Zhang Zehan, Li Hong
A novel hybrid algorithm, termed the finite element method-enhanced neural network (FEM-NN), is proposed for the convection-diffusion equations. In contrast to physicsinformed neural networks (PINNs), in which a large number of collocation points are required for loss minimization, only a small set of mesh nodes is utilized in the proposed method. The finite element stiffness matrix and load vector are incorporated into the loss function of a feedforward neural network, and new FEM-NN algorithms are constructed for both steady-state and unsteady-state problems. For the steady-state case, the hyperbolic tangent function is employed as the activation function, while for the unsteady case, the residuals are derived from the discretized backward Euler scheme, and the loss function is formulated by combining boundary and initial conditions with the residuals, the SiLU and GELU functions are adopted as activation functions, meanwhile, FEM solutions and FEMNN solutions are compared through numerical experiments. It is demonstrated that the FEM-NN algorithm is capable of accurately reconstructing the FEM solution in the steadystate regime, while higher accuracy and stronger generalization capability are exhibited in the unsteady regime. The method proposed in this paper integrates the physical consistency and data-driven adaptability, and it provides a novel numerical approach for convection-diffusion type partial differential equations.
