Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale generalization capabilities, as well as high computational costs. This paper proposes the Multi-input and Multi-scale Efficient Transformer (MMET), a novel framework designed to address the above challenges. MMET decouples mesh and query points as two sequences and feeds them into the encoder and decoder, respectively, and uses a Gated Condition Embedding (GCE) layer to embed input variables or functions with varying dimensions, enabling effective solutions for multi-scale and multi-input problems. Additionally, a Hilbert curve-based reserialization and patch embedding mechanism decrease the input length. This significantly reduces the computational cost when dealing with large-scale geometric models. These innovations enable efficient representations and support multi-scale resolution queries for large-scale and multi-input PDE problems. Experimental evaluations on diverse benchmarks spanning different physical fields demonstrate that MMET outperforms SOTA methods in both accuracy and computational efficiency. This work highlights the potential of MMET as a robust and scalable solution for real-time PDE solving in engineering and physics-based applications, paving the way for future explorations into pre-trained large-scale models in specific domains. This work is open-sourced at https://github.com/YichenLuo-0/MMET.

Modern recommender systems (RS) work by processing a number of signals that can be inferred from large sets of user-item interaction data. The main signal to analyze stems from the raw matrix that represents interactions. However, we can increase the performance of RS by considering other kinds of signals like the context of interactions, which could be, for example, the time or date of the interaction, the user location, or sequential data corresponding to the historical interactions of the user with the system. These complex, context-based interaction signals are characterized by a rich relational structure that can be represented by a multi-partite graph. Graph Convolutional Networks (GCNs) have been used successfully in collaborative filtering with simple user-item interaction data. In this work, we generalize the use of GCNs for N-partite graphs by considering N multiple context dimensions and propose a simple way for their seamless integration in modern deep learning RS architectures. More specifically, we define a graph convolutional embedding layer for N-partite graphs that processes user-item-context interactions, and constructs node embeddings by leveraging their relational structure. Experiments on several datasets from recommender systems to drug re-purposing show the benefits of the introduced GCN embedding layer by measuring the performance of different context-enriched tasks.