Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs): learning neural networks for designing neutral inclusions
This work addresses a long-standing challenge in composite materials design, offering a novel computational method for engineers and material scientists, though it appears incremental as it builds on existing PINN frameworks.
The authors tackled the inverse problem of designing neutral inclusions with arbitrary shapes in composite materials by introducing CoCo-PINNs, which integrate conformal mapping into physics-informed neural networks, achieving strong performance in two-dimensional cases and proving that training with a single linear field suffices for neutrality in untrained directions.
We focus on designing and solving the neutral inclusion problem via neural networks. The neutral inclusion problem has a long history in the theory of composite materials, and it is exceedingly challenging to identify the precise condition that precipitates a general-shaped inclusion into a neutral inclusion. Physics-informed neural networks (PINNs) have recently become a highly successful approach to addressing both forward and inverse problems associated with partial differential equations. We found that traditional PINNs perform inadequately when applied to the inverse problem of designing neutral inclusions with arbitrary shapes. In this study, we introduce a novel approach, Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs), which integrates complex analysis techniques into PINNs. This method exhibits strong performance in solving forward-inverse problems to construct neutral inclusions of arbitrary shapes in two dimensions, where the imperfect interface condition on the inclusion's boundary is modeled by training neural networks. Notably, we mathematically prove that training with a single linear field is sufficient to achieve neutrality for untrained linear fields in arbitrary directions, given a minor assumption. We demonstrate that CoCo-PINNs offer enhanced performances in terms of credibility, consistency, and stability.