Related papers: Flowing Through Layers: A Continuous Dynamical Sys…
We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme --with constant or decreasing step-- may explode and implicit…
The scaling law, a cornerstone of Large Language Model (LLM) development, predicts improvements in model performance with increasing computational resources. Yet, while empirically validated, its theoretical underpinnings remain poorly…
Transformers process tokens in parallel but are temporally shallow: at position $t$, each layer attends to key-value pairs computed based on the previous layer, yielding a depth capped by the number of layers. Recurrent models offer…
The aim of this paper is to provide a mathematical analysis of transformer architectures using a self-attention mechanism with layer normalization. In particular, observed patterns in such architectures resembling either clusters or uniform…
Transformers are increasingly adopted for modeling and forecasting time-series, yet their internal mechanisms remain poorly understood from a dynamical systems perspective. In contrast to classical autoregressive and state-space models,…
Recent theory and experiments have shown how the buildup of a high-concentration polymer layer at a one-dimensional solvent-air interface can lead to an evaporation rate that scales with time as $t^{-1/2}$ and that is insensitive to the…
In this paper, we study the stability of various difference approximations of the Euler-Korteweg equations. This system of evolution PDEs is a classical isentropic Euler system perturbed by a dispersive (third order) term. The Euler…
While modern representation learning relies heavily on global error signals, decentralized algorithms driven by local interactions offer a fundamental distributed alternative. However, the macroscopic convergence properties of these…
Motivated by the normal form of a fast-slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of the explicit Euler method. Tracking…
The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical…
Normalizing flows transform a simple base distribution into a complex target distribution and have proved to be powerful models for data generation and density estimation. In this work, we propose a novel type of normalizing flow driven by…
The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel.…
There are two distinct regimes commonly used to model traveling waves in stratified water: continuous stratification, where the density is smooth throughout the fluid, and layer-wise continuous stratification, where the fluid consists of…
Currently, Flow matching methods aim to compress the iterative generation process of diffusion models into a few or even a single step, with MeanFlow and FreeFlow being representative achievements of one-step generation based on Ordinary…
A multi-scale model for the evolution of the velocity gradient tensor in fully developed turbulence is proposed. The model is based on a coupling between a ``Restricted Euler'' dynamics [{\it P. Vieillefosse, Physica A, {\bf 14}, 150…
We study the well-posedness of a class of dynamical flow network systems describing the dynamical mass balance among a finite number of cells exchanging flow of a commodity between themselves and with the external environment. Systems in…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
This paper concerns the validity of the Prandtl boundary layer theory for steady, incompressible Navier-Stokes flows over a rotating disk. We prove that the Navier Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid…
Normalizing flows have grown more popular over the last few years; however, they continue to be computationally expensive, making them difficult to be accepted into the broader machine learning community. In this paper, we introduce a…
In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or $\alpha$-models. These models describe both nonlocal and local dynamics, with one…