Related papers: Generalized flows and singular ODEs on differentia…
We define a class of geometric flows on a complete K\"ahler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schr\"odinger equations…
It is shown for a simple ODE that it has many symmetry groups beyond its usual Lie group symmetries, when its generalized solutions are considered within the nowhere dense differential algebra of generalized functions.
This work aims to initiate a discussion on finding solutions to non-homoge\-neous differential equations in terms of generalized functions. For simplicity, we conduct the analysis within the specific context of the stationary Klein-Gordon…
The paper examines one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -- flat regions of solutions. A key…
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the…
We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models,…
Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying…
The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of…
We study the convergence of the derivative expansion for flow equations. The convergence strongly depends on the choice for the infrared regularisation. Based on the structure of the flow, we explain why optimised regulators lead to better…
We study the behaviour of solutions of ordinary differential equations of the second order with singular points, where the coefficients of the second-order derivative vanishes. In particular, we consider solutions entering a singular point…
In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely…
The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment…
There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations.…
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
Recently, there has been a surge of interest in incorporating neural networks into particle filters, e.g. differentiable particle filters, to perform joint sequential state estimation and model learning for non-linear non-Gaussian…
We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…
The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations…