Meta-Auto-Decoder:a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

Partial Orders on Right Inverse Semigroups

It is well known that there exists the smallest inverse semigroup congruence on an orthodox semigroup. We denote by Y the smallest inverse semigroup congruence on an orthodox semigroup. Let S be a right inverse semigroup. We construct partial orders on S by some kind of its subsemigroups and uncover that partial orders on S have close contact with partial orders on S/Y.

A New Relationship Between Partial Orders of Normal Matrices and Their Squares

A new relationship between partial orders of normal matrices and their squares is established.

PARTIAL REGULARITY FOR MINIMIZERS OF HIGHER ORDER QUASICONVEX FUNCTIONALS

We consider the following quasiconvex functional I(u)=∫ Gf(x,δu,D mu) d x where u is a vector valued function in W m,p (G),m>1 and p>2. The partial C m,a —regularity is proved for minimizers of I(u) under weaker conditions.

Additive Maps Preserving the Star Partial Order on B(H)

Let B（H） be the C^＊-algebra of all bounded linear operators on a complex Hilbert space H. It is proved that an additive surjective map φ on B（H） preserving the star partial order in both directions if and only if one of the following assertions holds. （1） There exist a nonzero complex number a and two unitary operators U and V on H such that φ（X） = a UXV or φ（X） = α UX^＊ V for all X ∈B（H）. （2） There exist a nonzero a and two anti-unitary operators U and V on H such that φ（X） = αUXV or φ（X） = aUX^＊ Vfor all X∈ B（H）.

The Lawson Partial Order on Wrpp Semigroups

In this paper,we introduce Lawson partial order ≤ w on wrpp semigroups.After obtaining some properties of ≤ w,we determine when ≤ w is(left;right) compatible with the multiplication.These results extend and enrich the related results of Lawson and GuoLuo on abundant semigroups,of Guo-Shum on rpp semigroups and of Liu-Guo on wrpp semigroups.

Common Fixed Point Result of Multivalued and Singlevalued Mappings in Partially Ordered Metric Space

In recent times the fixed point results in partially ordered metric spaces has greatly developed. In this paper we prove common fixed point results for multivalued and singlevalued mappings in partially ordered metric space. Our theorems generalized the theorem in [1] and extends the many more recent results in such spaces.

Experimental investigation on partially coherent higher-order non-diffractive beams

We experimentally study the generation of a partially coherent non-diffractive beam by focusing a partially coherent vortex beam with an axieon. The investigation results show that when the partially coherent vortex beam is focused by the axicon, the beam is transferred into a partially coherent higher-order non-diffractive beam. In the non-diffractive zone, the transverse intensity distribution of the partially coherent higher-order non-diffractive beam is invariant during propagation. In addition, the range of the non-diffractive zone is related to the coherence of the partially coherent vortex beam. The poorer the coherence of the partially coherent vortex beam, the shorter the range of the non-diffractive zone.

Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-Order Parabolic Partial Differential Equation

In this paper, the ADM method is used to construct the solution of the singular fourth-order partial differential equation.

The Existence of Coupled Solutions for a Kind of Nonlinear Operator Equations in Partial Ordered Linear Topology Space

The main purpose of this paper is to examine the existence of coupled solutions and coupled minimal-maximal solutions for a kind of nonlinear operator equations in partial ordered linear topology spaces by employing the semi-order method. Some new existence results are obtained.

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.

Some Fixed Point Results of Ciric-Type Contraction Mappings on Ordered <i>G</i>-Partial Metric Spaces

We introduce the concept of generalized quasi-contraction mappings in G-partial metric spaces and prove some fixed point results in ordered G-partial metric spaces. The results generalize and extend some recent results in literature.

Some new fixed point results under constraint inequalities in comparable complete partially ordered Menger PM-spaces

In this paper,we introduce the concept of comparable T-completeness of a partially ordered Menger PM-space and discuss the existence of fixed points for mappings satisfying certain conditions in the framework of a comparable T-complete partially ordered Menger PM-space.We obtain some new results which generalize many known ones in the literature.Moreover,we derive some consequent results and give an example to illustrate our main result.

<i>N</i>-Order Fixed Point Theory for N-Order Generalized Meir-Keeler Type Contraction in Partially Ordered Metric Spaces

This paper concerns N-order fixed point theory in partially ordered metric spaces. For the sake of simplicity, we start our investigations with the tripled case. We define tripled generalized Meir-Keeler type contraction which extends the definition of [Bessem Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517]. We then discuss the existence and uniqueness of tripled fixed point theorems in partially ordered metric spaces. For general cases, we generalized our results to the N-order case. The results will promote the study of N-order fixed point theory.

Continuity of Solution Mappings for Parametric Set Optimization Problems under Partial Order Relations

This paper mainly investigates the semicontinuity of solution mappings for set optimization problems under a partial order set relation instead of upper and lower set less order relations. To this end, we propose two types of monotonicity definition for the set-valued mapping introduced by two nonlinear scalarization functions which are presented by these partial order relations. Then, we give some sufficient conditions for the semicontinuity and closedness of solution mappings for parametric set optimization problems. The results presented in this paper are new and extend the main results given by some authors in the literature.

PARTIAL ELIMINTATION ALGORITHM FOR A LINEAR RECURRENCE SYSTEM R(n, m) OF ORDER m

This paper presents an algorithm for computing a linear recurrence system R(n, m) of order m for n equations on MIMD parallel system. This algorithm is not only easy to be programmed on a parallel computer system, but also reduces the data-waiting time due to compute-ahead strategy. The paper analyses how to achieve maximal load balancing when the algorithm is implemented on MIMD parallel system. By the end of the paper, an analysis on the speedup and parallel efficiency are given. The results indicate that the new parallel elimination algorithm has great improvement compared with the old ones.

NEWTON-LIK E METNODS IN PARTIALLY ORDERED BANACH SPACES

We provide convergence results and error estimates for Newton-like methods in generalized Banach spaces.The idea of a generalized norm is used whichis defined to be a map from a linear space into a partially ordered Banach space.Convergence results and error estimates are improved compared with the real norm theory.

Results of Associated Implication Algebra on a Partial Ordered Set

Some sufficient and necessary conditions that implication algebra on a partial ordered set is associated implication algebra are obtained, and the relation between lattice H implication algebra and associated implication algebra is discussed. Also, the concept of filter is proposed with some basic properties being studied.

Study of Fixed Point Theorems for Higher Dimension in Partially Ordered Metric Spaces

In this paper, we establish the existence and uniqueness of fixed points of operator , when n is an arbitrary positive integer and X is a partially ordered complete metric space. We have shown examples to verify our work. Our results generalize the recent fixed point theorems cited in [1]-[4] etc. and include several recent developments.

Endpoints of Multi-Valued Weak Contractions on the Metric Space of Partially Ordered Groups

The present work considers the endpoint in the abstract metric space. It firstly introduces the metric space of partially ordered groups and the metric space of partially ordered modules, respectively;and defines the convergence of sequences and the multi-valued weak contractions, etc., on the introduced space. And then, with the methods of functional analysis and abstract algebra, it successively establishes an endpoint theorem for the metric space of partially ordered groups and an endpoint theorem for the metric space of partially ordered modules. The contributions of this article extend the theory of cone metric space constructed by Huang and Zhang (2007) and some recent results on the fixed point and endpoint theory, such as the endpoint theorem given by Amini-Harandi (2010).