Haolin Li

Haolin Li, Yuyang Miao, Zahra Sharif Khodaei et al.

PINN models have demonstrated capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well-established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite domain, which conflicts with the finite boundaries typical of most solid structures; and b) the solution space utilised by PINN is Euclidean, which is inadequate for addressing the complex geometries often present in solid structures. This work presents a PINN architecture for general solid mechanics problems, referred to as the Finite-PINN model. The model is designed to effectively tackle two key challenges, while retaining as much of the original PINN framework as possible. To this end, the Finite-PINN incorporates finite geometric encoding into the neural network inputs, thereby transforming the solution space from a conventional Euclidean space into a hybrid Euclidean-topological space. The model is comprehensively trained using both strong-form and weak-form loss formulations, enabling its application to a wide range of forward and inverse problems in solid mechanics. For forward problems, the Finite-PINN model efficiently approximates solutions to solid mechanics problems when the geometric information of a given structure has been preprocessed. For inverse problems, it effectively reconstructs full-field solutions from very sparse observations by embedding both physical laws and geometric information within its architecture.

Haolin Li, Hoda Bidkhori

We introduce a novel framework for Federated Class Incremental Learning, called Federated Gaussian Task Embedding and Alignment (FedGTEA). FedGTEA is designed to capture task-specific knowledge and model uncertainty in a scalable and communication-efficient manner. At the client side, the Cardinality-Agnostic Task Encoder (CATE) produces Gaussian-distributed task embeddings that encode task knowledge, address statistical heterogeneity, and quantify data uncertainty. Importantly, CATE maintains a fixed parameter size regardless of the number of tasks, which ensures scalability across long task sequences. On the server side, FedGTEA utilizes the 2-Wasserstein distance to measure inter-task gaps between Gaussian embeddings. We formulate the Wasserstein loss to enforce inter-task separation. This probabilistic formulation not only enhances representation learning but also preserves task-level privacy by avoiding the direct transmission of latent embeddings, aligning with the privacy constraints in federated learning. Extensive empirical evaluations on popular datasets demonstrate that FedGTEA achieves superior classification performance and significantly mitigates forgetting, consistently outperforming strong existing baselines.