Yes / Weight initialization of neural networks has an important influence on the learning process, and the selection of initial weights is related to the activation interval of the activation function. It is proposed that an improved and extended weight initialization method for neural network with asymmetric activation function as an extension of the linear interval tolerance method (LIT), called ‘GLIT’ (generalized LIT), which is more suitable for higher-dimensional inputs. The purpose is to expand the selection range of the activation function so that the input falls in the unsaturated region, so as to improve the performance of the network. Then, a tolerance solution theorem based upon neural network system is given and proved. Furthermore, the algorithm is given about determining the initial weight interval. The validity of the theorem and algorithm is verified by numerical experiments. The input could fall into any preset interval in the sense of probability under the GLIT method. In another sense, the GLIT method could provide a theoretical basis for the further study of neural networks. / The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is partly supported by National Science Foundation of China under Grants (62073226, 61603262), Liaoning Province Natural Science Foundation (2020-KF-11-09, 2021-KF-11-05), Shen-Fu Demonstration Zone Science and Technology Plan Project (2020JH13, 2021JH07), Central Government Guides Local Science and Technology Development Funds of Liaoning Province (2021JH6).

First, basic aspects of interval analysis, roles of intervals and their applications are addressed. Then, various classes of interval matrices are described and their relations are depicted. This material forms a prelude to the unifying theme of the rest of the work - solving interval linear systems. Several methods for enclosing the solution set of square and overdetermined interval linear systems are covered and compared. For square systems the new shaving method is introduced, for overdetermined systems the new subsquares approach is introduced. Detecting unsolvability and solvability of such systems is discussed and several polynomial conditions are compared. Two strongest condi- tions are proved to be equivalent under certain assumption. Solving of interval linear systems is used to approach other problems in the rest of the work. Computing enclosures of determinants of interval matrices is addressed. NP- hardness of both relative and absolute approximation is proved. New method based on solving square interval linear systems and Cramer's rule is designed. Various classes of matrices with polynomially computable bounds on determinant are characterized. Solving of interval linear systems is also used to compute the least squares linear and nonlinear interval regression. It is then applied to real...