This study aims to achieve the goals described in the Research Overview mainly by addressing the following four issues:

(Issue 1) Establishment of a statistical inference method for operator-theoretic data analysis of dynamical systems We build a methodology for the development of estimation methods in accordance with various situations and for the identification of statistically significant dynamics by further developing the existing operator spectrum estimation method and the stochastic formulation of the dynamic mode decomposition constructed by Kawahara (principal investigator) et al.

(Issue 2) Construction of a mathematical system for extension to a system with continuous spectra The development of an analysis method for systems with continuous spectra is essential for dealing with complex phenomena. Compared with the case of a point spectrum, the mathematical treatment of a system is more difficult, and hence close examination based on a rigorous mathematical framework with a focus on geometry is required. Here, we extend the existing inference method to one that can be applied to cases with continuous spectra while also clarifying the associated assumptions, and we carry out the systematization. In particular, we focus on chaotic phenomena and on physically important situations such as systems in which dynamics with different spatiotemporal scales interact and proceed with the mathematical development on the relation between the obtained estimator and existing physical concepts, invariants, and the reduction of the mathematical models.

(Issue 3) Construction of a computational system for learning and prediction through the integrated use of mathematical model and extracted information We plan to construct a systematic methodology for prediction and learning that uses mathematical models and extracted information interactively through their dynamic characteristics. For example, a learning model such as a neural network may be regarded as a hierarchical nonlinear dynamical system, and in recent times a learning model using these characteristics has been reported and was the object of much attention. We plan to construct a framework for learning and inference using dynamic characteristics obtained by operatortheoretic analysis in an integrated manner while also utilizing the properties of the model as a dynamical system and to carry out the theoretical analysis of the model.

(Issue 4) Development by application to data analysis across various fields of science We will apply the framework obtained from the first three issues to dynamical phenomena across various fields and carry out its verification. First, based on the operator-theoretic analysis obtained from the issues mentioned above, the framework’s estimation method, and its mathematical interpretation, we will conduct data analysis to extract knowledge about principles that cannot be expressed by mathematical models in the domain. In addition, we shall apply the development methodology to rare phenomena that were difficult to predict in the past, to phenomena that are associated with complex factors, and to applications that require sophisticated predictions. We will also validate the usefulness of the methodology.

In the case of (Issue 1), we establish a statistical inference method based on operatortheoretic analysis and systematize its mathematical interpretation and then create and systematize an operator-theoretic data analysis. (Issue 2) is a mathematical extension of this analysis to complex phenomena, which is necessary to enable its application to phenomena of scientific interest. (Issue 3) is a methodological development in which the methods obtained by the previous tasks are used integrally with mathematical models for the prediction of complex phenomena or the extension of learning mechanisms. In (Issue 4), we apply the development method to issues across multiple domains for which it is important to analyse dynamic mechanisms, focusing on biological modelling issues. We also verify the usefulness of the method.

Given its academic uniqueness and significance, the three main direct impacts of the results of this research in terms of creating innovation in science and technology are as follows:

**1. It is a proposal for constructing a methodology and theoretical system for a new cross-disciplinary approach to science.**

In general, mathematical models have been constructed in every field of scientific research and such research has been progressing well up until now, barring a few exceptional cases. The framework developed in this study is based on the generic mathematical principles of operator expression of nonlinear dynamical systems and their statistical estimation. The constructed methodology can therefore be applied to dynamical complex phenomena across various fields with common interests as long as there is a mathematical model and measurement data for each domain. The extracted information obtained in this way can be linked to the knowledge of each domain by the developed theoretical system. In this way, the results of this research may be expected to influence not only the domains listed in the proposal but also other scientific and engineering fields.

**2. It creates a framework that integrates data-driven information extraction with mathematical models through dynamic characteristics.**

For socially important issues, such as disaster prevention and medical care (forecasting life phenomena), precise predictions and simulations are required even in the case of rare situations and complex situations, such as multi-scale phenomena. Because the framework developed in this study is based on the mathematical principles stated above, it extracts information (that is, the mechanism of the principles of dynamic phenomena) that can be interpreted in terms of mathematical models through the dynamic characteristics of the data generation mechanism. Therefore, in addition to the acquisition of scientific knowledge directly related to the principles in each domain, more precise predictions and simulations of complex phenomena can be made possible through the use of mathematical models integrated with (extrapolative) inference.

**3. It constructs and systematises data analysis methodologies based on rigorous mathematical principles.**

The purpose of this study is to construct a data analysis method that is directly linked to the physical interpretation of the mechanism of a phenomenon. Hence, it is important to closely examine the mathematical properties of the nonlinear dynamical system that generates the data. Therefore, mathematicians from various fields participated in this study and applied a wide range of mathematical techniques to this issue, including theories of mechanical systems that are directly related to the phenomenon of interest and the theories of functional analysis and operator algebra that are working in the background. As this study aims to construct a data analysis framework based on such rigorous mathematical considerations, these two factors may be said to contribute to its reliability and versatility.

Four groups worked together on this research: the Machine Learning and Mathematical Statistics Group (Kawahara, (IMI, Kyushu University / RIKEN AIP)), the Mathematics Group (Bannai (Keio University / RIKEN AIP)), the Nonlinear Physics Group (Nakao (Tokyo Institute of Technology)), and the Biological Modelling Group (Kurosawa (RIKEN iTHEMS)). The Machine Learning Group works from a data-driven perspective, the Nonlinear Physics Group works from a mathematical model perspective, and the Mathematics Group explores the mathematical principles that connect the two. The Biological Modelling Group verifies the applicability of the development methods throughout the research and provides feedback on methodologies and principles.

The Machine Learning and Mathematical Statistics Group

- Leader:
- Yoshinobu Kawahara（Professor, Institute of Mathematics for Industry, Kyushu University /Team Leader, RIKEN Center for Advanced Intelligence Research）
- Main research items:
- Creation of a complex dynamics computational platform based on operator-theoretic data analysis

The Mathematics Group

- Leader:
- Bannai Kenichi（Professor, Faculty of Science and Technology,Keio University / Team Leader, RIKEN Center for Advanced Intelligence Research）
- Main research items:
- Mathematical theory and construction of analytical methods for operator-theoretic data structures

The Nonlinear Physics Group

- Leader:
- Hiroya Nakao（Professor, School of Engineering, Tokyo Institute of Technology）
- Main research items:
- Establishment of a data-driven model reduction method for nonlinear dynamical systems with large degrees of freedom

The Biological Modelling Group

- Leader:
- Gen Kurosawa（Full-time researcher, RIKEN Mathematical Creation Program）
- Main research items:
- Construction of a mathematical model expressing dynamic life phenomena