An Implicit GNN Solver for Poisson-like problems
arxiv(2023)
摘要
This paper presents Ψ-GNN, a novel Graph Neural Network (GNN) approach
for solving the ubiquitous Poisson PDE problems with mixed boundary conditions.
By leveraging the Implicit Layer Theory, Ψ-GNN models an "infinitely" deep
network, thus avoiding the empirical tuning of the number of required Message
Passing layers to attain the solution. Its original architecture explicitly
takes into account the boundary conditions, a critical prerequisite for
physical applications, and is able to adapt to any initially provided solution.
Ψ-GNN is trained using a "physics-informed" loss, and the training process
is stable by design, and insensitive to its initialization. Furthermore, the
consistency of the approach is theoretically proven, and its flexibility and
generalization efficiency are experimentally demonstrated: the same learned
model can accurately handle unstructured meshes of various sizes, as well as
different boundary conditions. To the best of our knowledge, Ψ-GNN is the
first physics-informed GNN-based method that can handle various unstructured
domains, boundary conditions and initial solutions while also providing
convergence guarantees.
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