Analytical capabilities of the DeepSeek neural network: A review of results from solving test problems in applied mathematics and mathematical physics

This article describes the characteristic features and practical use of the DeepSeek neural network for analytical computations and the construction of exact solutions in various fields of applied mathematics and mathematical physics. The results of applying DeepSeek to numerous test problems, primarily of high difficulty level, are presented and analyzed. Special attention is given to the construction of exact analytical solutions for diverse mathematical equations (including algebraic, functional, integral, ordinary differential, and partial differential equations). A number of nonlinear equations of mathematical physics possessing complex non-invariant solutions are considered. Almost all test equations contain one or more free parameters or arbitrary functions, which significantly complicates the solution procedure. To adequately assess the potential capabilities of DeepSeek, test equations and problems were selected whose solution is based on the application of various methods and ideas (more than 60 equations and problems were considered in total). In some cases, the results obtained using DeepSeek are compared with those of specialized computer algebra systems, such as Mathematica and Maple. Overall, the results of the conducted testing show that the DeepSeek neural network possesses broad analytical capabilities and is useful for the work and education of researchers, university professors, postgraduate students, and engineers specializing in various fields of applied mathematics and mathematical physics.