This paper uses the physical information neural network (PINN) model to solve a 3D anisotropic steady-state heat conduction problem based on deep learning techniques. The model embeds the problem’s governing equations and boundary conditions into the neural network and treats the neural network’s output as the numerical solution of the partial differential equation. Then, the network is trained using the Adam optimizer on the training set. The output progressively converges toward the accurate solution of the equation. In the first numerical example, we demonstrate the convergence of the PINN by discussing the effect of the neural network’s number of layers, each hidden layer’s number of neurons, the initial learning rate and decay rate, the size of the training set, the mini-batch size, the amount of training points on the boundary, and the training steps on the relative error of the numerical solution, respectively. The numerical solutions are presented for three different examples. Thus, the effectiveness of the method is verified.

The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.

In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.