arXiv research

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

We provide a partial solution to the isoperimetric problem in the Heisenberg group.

Randomized matrix compression techniques, such as the Johnson-Lindenstrauss transform, have emerged as an effective and practical way for solving large-scale problems efficiently. With a focus on computational efficiency, however, forsaking solutions quality and accuracy becomes the trade-off. In this paper, we investi…

In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete non-compact Riemannian manifold M or Kahler-Ricci flow. We show that under a natural assumption, a new partial convexity property for smooth solutions to the heat equation is preserved.

Clarifies when solutions to stochastic PDEs stay near given subsets.

We construct Bäcklund transformations (BT) for the Gelfand-Dickey hierarchy (GD$_n$-hierarchy) on the space of $n$-th order differential operators on the line. Suppose $L=\partial_x^n-\sum_{i=1}^{n-1}u_i\partial_x^{(i-1)}$ is a solution of the $j$-th GD$_n$ flow. We prove the following results: (1) There exists a syste…

There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to s…

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under…

We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…

In this paper, partially invariant solutions (PISs) method is applied in order to obtain new four-dimensional Einstein Walker manifolds. This method is based on subgroup classification for the symmetry group of partial differential equations (PDEs) and can be regarded as the generalization of the similarity reduction m…

Neural networks solve SPDEs using Wiener chaos expansion.

Paper approximates backward heat equation using wave equations and Ricci flow.

Let $(M,g(t))$, $0\le t\le T$, $\partial M\neφ$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will pr…

It is shown that any smooth strictly convex global solution of $\det(\frac{\partial^{2}u}{\partial ξ_{i}\partial ξ_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial ξ_{i}} - d_0\right\},$ where $d_0$, $d_1$,...,$d_n$ are constants, must be a quadratic polynomial. This extends a well-known theorem of Jö…

Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.

The paper tackles individualized decision-making under unmeasured confounding, providing a novel minimax solution and a paradox.

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes …

Deep neural nets solve complex insurance math equations.

We consider least energy solutions to the nonlinear equation $-Δ_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is …

Solves boundary Yamabe problem with minimal boundary scenario.

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $\frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of M…

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that th…

Let $X=\mathbb{C}\timesΣ$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $X$ with finite analytic energy. The spin bundle $S^+\to X$ splits as $L^+\oplus L^-$. When $2-2g\leq c_1(S^+)[Σ]<0$, the moduli space is in b…

The authors prove that the logarithmic Monge-Ampère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time $t=0$. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation {equation*} \det D^{2}u=\exp\{n(-u+1/2\sum_{i=…

On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $ε$ tends to zero of solutions $u_ε: M \to \mathbb{C}$ to the equation $Δu_ε + ε^{-2}(1 - |u_ε|^{2})u_ε = 0$ with the boundary condition $\partial_νu_ε = 0$ on $\partial M$. Assuming an energy upper bound on the solutions and a…

Paper explores VRM for PSMLC with partially labeled medical images.

New methods solve complex PDEs with mixed boundary conditions.

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold $(M^n, g)$ endowed with the complementary orthogonal distributions $D_1$ and $D_2$. We provide conditions for symmetric $(0,2)$-tensors $T$ of a simple form (defined on $M$) to admit metrics $\tilde g$, conformal to …

The paper explores partial identifiability in nonnegative matrix factorization under specific conditions.

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time existence and uniqueness theorem for a $G$-invariant solution $g(t)$ satisfying the …

New methods prove existence of rotating shapes moving in space.

Meta-learning base distributions for efficient PDE solutions.

This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the Hölder-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has time-derivative bounded from above. On a Kähler manifold, this Hölder estimate works when the …

In the theory of minimal submanifold, the following problem is fundamental: when does a given Riemannian manifold admit (or does not admit) a minimal isometric immersion into an Euclidean space form of arbitrary dimension? A partial solution of this problem was obtained by B.Y. Chen as an application of his fundamental…

An unsupervised deep learning method solves PIDEs for option pricing.

We show uniqueness for overdetermined elliptic problems defined on topological disks $Ω$ with $C^2$ boundary, i.e., positive solutions $u$ to $Δu + f(u)=0$ in $Ω\subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vecη} = cte$ along $\partial Ω$, $\vecη$ the unit outward normal along $\partialΩ$ under the…

New method learns frequency-dependent partial correlations.

Study on singularities of solutions to Hamilton-Jacobi equations on manifolds.

It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one o…

Study on Tukey depth in machine learning using Hamilton-Jacobi equations.

We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(Δ_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^α…

Reconstructing a planar domain from its Dirichlet-to-Neumann data

We prove the statistical consistency of kernel Partial Least Squares Regression applied to a bounded regression learning problem on a reproducing kernel Hilbert space. Partial Least Squares stands out of well-known classical approaches as e.g. Ridge Regression or Principal Components Regression, as it is not defined as…

This paper analyzes a class of infinite-time-horizon stochastic games with singular controls motivated from the partially reversible problem. It provides an explicit solution for the mean-field game (MFG) and presents sensitivity analysis to compare the solution for the MFG with that for the single-agent control proble…

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…

Deep learning approximates PDE evolution operators from solution data.

Two-dimensional Riemannian manifolds uniquely determined by boundary data.