The consumption of solving large-scale linear equations is one of the most critical issues in numerical computation. An innovative method is introduced in this study to solve linear equations based on deep neural networks. To achieve a high accuracy, we employ the residual network architecture and the correction iteration inspired by the classic iteration methods. By solving the one-dimensional Burgers equation and the two-dimensional heat-conduction equation, the precision and effectiveness of the proposed method have been proven. Numerical results indicate that this DNN-based technique is capable of obtaining an error of less than 10–7. Moreover, its computation time is less sensitive to the problem size than that of classic iterative methods. Consequently, the proposed method possesses a significant efficiency advantage for large-scale problems.

In this paper, we propose a novel second‐order explicit midpoint method to address the issue of energy loss and vorticity dissipation in Eulerian fluid simulation. The basic idea is to explicitly compute the pressure gradient at the middle time of each time step and apply it to the velocity field after advection. Theoretically, our solver can achieve higher accuracy than the first‐order solvers at similar computational cost. On the other hand, our method is twice and even faster than the implicit second‐order solvers at the cost of a small loss of accuracy. We have carried out a large number of 2D, 3D and numerical experiments to verify the effectiveness and availability of our algorithm.

λ A DNN-based algorithm that solves the multi-diagonal linear equations is proposed. λ We employed an iteration method that decreased the error of the numerical solution to 10− 7. λ The computational efficiency of the proposed method is 2 to 10 times of the classic algorithms.

The pressure Poisson equationis usually the most time-consuming problem in fluid simulation. To accelerate its solving process, we propose a deep neural network-based numerical method, termed Deep Residual Iteration Method (DRIM), in this paper. Firstly, the global equation is decomposed into multiple independent tridiagonal sub-equations, and DRIM is capable of solving all the sub-equations simultaneously. Moreover, we employed Residual Network and a correction iteration method to improve the precisionof the solution achieved by the neural network in DRIM. The numerical results, including the Poiseuille flow, the backwards-facing step flow, and driven cavity flow, have proven that the numerical precision of DRIM is comparable to that of classic solvers. In these numerical cases, the DRIM-based algorithm is about 2 to 10 times faster than the conventional method, which indicates that DRIM has promising applications in large-scale problems.