Related papers: Pattern formation with pde2path -- a tutorial

This work introduces InJecteD, a framework for interpreting Denoising Diffusion Probabilistic Models (DDPMs) by analyzing sample trajectories during the denoising process of 2D point cloud generation. We apply this framework to three…

We investigate the scattering matrix in mass-deformed N>=4 Chern-Simons models including as special cases the BLG and ABJM theories of multiple M2 branes. Curiously the structure of this scattering matrix in three spacetime dimensions is…

In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive…

We propose a method for reconstructing 3D shapes from 2D sketches in the form of line drawings. Our method takes as input a single sketch, or multiple sketches, and outputs a dense point cloud representing a 3D reconstruction of the input…

We develop a consistent method for estimating the parameters of a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. Path signatures are iterated integrals of a given path with the…

Two-dimensional simulations of the coarsening process of the isotropic/smectic-A phase transition are presented using a high-order Landau-de Gennes type free energy model. Defect annihilation laws for smectic disclinations, elementary…

Parameter estimation for non-stationary stochastic differential equations (SDE) with an arbitrary nonlinear drift, and nonlinear diffusion is accomplished in combination with a non-parametric clustering methodology. Such a model-based…

Mechanochemical processes on surfaces such as the cellular cortex or epithelial sheets, play a key role in determining patterns and shape changes of biological systems. To understand the complex interplay of hydrodynamics and material flows…

Molecular dynamics simulation has been used to model pattern formation in three-dimensional Rayleigh--Benard convection at the discrete-particle level. Two examples are considered, one in which an almost perfect array of hexagonally-shaped…

Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a…

Semiconductor nanostructures based on two dimensional electron gases (2DEGs) have the potential to provide new approaches to sensing, information processing, and quantum computation. Much is known about electron transport in 2DEG…

Structure formation in 1+1 dimensions is considered, with emphasis on the effects of shell-crossing. The breakdown of the perturbative expansion beyond shell-crossing is discussed, and it is shown, in a simple example, that the perturbative…

This paper is the second in a two-part exposition on {\it surface-directed spinodal decomposition} (SDSD), i.e., the interplay of kinetics of wetting and phase separation at a surface which is wetted by one of the components of a binary…

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of…

Pattern forming systems allow for a wealth of states, where wavelengths and orientation of patterns varies and defects disrupt patches of monocrystalline regions. Growth of patterns has long been recognized as a strong selection mechanism.…

In this paper mathematical models for the evolutionary conserved Notch-Delta pathway are developed and analyzed in order to better understand how two neighboring biological cells can become different. We pursue a structure-based…

We investigate the iterative construction of discrete Laplacians on 2D square lattices, revealing emergent fractal-like patterns shaped by modular arithmetic. While classical 2222-style iterations reproduce known structures such as the…

Patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical wavelength or domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally…

High-dimensional point cloud data arise across many scientific domains, especially single-cell biology. The shapes or topologies of these datasets determine the types of information that can be extracted. For example, clustered data…

We study the dynamic after a smooth quench across a continuous transition from the disordered phase to the ordered phase. Based on scaling ideas, linear response and the spectrum of unstable modes, we develop a theoretical framework, valid…