Bin Dong(董彬) Long CV, Short CV

Professor

Beijing International Center for Mathematical Research (BICMR),

Center for Machine Learning Research (CMLR)

Peking University

Office: BICMR77101; Phone: +8610-62744091
Email: dongbin {at} math {dot} pku {dot} edu {dot} cn

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About -- Biography -- Events -- Publications -- Teaching

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Research Philosophy

My research philosophy has consistently been driven by the dual aspiration of devising effective models and algorithms to address real-world challenges while distilling novel mathematical problems during this process. In essence, I aim to contribute to both practical applications and the advancement of mathematics itself. This dualism has served as the cornerstone of my research endeavors.

 

Areas of Focus

My primary research areas converge on biomedical imaging, particularly in image reconstruction, image analysis, and auxiliary decision-making in clinical diagnosis leveraging images and other medical data.

 

Earlier Career

During my doctoral and postdoctoral years, I have developed various models and algorithms within the frameworks of Partial Differential Equations (PDEs) and wavelets. These include designing PDE models for cerebral aneurysm segmentation to facilitate subsequent geometric measurement (Ref); integrating the notion of PDEs, level-set methods and wavelet transforms to create computational tools for cardiovascular plaque quantification (Ref); formulating CT image reconstruction models by exploiting the property of sparse approximation of wavelet frame transforms (Ref1, Ref2, Ref3); and optimizing radiation therapy for cancer mathematically (Ref) which was later reported by phys.org.

 

Unveiling the Connection: PDEs and Wavelets

Throughout the course of addressing these applied problems, I became increasingly aware of the striking algorithmic resemblance between PDEs and wavelets. This led to an investigative journey with collaborators to explore any intrinsic relationships between these two paradigms, which had evolved independently in the field of imaging for nearly three decades. We discovered profound connections, unsettling traditional views and establishing an asymptotic convergence relationship between wavelet-based optimization models and PDE models. (Ref1, Ref2, Ref3, Ref4) This theoretical insight effectively bridged the geometric interpretations of differential operators with the sparse approximations and multi-scale nature of wavelets, inspiring new models that amalgamated the strengths of both PDEs and wavelets. In particular, this insight has led to new wavelet frame-based image segmentation models for medical images (Ref). Notably, our collaborative paper on wavelet frame-based segmentation of ultrasound videos (Ref) with my colleagues at Shanghai Jiao Tong University and National University of Singapore gained media attention, including coverage by SIAM News, Science Daily, etc.

 

Journey into Machine Learning

My initial venture into machine learning commenced in 2014 resulting in a sole-authored paper that introduced wavelet frames into semi-supervised learning models on graphs and point clouds (Ref), as well as a work on sparse linear discriminant analysis with my colleagues from the statistics program at the University of Arizona (Ref). However, it was not until 2015-2016 that I fully grasped the revolutionary impact of deep learning, particularly on biomedical imaging problems. It was also during this period that our earlier works on unveiling the connections between PDEs and wavelets contributed to a unified perspective on these mathematical models, which later informed our mathematical understanding of deep neural networks.

Subsequently, I led my team to pivot our focus towards deep learning methodologies. The core motive has been to explore the algorithmic relations between traditional mathematical methods like PDEs and wavelets and the data-driven models in deep learning. This exploration culminated in two important works of mine between 2017 and 2018: the establishment of the link between numerical differential equations and deep neural network architectures, leading to the development of ODE-Net (Ref) and PDE-Net (Ref1, Ref2), which were empirically validated on large-scale datasets. Through this and several subsequent research studies, we gradually developed a novel generic methodology for synergistically integrating mathematical modeling and machine learning techniques. We have been constantly practicing and refining this generic methodology in my research endeavors ever since (e.g. examples in Detour to Scientific Computing).

The core focus of my presentation at the 2022 International Congress of Mathematicians (ICM) was to elucidate the intricate connections between PDEs and wavelets, as well as the links between discrete differential equations and deep neural networks. These bridges have intriguing implications for the fields of computational imaging and scientific computing, which were thoroughly discussed in my talk and the corresponding proceedings and notes (Ref1, Ref2).


Detour to Scientific Computing

Upon understanding deep neural networks through the lens of numerical differential equations, it became natural to tackle challenges in scientific computing, specifically, PDE-based forward and inverse problems. For the forward problems, we blended numerical methods for conservation laws and multigrid methods with deep reinforcement learning and meta-learning. (Ref1, Ref2) More recently, we discovered a connection between meta-learning and reduced-order modeling, paving the way for a new technique in approximating the solution manifold of parametric PDEs. (Ref1, Ref2) For inverse problems, our focus has been primarily on imaging, where we proposed several ways of melding deep learning techniques with traditional mathematical algorithms (Ref1, Ref2, Ref3, Ref4).

 

Clinical Applications and Translational Research

Since 2019, I have embarked on an extensive collaboration with Peking University Cancer Hospital, focusing on clinical auxiliary diagnostic issues. This has resulted in a series of exciting progresses in medical image analysis, contributing valuable tools for clinical practice that have seen successful adoption and some degree of translational implementation.(Ref1, Ref2, Ref3, Ref4)

 

Future Research Avenues

In my academic trajectory, I have encountered two major challenges that will guide my research focus for the next n+1 years.

   AI for Computational Imaging: Addressing the 'Boutique' Nature of Biomedical Imaging Models:

The first challenge is rooted in the need for a unified and robust model in the realm of biomedical imaging. Currently, while the fusion of machine learning with first-principles modeling has proven powerful, its utility often rests on designing specific models for specific scenariosa "boutique" approach that is neither efficient nor universally applicable. In light of this, my future research aims to construct a foundational model for computational imaging. This endeavor will focus on two key tasks: firstly, to seamlessly integrate the three stages of computational imaging, namely sensing, reconstruction, and analysis; and secondly, to amalgamate various imaging modalities into a comprehensive framework. This unified model aims to better address real-world challenges encountered in biomedical imaging.

   AI for Mathematics: Building AI Copilot for Theoretical Research

The second challenge stems from the observation that advances in artificial intelligence technology have outpaced the development of underlying theory. Traditional theoretical research has been effective in guiding the design of algorithms, but it is increasingly lagging in the realm of AI. One key bottleneck is the inefficiency of theory-driven research, where researchers often find it difficult to flexibly apply a breadth of mathematical knowledge or to innovatively create new mathematical concepts and tools. To address this, my future work plans to develop an AI assistant designed to facilitate mathematical theory research. This assistant would serve dual functions: to inspire mathematical thinking and to automate several aspects of theoretical work. This includes, but is not limited to, the verification of theorem proofs, automated theorem proving, and semantic search of mathematical concepts and results. While we've made some initial (but exciting) steps in this direction in collaboration with colleagues from Hong Kong University, specifically in the application of machine learning tools to explore affine Deligne-Lusztig varieties (Ref), there is still much to be done to fully realize the potential of AI as a tool for advancing mathematical theory.

 

By focusing on these two pivotal challenges, I aim to modestly contribute to both the realm of biomedical imaging and the advancement of mathematics itself.

 

Last updated: 10/2023.