Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the \textbf{AMG} method, a \textbf{M}ulti-\textbf{G}raph neural operator approach designed for efficiently solving PDEs on \textbf{A}rbitrary geometries. AMG leverages advanced graph-based techniques and dynamic attention mechanisms within a novel GraphFormer architecture, enabling precise management of diverse spatial domains and complex data interdependencies. By constructing multi-scale graphs to handle variable feature frequencies and a physics graph to encapsulate inherent physical properties, AMG significantly outperforms previous methods, which are typically limited to uniform grids. We present a comprehensive evaluation of AMG across six benchmarks, demonstrating its consistent superiority over existing state-of-the-art models. Our findings highlight the transformative potential of tailored graph neural operators in surmounting the challenges faced by conventional PDE solvers. Our code and datasets are available on \url{https://github.com/lizhihao2022/AMG}.

Federated self-supervised learning (FSSL) has recently emerged as a promising paradigm that enables the exploitation of clients' vast amounts of unlabeled data while preserving data privacy. While FSSL offers advantages, its susceptibility to backdoor attacks, a concern identified in traditional federated supervised learning (FSL), has not been investigated. To fill the research gap, we undertake a comprehensive investigation into a backdoor attack paradigm, where unscrupulous clients conspire to manipulate the global model, revealing the vulnerability of FSSL to such attacks. In FSL, backdoor attacks typically build a direct association between the backdoor trigger and the target label. In contrast, in FSSL, backdoor attacks aim to alter the global model's representation for images containing the attacker's specified trigger pattern in favor of the attacker's intended target class, which is less straightforward. In this sense, we demonstrate that existing defenses are insufficient to mitigate the investigated backdoor attacks in FSSL, thus finding an effective defense mechanism is urgent. To tackle this issue, we dive into the fundamental mechanism of backdoor attacks on FSSL, proposing the Embedding Inspector (EmInspector) that detects malicious clients by inspecting the embedding space of local models. In particular, EmInspector assesses the similarity of embeddings from different local models using a small set of inspection images (e.g., ten images of CIFAR100) without specific requirements on sample distribution or labels. We discover that embeddings from backdoored models tend to cluster together in the embedding space for a given inspection image. Evaluation results show that EmInspector can effectively mitigate backdoor attacks on FSSL across various adversary settings. Our code is avaliable at https://github.com/ShuchiWu/EmInspector.