Keywords: physics-informed neural network, direct method, Galerkin method, initial boundary value problems, machine learning, impose initial and boundary conditions.

This paper introduces a novel approach called reduced-dimension physics-informed neural network (rd-PINN) for solving initial boundary value problems (IBVPs). The goal of the proposed rd-PINN is to transform the partial differential equation (PDE) into a system of ordinary differential equations (ODEs). Particularly, the numerical solution is formulated in the form of a linear combination of approximation functions and coefficients, wherein the approximation functions are admissible functions and the coefficients are functions of time to be determined. Accordingly, solving the original IBVP is transferred to the task of finding coefficient functions that satisfy the obtained ODEs. To solve these ODEs, a multi-network structure is designed to parameterize coefficients. Besides, we also proposed a framework that is used to automatically impose initial conditions. The advantages of rd-PINN over the original PINN in terms of solution accuracy and training cost are demonstrated through several numerical examples with different types of PDEs, boundary conditions, and initial conditions.

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