Deep Surrogate Model For Solving Partial Differential Equation With Multipoint Sources

Partial differential equations(PDEs)with multi-point sources are commonly used in fields such as electrostatics,mechanical engineering,and groundwater problems.Their source terms are composed of multiple Dirac delta functions.For example,in groundwater problems,point sources can be used to describe pumping wells,while PDEs with multi-point sources describe the head’s changes caused by pumping wells.Due to the possible differences in the quantity,location,and pumping capacity of pumping wells,PDEs with multi-point sources can be considered as a parameterized PDE.A single PDE can be solved by traditional numerical methods,but these methods are not particularly suitable for solving parameterized PDE.In recent years,deep learning technology has been fully developed in various fields,and has also received increasing attention in fields such as scientific computing.Due to the powerful expressive power of neural network,how to integrate it with traditional numerical methods to provide a new perspective for solving PDEs with multi-point sources,is the focus of the thesis.In this thesis,we use a special convolutional neural network,namely U-Net architecture,to model the functional relationship between the source term and its solution of PDEs,which is called deep surrogate model.The model uses the discretization of PDEs as physical information to constrain the network,while encoding boundary conditions into the network in term of hard constraints.The training data only needs to be sampled based on the quantity,location,and strength of point sources,and no additional labels are required.In addition,the effect of different training strategies on the performance of the deep surrogate model are also investigated,and we find that the training strategy proposed in this paper has obvious superiority over the conventional one.Finally,numerical experiments are provided in different scenarios,and numerical results verify the accuracy and efficiency of the proposed deep surrogate model.