A New Era in Mathematical Science: The Synergy of Numerical Analysis and Machine Learning

Reference No.	2023a014
Type/Category	Grant for Young Researchers and Students-Short-term Joint Research
Title of Research Project	A New Era in Mathematical Science: The Synergy of Numerical Analysis and Machine Learning
Principal Investigator	Noboru Isobe（Graduate school of mathematical sciences, the university of Tokyo, ・Ph.D student）
Research Period	November 2,2023. ～ November 3,2023.
Keyword(s) of Research Fields	Numerical Analysis, Computation, Machine Learning, Deep Learning, Mathematical Analysis, Differential Equations, Mathematical Modeling
Abstract for Research Report	This joint research aims to develop reliable and versatile simulation techniques that combine traditional numerical methods and machine learning techniques, which have been rapidly growing in recent years, in a complementary manner. Traditionally, numerical analysis and numerical methods have focused on " approximating" individual phenomena and mathematical models. On the other hand, machine learning techniques, with deep learning at their core, are making it possible to compute a wide range of phenomena by "learning" based on mathematical optimization as long as there is a vast amount of data. Combining the ingenuity of "approximation" and the versatility of "learning" in these two technologies will make it possible to develop general-purpose simulation technology with high speed and high accuracy. However, there are still barriers to combining these technologies, which have different backgrounds and value criteria. This barrier is because each technology is described ad hoc manner in line with the problem. In this joint research, we attempt to abstract and describe both "approximation" and "learning" in the language of mathematical analysis through the collaboration of researchers in numerical computation, machine learning, and the mathematical sciences, which are positioned at the boundary between the two. For example, the finite element method in numerical analysis is a finite-dimensional approximation of function spaces. In contrast, the universal approximation capability of a deep neural network can be described as the tightness in a specific function space. We can find similarities between the two through the mediation of these mathematical sciences. Using such mathematical analysis as a catalyst, we can expect a new mathematical concept that seamlessly connects the two as an outcome. Furthermore, this concept will be the seed from which simulations of complex and difficult-to-observe phenomena, such as semiconductor manufacturing, will be possible through machine learning from known simple simulations.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Noboru Isobe（Graduate School of Mathematical Sciences, The University of Tokyo・D1） Daisuke Inoue（Toyota Central R&D Labs,. Inc.・Researcher） Daisuke Tagami（Institute of Mathematics for Industry, Kyushu University・Associate Professo） Masanobu Horie（RICOS Co. Ltd.・Head of Foundation Research Department） Takaharu Yaguchi（Department of Computational Science, Graduate School of System Informatics, Kobe University・Associate Professor） Takeshi Koshizuka（Department of Computer Science, The University of Tokyo・D1） Yuka Hashimoto（NTT/RIKEN AIP・Researcher） Kengo Nakai（Tokyo University of Marine Science and Technology・Assistant Professor） Tokuhiro Eto（Graduate School of Mathematical Sciences, The University of Tokyo・D3）