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.

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations

Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.

An Oscillation Criterion for Second Order Superlinear Differential Equations

A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

On the Asymptotic Behavior of Certain Second Order Differential Equations

Two criteria for asymptotic behavior of certain second order differential equations are obtained. The results extend and improve Feng's theorem.

Oscillation of Second Order Delay Differential Equations with Nonlinear Neutral Term

By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.

Approximate Controllability of Second-Order Neutral Stochastic Differential Equations with Infinite Delay and Poisson Jumps

The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.

Oscillation of Second Order Nonlinear Neutral Differential Equations with Mixed Neutral Term

In this paper, we obtained some sufficient conditions for the oscillation of all solutions of the second order neutral differential equation of the form where , and . Examples are provided to illustrate the main results.

ON GLOBAL MEROMORPHIC SOLUTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.

SOLVING SECOND ORDER DIFFERENTIAL EQUATIONS IN QUANTUM MECHANICS BYORDER REDUCTION

Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.

SOLVABILITY IN HARDY SPACE FOR SECOND ORDER ELLIPTIC EQUATIONS

We obtain a priori estimates and solvability in Hardy type space in a bounded domain of Rn for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case of the classical LP（1 〈 p 〈 ∞） theory for second order elliptic equations. Our approach is based on a standard technique of perturbation rather than that of integral representation formula.

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.

Pseudo Almost Periodic Solutions for Second Order Abstract Differential Equations

On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract differential equation ( u″(t)+A 1u′(t)+A 0u(t)=0, t≥0 ) is strongly well posed, the necessary conditions for their solutions to be pseudo almost periodic are derived.

Oscillation Theorems for Second Order Damped Nonlinear Differential Equations

Several oscillation criteria are given for the second order nonlinear differential equation with damped term of the form [α(t)(y'(t))σ]' +p(t)(y'(t))σ+ q(t)f(y(t)) = 0, where α∈C(R, (0,∞)), p(t) and q(t) are allowed to change sign on [t0, ∞), and f∈C1 (R, R) such that xf(x) > 0 for x ≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.

Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping

The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique. Some well known results in the literature are extended. Moreover, two examples are given to illustrate the theoretical analysis.

ON LEVIN TYPE COMPARISON THEOREMS FOR CERTAIN SECOND ORDER DIFFERENTIAL EQUATIONS

In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.

<i>L</i>-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations

An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented.