…

This paper investigates the problem of synthesizing joint distributions in the finite-length regime. For a fixed blocklength $n$ and an upper bound on the distribution approximation $\epsilon$, we prove a capacity result for fixed-length…

Many real-world scale-free networks, such as neural networks and online communication networks, consist of a fixed number of nodes but exhibit dynamic edge fluctuations. However, traditional models frequently overlook scenarios where the…

We develop a diagrammatic theory for transport of waves in disordered media with weak nonlinearity. We first represent the solution of the nonlinear wave equation as a nonlinear Born series. From this, we construct nonlinear ladder and…

To model time-varying nonlinear temporal dynamics in sequential data, a recurrent network capable of varying and adjusting the recurrence depth between input intervals is examined. The recurrence depth is extended by several intermediate…

Experimentally, the phase of the amplitude for electron transmission through a quantum dot (transmission phase) shows the same pattern between consecutive resonances. Such universal behavior, found for long sequences of resonances, is…

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The…

A prior work (see Chapter 8 of the book, ``Graphs and Networks: Transfinite and Nonstandard,'' Birkhauser-Boston, Cambridge, Mass., USA, 2004) examined the propagation of an electromagnetic wave on a transfinite transmission line,…

The time-dependent diffusion spreadability $\mathcal{S}(t)$ is a powerful dynamical probe of the microstructure of two-phase heterogeneous media across length scales [Torquato, S., \emph{Phys. Rev. E.}, 104 054102 (2021)]. It has been shown…

Recurrent neural networks have been extensively studied in the context of neuroscience and machine learning due to their ability to implement complex computations. While substantial progress in designing effective learning algorithms has…

Continuous-time signals are well known for not being perfectly localized in both time and frequency domains. Conversely, a signal defined over the vertices of a graph can be perfectly localized in both vertex and frequency domains. We…

We present a detailed discussion of scalar wave propagation and light intensity transport in three dimensional random dielectric media with optical gain. The intrinsic length and time scales of such amplifying systems are studied and…

We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-$1$ norm between the…

We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the…

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its…

State space models (SSMs) have shown remarkable empirical performance on many long sequence modeling tasks, but a theoretical understanding of these models is still lacking. In this work, we study the learning dynamics of linear SSMs to…