Doklady Mathematics最新文献

The accuracy of solving partial differential equations using physics-informed neural networks (PINNs) significantly depends on their architecture and the choice of hyperparameters. However, manually searching for the optimal configuration can be difficult due to the high computational complexity. In this paper, we propose an approach for optimizing the PINN architecture using a differential evolution algorithm. We focus on optimizing over a small number of training epochs, which allows us to consider a wider range of configurations while reducing the computational cost. The number of epochs is chosen such that the accuracy of the model at the initial stage correlates with its accuracy after full training, which significantly speeds up the optimization process. To improve efficiency, we also apply a surrogate model based on a Gaussian process, which reduces the number of required PINN trainings. The paper presents the results of optimizing PINN architectures for solving various partial differential equations and offers recommendations for improving their performance.

The problem of searching for a mathematical expression is formulated. Classes of mathematical problems where this problem is in demand are presented. A general approach to solving this problem based on machine learning using symbolic regression methods is given. This approach allows finding not only the parameters of the desired mathematical expression, but also its structure. The general problem of machine learning by methods of symbolic regression is described and a unified approach to overcoming it based on the application of the principle of small variations of the basis solution is given.

The paper considers a new method for interpolation of nonlinear functions on an interval, the so-called p-factor interpolation method. By using a Newton interpolation polynomial as an example, it is shown that, in the case of degeneration of the approximated function (f(x)) in the solution, classical interpolation does not provide the necessary accuracy for finding an approximate solution to the equation (f(x) = 0), in contrast to the nondegenerate regular case. In turn, the use of p-factor interpolation polynomials for approximating functions in order to obtain the desired approximate solution to the equation provides the necessary order of accuracy in the argument during the calculations. The results are based on constructions of p-regularity theory and the apparatus of p-factor operators, which are effectively used in the study of degenerate mappings.

Measuring a company’s ethics is an important element in the mechanism of regulating the behavior of market participants, as it allows consumers and regulators to make better decisions, which has a disciplining effect on companies. We tested various methods for machine analysis of feedback from Russian bank consumers and developed an ethics index that allows us to calculate a quantitative assessment of the ethics of three hundred Russian banks based on consumer feedback for different time periods from 2005 to 2022. We used a bag-of-words method based on the Moral Foundations Dictionary (MFD) and BERT model training based on a 3000- and 10 000-sentence sample marked up by experts. The resulting index was validated based on the number of arbitration cases from 2005 to 2022 (more ethical companies are involved in fewer arbitration cases as a defendant). As a result, only the BERT model was validated, whereas the MFD-based model was not. The ethics index would be useful as a metric alternative to popular ESG ratings for both theoretical research on company behavior and practical tasks of managing company reputation and forming policies of regulating the behavior of market participants.

“Ark of Knowledge” is a digital project developed by Lomonosov Moscow State University. It provides access to fundamental knowledge in Russian and should play a key role in the preservation and dissemination of Russia’s cultural and scientific heritage. “Ark of Knowledge” is an ontological information system. The article discusses modern ideas about ontology, the stages of creation and ontological features of the Great Russian Encyclopedia and Wikidata, as well as the design of an information system and its use for language models training. The initial working prototype of this information system is briefly described. Work on creating the system is being carried out by researchers and programmers from the Knowledge Engineering Laboratory of the Institute for Mathematical Research of Complex Systems of Lomonosov Moscow State University, as well as researchers from the Faculties of Philology, Mechanics and Mathematics, Computational Mathematics and Cybernetics, and the Branch of Moscow State University in Sevastopol.

We propose a novel method for rapid pattern analysis of high-dimensional numerical data, termed tunnel clustering. The main advantages of the method are its relatively low computational complexity, endogenous determination of cluster composition and number, and a high degree of interpretability of final results. We present descriptions of three different variations: one with fixed hyperparameters, an adaptive version, and a combined approach. Three fundamental properties of tunnel clustering are examined. Practical applications are demonstrated on both synthetic datasets containing 100 000 objects and on classical benchmark datasets.

The paper considers boundary value problems generated by an ordinary differential expression of the nth order and arbitrary boundary conditions with linear dependence on the spectral parameter both in the equation and the boundary conditions. Classes of problems are defined, which are called regular and strongly regular. Linear operators in the space (H = {{L}_{2}}[0,1] oplus {{mathbb{C}}^{m}},;m leqslant n,) are assigned to these problems, and the corresponding adjoint operators are constructed in explicit form. In the general form, we solve the problem of selecting “superfluous” eigenfunctions, which was previously studied only for the special cases of second- and fourth-order equations. Namely, a criterion is found for selecting m eigen- or associated (root) functions of a regular problem so that the remaining system of root functions forms a Riesz basis or a Riesz basis with parenthesis in the original space ({{L}_{2}}[0,1]).

The problem of approximating a continuous real function of one real variable defined on an interval using a band-limited function based on Tikhonov’s regularization method is considered. Numerical estimates of the accuracy of such approximations are calculated for a model trigonometric function. We analyze why a theoretical estimate for the approximation accuracy of a continuous function by band-limited functions is difficult to achieve numerically. The problem of estimating the spectrum of a signal defined on a finite interval is discussed.

It is shown that the multiple nonconstant relaxation time lattice Boltzmann equation for five discrete velocities is equivalent in the diffusion limit to a nonlinear anisotropic diffusion equation. The proposed model is applied to speckle and Gaussian noise removal problem.

This paper examines seismic wave propagation in a full three-dimensional case. In practice, the stress-strain state of a geological medium during seismic exploration is frequently described using acoustic and linear elastic models. The governing systems of partial differential equations of both models are linear hyperbolic. A computational algorithm for them can be constructed by applying a grid-characteristic approach. In the case of multidimensional problems, an important role is played by dimensional splitting. However, the final three-dimensional scheme fails to preserve the achieved high order even in the case of extended spatial stencils used to solve the resulting one-dimensional problems. In this paper, we propose an approach based on multistage operator splitting schemes, which made it possible to construct a three-dimensional grid-characteristic scheme of the third order. Several test problems are solved numerically.