Deep generalized Green's function

The Green's function has ubiquitous and unparalleled usage for the efficient solving of partial differential equations (PDEs) and analyzing systems governed by PDEs. However, obtaining a closed-form Green's function for most PDEs on various domains is often impractical. Several approaches attempt to address this issue by approximating the Green's function with numerical methods, but several challenges remain. We introduce the Deep Generalized Green's Function (DGGF), a deep-learning approach that overcomes the challenges of problem-specific modeling, long computational times, and large data storage requirements associated with other approaches. Our method efficiently solves PDE problems using an integral solution format. It outperforms direct methods, such as FEM and physics-informed neural networks (PINNs). Additionally, our method alleviates the training burden, scales effectively to various spatial dimensions, and is demonstrated across a range of PDE types and domains. Unlike the direct Gaussian approximation of a Dirac delta function, our method can be used to solve PDEs in higher dimensions. Because our method directly addresses the singularity, it can be used to solve different PDEs without prior knowledge. Unlike BI-GreenNet, which is limited to PDEs with known expressions of the singular part of the Green's function, our method does not require prior knowledge of the singularity. The results confirm the advantages of DGGFs and the benefits of Generalized Greens Functions as a novel and effective approach to solving PDEs without suffering from singularities.

Duke Scholars

Author Vahid Tarokh Electrical and Computer Engineering

Author Willie John Padilla Electrical and Computer Engineering