Bin Dong(董彬) Long CV, Short CV
Professor
Beijing International Center for Mathematical
Research (BICMR),
Center for Machine Learning Research (CMLR)
Office: BICMR，77101; 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
research has primarily focused on biomedical imaging, with a particular
emphasis on image reconstruction, image analysis, and using images and medical
data to aid in clinical decision-making.
More recently, I have also begun working extensively in machine learning
and artificial intelligence, particularly in the development of foundation
models for biomedical data analysis, automated reasoning, and scientific
computing.
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).
Branching into 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 some 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:
I
have observed a dire 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 scenarios—a "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.
The
quest for a foundational model in computational imaging is intrinsically linked
to the universal principles governing wave-matter interactions, which underpin
virtually all imaging modalities. These interactions can be elegantly described
through PDEs. Recognizing this, our research endeavors to lay the groundwork
for a foundational model that encompasses the diverse spectrum of PDEs inherent
in imaging processes (and beyond). Our current trajectory has seen us making
promising strides in the one-dimensional domain (Ref1,Ref2), marking the initial phase of
a comprehensive journey towards achieving a holistic model for computational
imaging.
―
AI for Mathematics: Building AI Copilot
for Theoretical Research
The
theme of AI for Mathematics (AI4M) is rooted in a dynamic, bi-directional
relationship between artificial intelligence and mathematics, where each field
not only leverages the other but also fosters mutual advancement.
Traditionally,
mathematics has served as the theoretical backbone for AI, guiding the design
of algorithms and models. However, as AI technology has advanced at an
unprecedented pace, the development of theoretical foundations has struggled to
keep up, often constrained by the inefficiency of strictly theory-driven
approaches. In response, a new paradigm emerges: AI tools designed to support
mathematical research itself, opening pathways for more adaptive and creative
mathematical exploration. 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.
Conversely,
AI’s limitations in reasoning and formalization can benefit from
mathematical rigor, leveraging both formal and informal data and drawing from
the structured training and thinking processes intrinsic to mathematical
practice. Together, these intertwined advancements form a symbiotic
relationship, enabling AI and mathematics to evolve collaboratively in an
upward, spiral progression.
To
accumulate high-quality data and enhance AI's reasoning capabilities,
digitalizing mathematics in a formal language, such as Lean, becomes essential.
By translating mathematical concepts and proofs into a structured,
machine-readable format, we can create a foundation for both human-annotated
and AI-generated datasets that embody the rigor of mathematical reasoning. Our
recent efforts have been partially focused on expediting this digitalization
process by equipping the community with convenient tools, aiming to simplify
and accelerate the conversion of mathematical concepts and proofs into
formalized, machine-readable structures (Ref1,Ref2).
Building
on this formalized data, a focused approach to training AI’s reasoning
skills is necessary, drawing on domain-specific knowledge in mathematics to
create a tailored curriculum. Reinforcement Learning (RL) emerges as a
promising general strategy for training AI in this context; however, effective
application requires careful design of problem-solving tasks, reward
structures, and incremental challenges to mimic the stages of mathematical
discovery and intuition. Such a curriculum should emulate the processes
mathematicians use to learn and reason, gradually guiding AI systems toward a
refined understanding and application of complex mathematical reasoning. This
strategy holds the potential to advance AI's capacity for formal reasoning
while further strengthening the collaborative development of AI and
mathematics.
Last updated: 10/2024.