Related papers: Developmental Partial Differential Equations
Solving partial differential equations (PDEs) on complex domains can present significant computational challenges. The Diffuse Domain Method (DDM) is an alternative that reformulates the partial differential equations on a larger, simpler…
In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A function that has a prescribed value on the domain in which a differential…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates…
There are three fundamental physical processes that gives rise to the morphology of a surface: deposition, surface diffusion and desorption. The characteristics of the interfaces generated by the combination of deposition and surface…
Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice…
A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly…
Differential equations need boundary conditions (BC's) for their solution. It is commonly acknowledged that differential equations and BC's are representative of independent physical processes, and no correlations between them is required.…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
Diffusion is the macroscopic manifestation of disordered molecular motion. Mathematically, diffusion equations are partial differential equations describing the fluid-like large-scale dynamics of parcels of molecules. Spatially…
We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that,…
In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence is changing sign, as in models for metamaterials.We focus on the construction of a fundamental…
The data-driven models allow one to define the model structure in cases when a priori information is not sufficient to build other types of models. The possible way to obtain physical interpretation is the data-driven differential equation…
We propose DiffusionRollout, a novel selective rollout planning strategy for autoregressive diffusion models, aimed at mitigating error accumulation in long-horizon predictions of physical systems governed by partial differential equations…
We demonstrate a method for filtering images defined on curved surfaces embedded in 3D. Applications are noise removal and the creation of artistic effects. Our approach relies on in-surface diffusion: we formulate Weickert's edge/coherence…
The role of step edge diffusion (SED) in epitaxial growth is investigated. To this end we revisit and extend a recently introduced simple cubic solid-on-solid model, which exhibits the formation and coarsening of pyramid or mound like…
This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate…
Numerical solutions of partial differential equations (PDEs) on manifolds continues to generate a lot of interest among scientists in the natural and applied sciences. On the other hand, recent developments of 3D scanning and computer…
For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied in this paper. The velocity of the evolving surface is not given explicitly, but depends on the…
In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…