Physics-Informed Sparse Networks with Adaptive Sampling Method for Solving Pdes

Abstract

The paper introduces an enhanced deep learning model to address limitations in traditional numerical and deep learning methods for solving Partial Differential Equations (PDEs). It proposes augmenting feedforward neural networks with radial basis function (RBF) layers, enabling adaptive adjustment of central point positions to enhance accuracy and efficiency in solving PDEs. Furthermore, it integrates the residual-based adaptive refinement (RAR) method for low-dimensional cases and residual-based adaptive distribution (RAD) method for high-dimensional cases to further enhance the model's solving capabilities. Numerical experiments on Poisson and Black-Scholes equations demonstrate the effectiveness of the proposed method. By applying the model to financial derivatives pricing, its practical feasibility is validated. The introduction of the RBF layer enhances solution accuracy and generalization, while the RAR and RAD methods optimize sampling allocation, especially in high-dimensional scenarios. The model's successful application in option pricing underscores its potential in financial engineering.

Keywords: Deep learning, Partial differential equation, physics-informed neural networks