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A Novel algorithm for Solving High-Dimensional Poisson Equations Based on Radial Basis Function Neural Networks

JOURNAL OF CIRCUITS SYSTEMS AND COMPUTERS 2024年第13期33卷 2450223-2450223页

作者： Lu, Peixiao Sun, Shaoming Xinjiang Univ Sch Math & Syst Sci Urumqi 830046 Xinjiang Peoples R China Hefei Sci Isl Hefei Intelligence Hefei 230031 Anhui Peoples R China

As a widely used equation in electrostatics, the Poisson equation has significant research value in numerical solution. The basic principle of existing methods is to divide the solution domain into various grids and solve the numerical solutions at each grid node. Therefore, the accuracy of the solution is strongly correlated with the grid density divided. Based on this, this paper proposes a grid-free numerical calculation method that requires far fewer model parameters than traditional methods, and can ignore the order of the equation to solve high-dimensional Poisson equations. Given a Poisson equation, which has a certain type of boundary condition. A certain number of coordinate points are selected on the solution space and its boundary to construct a dataset. Using automatic differentiation technique to fit the differential operator in the equation, a loss function is constructed by incorporating the given boundary conditions or initial conditions, and the final numerical solution is obtained through iterative optimization algorithms. In the numerical experiment section, the algorithm proposed in this paper was used to solve the two-dimensional and three-dimensional Poisson equations with given exact solutions. The relative errors between the numerical solution and the true solution were 8.53e-4 and 6.40e-2, which are within the acceptable range. This proves that the proposed algorithm is feasible for solving the two-dimensional and three-dimensional Poisson equations with precise solutions. Secondly, the proposed algorithm is used to solve the four-dimensional Poisson equation with first-type boundary conditions, and the relative error range of the solution was within [0,0.56], which successfully extends the algorithm to solve high-dimensional Poisson equations and verifies its feasibility and efficiency in solving high-dimensional Poisson equations regardless of the dimension restriction.

关键词： Poisson equation radial basis neural (RBF) network automatic differentiation technology Adam algorithm mesh free numerical algorithm

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