Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs): learning neural networks for designing neutral inclusions

2025-01-14

AI Summary

Designing arbitrarily shaped neutral inclusions in composite materials—longstanding due to geometric complexity and interfacial imperfections—remains challenging for conventional analytical and numerical methods. Method: This paper introduces a novel framework integrating conformal mapping from complex analysis with physics-informed neural networks (PINNs). It is the first to embed conformal mappings into the PINN architecture, enabling construction of forward/inverse PDE solvers rigorously accommodating imperfect interface conditions. The method employs conformal coordinate transformation, complex-variable modeling, and neural parameterization of interface properties. Contribution/Results: We rigorously prove that training under a single uniform far-field loading suffices to achieve omnidirectional neutrality—overcoming fundamental convergence and generalization bottlenecks of traditional PINNs in inverse design. The approach significantly enhances prediction accuracy, robustness in geometric reconstruction, and physical consistency, enabling high-fidelity neutral inclusion design for arbitrary 2D shapes.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Neutral Inclusions
• Material Science
• Shape Adaptability

Innovation

• CoCo-PINNs
• Neutral Inclusions Design
• Adaptability and Generalization