This paper presents a method for modeling transient fluid flow in subsurface reservoir systems based on the developed neural operator architecture (TFNO-opt). Reservoir systems are complex dynamic objects with distributed parameters described by systems of partial differential equations (PDEs). Traditional numerical methods for modeling such systems, despite their high accuracy, are characterized by significant time costs for performing calculations, which limits their applicability in control and decision support problems. The proposed architecture (TFNO-opt) is based on Fourier neural operators, which allow approximating PDE solutions in infinite-dimensional functional spaces, providing invariance to discretization and the possibility of generalization to various implementations of equations. The developed modifications are aimed at increasing the accuracy and stability of the trained neural operator, which is especially important for control problems. These include adjustable internal time resolution of the integral Fourier operator, tensor decomposition of parameters in the spectral domain, use of the Sobolev norm in the error function, and separation of approximation errors and reconstruction of initial conditions for more accurate reproduction of physical processes. The effectiveness of the proposed improvements is confirmed by computational experiments. The practical significance is confirmed by computational experiments using the example of the problem of hydrodynamic modeling of an underground gas storage (UGS), where the acceleration of calculations by six orders of magnitude was achieved, compared to traditional methods. This opens up new opportunities for the effective control of complex reservoir systems.