Deep Surrogate Model For Learning Green’s Function Of Reaction-diffusion Equations

Partial differential equations are widely used to simulate various physical phenomena such as heat diffusion,wave conduction,hydrodynamics,elastodynamics,electrodynamics and so on.Given the significance of the simulation of physical phenomena in engineering and science,the solution of partial differential equations has become a hot topic of research in the field of applied mathematics,and over time researchers have developed many numerical algorithms for solving partial differential equations.Recently,due to the significant increase in computing power of computers,especially GPUs,surrogate models based on deep learning have gained the attention of a wide range of researchers.In order to solve partial differential equations using deep learning techniques,a sufficiently rich dataset needs to be constructed from analytical solutions of partial differential equations or high precision approximate solutions generated by numerical methods.Since only a small number of partial differential equations have analytical solutions,solving partial differential equations by high precision numerical methods is the only option to build a dataset for training deep learning models.As deep learning models are significantly dependent on the size of the dataset,the richer the dataset,the more powerful the trained model will be.The process of solving partial differential equations and thus constructing datasets through numerical methods is tedious and time-consuming.To address this serious contradiction,this paper proposes to use a loss function evolved from the finite difference method to train deep learning models,instead of using a data-driven approach by constructing a dataset from the numerical solutions of numerical methods such as finite element,finite volume or finite difference solutions in advance.It can be demonstrated through sufficient numerical experiments that deep learning models trained using the loss functions proposed in this paper can achieve results comparable to those of the data-driven method,eliminating the need to apply deep learning techniques with the process of constructing data sets in the process of solving partial differential equations and saving a significant amount of computational time.Further by using the alternative model results via Green’s representation theorem more clusters of partial differential equations consisting of different right end terms and boundary conditions can be solved.