Shanshan Xiao

Shanshan Xiao, Ye Li, Zhongyi Huang et al.

Solving Partial Differential Equation (PDE) interface problems on varying domains is a critical task in design and optimization, yet it remains computationally prohibitive for traditional solvers. Although operator learning has shown promise on fixed geometries, its potential for geometry-dependent interface problems has been largely unexplored. To bridge this gap, we propose an extension-based neural operator framework applicable to general linear interface problems. A key innovation of our method is the integration of the Tailored Finite Point Method (TFPM) with our base network, which reduces memory consumption and effectively alleviates the curse of dimensionality. On the theoretical front, we establish the continuity of the Helmholtz operator with respect to domain perturbations and provide rigorous error estimates for the proposed encodings. Comprehensive numerical experiments demonstrate that our framework achieves state-of-the-art accuracy and robustness. Consequently, this work provides a powerful, data-efficient tool for varying-domain simulations, offering new possibilities for real-time shape optimization.

Shanshan Xiao, Jiawei Zhang, Yifa Tang

Incorporating neural networks for the solution of Ordinary Differential Equations (ODEs) represents a pivotal research direction within computational mathematics. Within neural network architectures, the integration of the intrinsic structure of ODEs offers advantages such as enhanced predictive capabilities and reduced data utilization. Among these structural ODE forms, the Lagrangian representation stands out due to its significant physical underpinnings. Building upon this framework, Bhattoo introduced the concept of Lagrangian Neural Networks (LNNs). Then in this article, we introduce a groundbreaking extension (Genralized Lagrangian Neural Networks) to Lagrangian Neural Networks (LNNs), innovatively tailoring them for non-conservative systems. By leveraging the foundational importance of the Lagrangian within Lagrange's equations, we formulate the model based on the generalized Lagrange's equation. This modification not only enhances prediction accuracy but also guarantees Lagrangian representation in non-conservative systems. Furthermore, we perform various experiments, encompassing 1-dimensional and 2-dimensional examples, along with an examination of the impact of network parameters, which proved the superiority of Generalized Lagrangian Neural Networks(GLNNs).