Deep Generalized Green's Function
Keywords: Partial Differential Equation; Deep Learning; Green's Function; Boundary Element Method
Abstract: The Green's function has ubiquitous usages in efficient problem-solving of partial differential equations (PDEs) and analyzing the system governed by the PDE. However, obtaining a closed-form Green's function for most PDEs on various domains is almost impractical. Numerical Green's function (NGF) seeks an approximate Green's function using traditional methods, like finite element analysis, which is particularly useful for complex problems like fracture mechanics and dynamic scattering. In this study, we introduce the Deep Generalized Green's Function (DGGF), a deep-learning approach addressing the challenges associated with NGF: problem-specific modeling, time consumption, and data storage demand. Our method is demonstrated to efficiently solve PDE problems with the integral format of solutions compared to direct methods, for instance, FEM and physics-informed neural networks (PINNs). Besides, our method relieves the training burden and scales to higher dimensions than any method with direct Gaussian approximation of a Dirac delta function. Since our method directly addresses the singularity, it applies to different PDEs without prior knowledge, whereas BI-GreenNet is limited to PDEs with known expressions of the singular part of the Green's function. The numerical results confirm the advantages of DGGFs and the benefits of Generalized Greens Functions as a novel alternative approach to solving PDEs without suffering from singularities.
Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Submission Number: 7942
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