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Register a new userFourier-Taylor Parameterization of Unstable Manifolds for Parabolic Partial Differential Equations: Formalization, Implementation, and Rigorous Validation
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In this paper we study high order expansions of chart maps for local finite dimensional unstable manifolds of hyperbolic equilibrium solutions of scalar parabolic partial differential equations. Our approach is based on studying an infinitesimal invariance equation for the chart map that recovers the dynamics on the manifold in terms of a simple conjugacy. We develop formal series solutions for the invariance equation and efficient numerical methods for computing the series coefficients to any desired finite order. We show, under mild non-resonance conditions, that the formal series expansion converges in a small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument proves convergence in larger neighborhoods. We implement the method for a spatially inhomogeneous Fishers equation and numerically compute and validate high order expansions of some local unstable manifolds for morse index one and two. We also provide a computer assisted existence proof of a saddle-to-sink heteroclinic connecting orbit.
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Let ${u(t,,x)}_{tge 0, xin mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure $f$ and satisfies Dalangs condition. We prove two general functional central limit theorems for occupation fields of the form $N^{-d} int_{mathbb{R}^d} g(u(t,,x)) psi(x/N), mathrm{d} x$ as $Nrightarrow infty$, where $g$ runs over the class of Lipschitz functions on $mathbb{R}^d$ and $psiin L^2(mathbb{R}^d)$. The proof uses Poincare-type inequalities, Malliavin calculus, compactness arguments, and Paul Levys classical characterization of Brownian motion as the only mean zero, continuous Levy process. Our result generalizes central limit theorems of Huang et al cite{HuangNualartViitasaari2018,HuangNualartViitasaariZheng2019} valid when $g(u)=u$ and $psi = mathbf{1}_{[0,1]^d}$.
Our aim in this paper is to establish stable manifolds near hyperbolic equilibria of fractional differential equations in arbitrary finite dimensional spaces.
We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded domain in $mathbf{R}^{n}$. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler equation as an ODE on an infinite dimensional manifold of volume preserving maps in $W^{k, q}$, $(k>1+frac{n}{q})$, the unstable (and stable) manifolds of $v_{0}$ are constructed under certain spectral gap condition which is verified for both 2D and 3D examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of $v_{0}$ in the sense that arbitrarily small $W^{k, q}$ perturbations can lead to $L^{2}$ growth of the nonlinear solutions.
We investigate methods for learning partial differential equation (PDE) models from spatiotemporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyze the performance in utilizing previous methods to denoise data for the task of discovering the governing system of partial differential equations (PDEs). We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e., the advection-diffusion, classical Fisher-KPP, and nonlinear Fisher-KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the deep learning-based techniques. Based on the data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. By using automatic differentiation, the PDEs system could be explicitly encoded into NNs and consequently, the sum of mean squared residuals from PDEs could be minimized with respect to the NN parameters. In this study, the residual in the loss function could be constructed validly by using the automatic differentiation because of the relationship between the surface differential operators $ abla_S/Delta_S$ and the standard Euclidean differential operators $ abla/Delta$. We first consider the unit sphere as surface to investigate the numerical accuracy and convergence of the PINNs with different training example sizes and the depth of the NNs. Another examples are provided with different complex manifolds to verify the robustness of the PINNs.
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