Haomiao Li | Searcharxiv

arXiv2026

Symplectic Hamiltonian Direct Discontinuous Galerkin Method for Wave Propagation

This paper presents a symplectic Hamiltonian direct discontinuous Galerkin (DDG) method for approximating wave propagation problems, including the linear and semilinear wave equations. Within an auxiliary-variable-free DG framework, we prove that the symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure. It follows that methods such as the symmetric interior penalty method and the symmetric DDG (SDDG) method admit a discrete Hamiltonian structure, whereas schemes including the Baumann--Oden, DDG, and BR2 methods do not possess this property. Exploiting this structure, we construct fully discrete symplectic schemes by combining the SDDG spatial discretization with symplectic time integrators. We further derive error estimates for the SDDG method applied to semilinear wave equations, showing the optimal convergence rate for the displacement and the suboptimal convergence rate for the velocity. Numerical experiments validate the theoretical convergence rates and demonstrate that the symplectic Hamiltonian DDG method achieves superior long-time energy conservation and accuracy.

math.NA↗

arXiv2025

PinFM: Foundation Model for User Activity Sequences at a Billion-scale Visual Discovery Platform

User activity sequences have emerged as one of the most important signals in recommender systems. We present a foundational model, PinFM, for understanding user activity sequences across multiple applications at a billion-scale visual discovery platform. We pretrain a transformer model with 20B+ parameters using extensive user activity data, then fine-tune it for specific applications, efficiently coupling it with existing models. While this pretraining-and-fine-tuning approach has been popular in other domains, such as Vision and NLP, its application in industrial recommender systems presents numerous challenges. The foundational model must be scalable enough to score millions of items every second while meeting tight cost and latency constraints imposed by these systems. Additionally, it should capture the interactions between user activities and other features and handle new items that were not present during the pretraining stage. We developed innovative techniques to address these challenges. Our infrastructure and algorithmic optimizations, such as the Deduplicated Cross-Attention Transformer (DCAT), improved our throughput by 600% on Pinterest internal data. We demonstrate that PinFM can learn interactions between user sequences and candidate items by altering input sequences, leading to a 20% increase in engagement with new items. PinFM is now deployed to help improve the experience of more than half a billion users across various applications.

cs.LG↗