Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional numerical methods have been developed and widely used for their solutions. Inspired by rapidly growing impact of deep learning on scientific and engineering research, in this paper we propose a novel neural network, GF-Net, for learning the Green's functions of linear reaction-diffusion equations in an unsupervised fashion. The proposed method overcomes the challenges for finding the Green's functions of the equations on arbitrary domains by utilizing physics-informed approach and the symmetry of the Green's function. As a consequence, it particularly leads to an efficient way for solving the target equations under different boundary conditions and sources. We also demonstrate the effectiveness of the proposed approach by experiments in square, annular and L-shape domains.
翻译:局部差异方程式常常被用来模拟各种物理现象,如热传播、波波传播、流体动态、弹性、电动和图像处理等,而且已经制定许多分析方法或传统数字方法,并广泛用于其解决办法;由于深层学习对科学和工程研究的迅速增长影响,本文件提出一个新的神经网络GF-Net,用于以不受监督的方式学习Green对线性反应扩散方程式的功能;拟议方法通过利用物理知情方法和Green功能的对称,克服了在任意领域寻找Green方程式功能的挑战;结果,它特别导致在不同边界条件和来源下有效解决目标方程式的方法;我们还展示了在广场、废弃和L-shape领域进行实验的拟议方法的有效性。