arXiv research

A scalable deep learning framework accelerates training of large neural networks for solving 3D Poisson equations.

Derives a Hamiltonian model for 3D axially symmetric magnetohydrodynamics.

We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on di…

Derives equations for interacting Lie-Poisson systems using 2-cocycle extensions.

The study characterizes and proves properties of 3D Poisson quasi-Nijenhuis manifolds.

The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…

Proves local solvability for $G_2$-structures with Poisson equations.

New solutions to 3D integrability equations using quantum cluster algebras.

The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…

The aim of this paper is to present the stochastic Poisson equations associated to Lie algebroids. The stochastic Poisson equations associated to a refinement of a concrete principal bundle are determined.

Study of 3d-3d correspondence involving $q$-Weyl algebra and 3d-index.

Study interior estimates for solutions of Poisson equation on Riemann surfaces.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demo…

New hierarchies and equations derived from Poisson structures.

In this paper, we study the Poisson equation and heat equation in a model matrix geometry $M_n$. Our main results are about the Poisson equation and global behavior of the heat equation on $M_n$. We can show that if $c_0$ is the initial positive definite matrix in $M_n$, then $c(t)$ exists for all time and is positive …

Symplectic groupoids create Poisson integrators for complex systems.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…

Study improves Poisson equation solutions on various manifolds.

We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.

Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.

Study bends 2D surfaces in 3D space using special equations.

Formula connects $G_2$-structure geometry to Poisson equation.

Study Einstein warped-product manifolds with specific curvature conditions.

The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.

Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to…

NWoS solves high-dimensional Poisson equations using neural networks.

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…

V1 cortex reconstructs images as Poisson equation solutions with varying weights.

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\varepsilon})$ for some $\varepsilon >0,$ where $r$ is the distance function to a fixed point, then t…

In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.

We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets …

A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.

Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of mot…

The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.

Invariant reduction preserves Poisson structures in PDEs.

In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our cons…

The Poisson equation is commonly encountered in engineering, for instance in computational fluid dynamics (CFD) where it is needed to compute corrections to the pressure field to ensure the incompressibility of the velocity field. In the present work, we propose a novel fully convolutional neural network (CNN) architec…

The paper studies $G_2$-Poisson equations on 7-spheres and classifies invariant solutions.

The paper explores squircles and their 3D applications.

Poisson learning improves graph-based semi-supervised learning at very low label rates.

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimension…

Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.

Physics-constrained neural nets solve EM fields of charged particle beams.

Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of rigid motions of ${\Bbb R}^3$. Looking the adjoint representation of ${\rm SE}(3)…

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vect…

Solves optimal stopping problem with Poisson constraints using jumps.