We deal with interval linear systems of equations. We present a new operator, which generalizes the interval Gauss-Seidel method. Also, based on the new operator and properties of the well-known methods, we propose a new algorithm, called the magnitude method. We illustrate by numerical examples that our approach overcomes some classical methods with respect to both time and sharpness of enclosures.
The work is devoted to the construction of a new type of intervals -- functional intervals. These intervals are built on the idea of expanding boundaries from numbers to functions. Functional intervals have shown themselves to be promising for further study and use, since they have more rich algebraic properties compared to classical intervals lamy. In the work, linear functional arithmetic was constructed from one variable. This arithmetic was applied to solve such problems of interval analysis, as minimization of a function on an interval and finding zeros of a function on an interval. Resul…
In our work, we consider the linear least squares problem for $m\times n$-systems of linear equations $Ax = b$, $m\geq n$, such that the matrix $A$ and right-hand side vector $b$ can vary within an interval $m\times n$-matrix and an interval $m$-vector respectively. We have to compute, as sharp as possible, an interval enclosure of the set of all least squares solutions to $Ax = b$ when $A$ and $b$ independently vary within their interval bounds. Our article is devoted to the development of the so-called PPS-methods (based on Partitioning of the Parameter Set) to solve the above problem. We re…
In this paper, two numerical approaches based on the Newton iteration method with spectral algorithms are introduced to solve the Thomas-Fermi equation. That Thomas-Fermi equation is a nonlinear singular ordinary differential equation (ODE) with boundary condition in infinite. In these schemes, the Newton method is combined with a spectral method where in one of those, by Newton method we convert nonlinear ODE to a sequence of linear ODE then, solve them using the spectral method. In another one, by the spectral method the nonlinear ODE be converted to system of nonlinear algebraic equations, …
Reliable global optimization is dedicated to finding a global minimum in the presence of rounding errors. The only approaches for achieving a numerical proof of global optimality are interval branch and bound methods that interleave branching of the search-space and pruning of the subdomains that cannot contain an optimal solution. It is of the utmost importance: i) to compute sharp enclosures of the objective function and the constraints on a given subdomain; ii) to find a good approximation (an upper bound) of the global minimum. State-of-the-art solvers are generally integrative methods, th…
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. In this paper, the poss…
In this paper, we propose a deep neural network approximation to the evolution operator for time dependent PDE systems over long time period by recursively using one single neural network propagator, in the form of POD-DeepONet with built-in causality feature, for a small time interval. The trained DeepPropNet of moderate size is shown to give accurate prediction of wave solutions over the whole time interval.
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adam…
An analysis of the derivative-free loss method for solving PDEs
Department of Mathematics and Statistics, University at Albany, State University of New York（数学与统计学系，阿尔巴尼大学，纽约州立大学）；Department of Mathematics, Dartmouth College（数学系，达特茅斯学院）
Solving a system of nonlinear inequalities is an important problem for which conventional numerical analysis has no satisfactory method. With a box-consistency algorithm one can compute a cover for the solution set to arbitrarily close approximation. Because of difficulties in the use of propagation for complex arithmetic expressions, box consistency is computed with interval arithmetic. In this paper we present theorems that support a simple modification of propagation that allows complex arithmetic expressions to be handled efficiently. The version of box consistency that is obtained in this…
DeltaPhi: Physical States Residual Learning for Neural Operators in Data-Limited PDE Solving
College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）；The State Key Lab of Brain-Machine Intelligence, Zhejiang University（浙江大学脑机智能国家重点实验室）
Shanghai Jiao Tong University（上海交通大学）；Shanghai Center for Brain Science and Brain-Inspired Technology（上海脑科学与类脑研究中心）；Peking University（北京大学）；AI for Science Institute（北京科学智能研究院）
In this paper, we propose a a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear PDEs, we use a DNN to parameterize the Green's function and obtain the neural operator to approximate the solution according to the Green's method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the…
