Related papers: Lecture notes for pseudodifferential operators and…

Learning dynamics governed by differential equations is crucial for predicting and controlling the systems in science and engineering. Neural Ordinary Differential Equation (NODE), a deep learning model integrated with differential…

Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable…

Applying deep learning (DL) for annotating surgical instruments in robot-assisted minimally invasive surgeries (MIS) represents a significant advancement in surgical technology. This systematic review examines 48 studies that and advanced…

We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…

Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional…

We give some details about the stationary phase lemma. We first prove a special case where the high order terms are derived explicitly. Based on that, we prove a more general case by using Morse lemma.

This note provides a detailed algorithm to the application of local (perturbation) analysis of differential equations which is normally taught at graduate math courses. Exercise books often present more abstract and simplified versions of…

In this work, we propose an Operator Learning (OpL) method for solving boundary value inverse problems in partial differential equations (PDEs), focusing on recovering diffusion coefficients from boundary data. Inspired by the classical…

This paper introduces a meta-learning approach for parameterized pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed…

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the…

This paper is based on talks delivered in summer 2008 at the Conference on Motives, QFT and Pseudodifferential Operators in Boston, and at the Trimester programme Geometry and Physics, Hausdorff Institute for Mathematics in Bonn The paper…

Modeling sequential patterns from data is at the core of various time series forecasting tasks. Deep learning models have greatly outperformed many traditional models, but these black-box models generally lack explainability in prediction…

These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of…

This is a series of lecture notes, with embedded problems, aimed at students studying differential topology. Many revered texts, such as Spivak's "Calculus on Manifolds" and Guillemin and Pollack's "Differential Topology" introduce forms by…

In this paper, we study the boundedness of pseudodifferential operators with symbols in the H\"ormander class $S^0_{\rho,\rho}$ on $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$, and consider the relation between $\alpha$ and $\rho$. In…

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces both in the quasi-analytic and in the…

Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having…

Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a…

Let M be a holonomic module over the Weyl algebra K[t]<\partial_t>, K a field of characteristic zero. We prove a stationary phase formula which expresses the formalization of the germ at infinity of the Fourier transform of M in terms of a…

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators…