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His PhD adviser from Princeton University, Professor E. Weinan, once told him that applied mathematics research must always be connected to the broader scientific community, and Dr Li Qianxiao has held on to this principle ever since.
“(It) has driven my research and reminded me that I must constantly learn from and talk with scientists from other fields to formulate good research questions,” said the assistant professor from the Department of Mathematics at the NUS Science Faculty.
Dr Li, 37, has made significant contributions to the mathematical foundations of deep learning, a crucial technology that underpins the success and scaleability of modern AI systems.
His work provided rigorous mathematical frameworks necessary for understanding and improving deep learning methodologies, and laying the groundwork for future innovations in the field.
“Deep learning models use layers of ‘artificial neurons’ expressed as simple mathematical functions or operations, that process information in a way inspired by how the human brain works. These models are trained on large amounts of data to perform certain tasks that only humans used to be able to do in the past,” Dr Li explained.
AI technologies such as image recognition, translation and language models all have deep learning as a crucial component, he added.
With his team, Dr Li pioneered the understanding of how expressiveness – the ability to learn arbitrary relationships – arises in deep neural networks, and the work turned up intricate connections between this modern technology and classical mathematics.
“My work tries to understand how these ‘artificial neurons’ can express very complex relationships between things, and how best to improve the efficiency of training them. This is important as the scale of modern applications becomes larger,” he said.
“A newly released language model has more than 70 billion parameters to train, and understanding these basic principles also helps us know how best to apply this technology to new applications, like materials science and engineering.”
Mathematics to Dr Li is less about numbers and more about logic and concepts.
He said: “The famous mathematician Henri Poincare once said that, akin but somewhat opposite to poetry, mathematics is ‘the art of giving the same name to different things’.
“In fact, the dynamics of fluids, polymers and the training process of large neural networks are described by very similar mathematical models.
“In this sense, mathematics bridges our understanding across different domains, (and helps us) find commonalities and often creative ways to understand and solve problems.”
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