陽明交通大學應用數學系

Abstract. Transformer-based machine learning models have achieved state-of-the-art performance in many areas. However, the quadratic complexity of the self-attention mechanism in Transformer models with respect to the input length hinders the applicability of Transformer-based models to long sequences or large images. To address this, we present Fast Multipole Attention (FMA), a new attention mechanism that uses a divide-and-conquer strategy to reduce the time and memory complexity of attention from $O(n^2)$ to $O(n \log n)$ or $O(n)$, while retaining a global receptive field. The hierarchical approach groups queries, keys, and values into $O(\log n)$ levels of resolution, where groups at greater distances are increasingly larger in size and the weights to compute group quantities are learned. As such, the interaction between tokens far from each other is considered in lower resolution in an efficient hierarchical manner. This multi-level divide-and-conquer strategy is inspired by fast summation methods from n-body physics and the Fast Multipole Method. We perform evaluation on language modeling and image processing tasks and compare our FMA model with other efficient attention variants on medium-size datasets. We find empirically that the Fast Multipole Transformer outperforms other efficient transformers in terms of memory size and accuracy. For large language models, the FMA mechanism has the potential to enable greater sequence lengths, taking the full context into account in an efficient, naturally hierarchical manner during training and when generating long sequences.

Abstract

A domino (resp., lozenge) tiling is a covering of a region on the square (resp., triangular) lattice using dominoes (resp., lozenges) without gaps or overlaps. In this talk, I will introduce the physical background of these objects and several classical techniques for enumerating tilings of specific regions. Finally, I will present recent results regarding the symmetry classes of domino tilings of the Aztec diamonds. This talk does not assume any prior background; undergraduate students are welcome.

Abstract. The heat equation with a random potential driven by space-time white noise serves as a basic model for continuum directed polymers in random media, a topic of longstanding interest in the physics literature. While this model generates systems exhibiting scale invariance and universality, establishing their fundamental properties has posed deep mathematical challenges. In particular, although classical stochastic analysis can construct the corresponding Gibbs measures for polymers as two-dimensional curves, extending these results to three dimensions introduces substantial complexity due to the critical, non-Gaussian behaviour.

This talk will present the basic ideas of continuum directed polymers in random media, focusing on their behaviour as three-dimensional curves.

Abstract. Newton-type methods are among the most effective tools for solving large-scale nonlinear systems arising in scientific computing. Despite their fast local convergence, their global behavior can be unpredictable, with common issues such as overshooting, stagnation, and sensitivity to problem scaling—especially in stiff or highly unbalanced systems.

In this talk, we present a dynamical systems perspective for understanding these behaviors by interpreting Newton iterations as discrete approximations of an underlying continuous-time flow. This viewpoint provides an intuitive characterization of nonlinear imbalance in terms of stiffness, offering insight into why classical globalization strategies, particularly line search, may become ineffective or overly restrictive.

Motivated by this perspective, we revisit line search methods and introduce improved strategies, including curve search techniques, that better align with the intrinsic dynamics of the nonlinear system. We further show that nonlinear preconditioning can be naturally interpreted as a transformation that reduces stiffness and restores balance, leading to improved robustness and convergence.

Numerical examples from nonlinear PDEs illustrate how this framework not only enhances performance, but also provides a unified viewpoint for understanding globalization, acceleration, and stabilization in Newton-type methods.

Abstract
Artificial intelligence (AI) has become increasingly integrated into modern radiology, offering tools that support image interpretation, quantitative analysis, workflow prioritization, and clinical decision-making. Recent advances in machine learning and deep learning have enabled the detection of subtle imaging patterns and the efficient analysis of large-scale imaging data, helping to enhance diagnostic accuracy and improve workflow efficiency. In current radiology practice, AI is already being applied in areas such as lesion detection, organ segmentation, triage, and risk prediction.
Pancreatic cancer remains one of the most lethal malignancies, in part because of its subtle imaging features and the difficulty of early diagnosis on routine CT. In this talk, we will review the emerging role of AI in pancreatic cancer detection using CT imaging, with a focus on recent advances in AI-based detection models and their potential to facilitate earlier diagnosis. We will also briefly discuss the broader real- world applications of AI in radiology and consider the challenges and opportunities for translating these technologies into clinical practice.

Abstract

The Kadomtsev-Petviashvili II (KPII) equation is one of the few physically relevant integrable systems in more than one spatial dimension. In this talk, we present an overview of the inverse scattering theory and the stationary phase method, and explain how these tools are used to derive the long-time asymptotic behavior of solutions.