Singularities in geophysical inverse problems: an approach using Lie groups and singular Bayesian learning theory

Reference No.	2026a040
Type/Category	Grant for Project Research- Short-term Visiting Researcher
Title of Research Project	Singularities in geophysical inverse problems: an approach using Lie groups and singular Bayesian learning theory
Principal Investigator	Keisuke Yano（The Research Organization of Information and Systems The Institute of Statistical Mathematics・Associate Professor）
Research Period	February 15,2027. - February 27,2026.
Keyword(s) of Research Fields	Lie group, symmetry, singular Bayesian learning theory
Abstract for Research Report	In geophysics, high-dimensional inverse problems often arise, such as fault slip estimation. In such inverse problems, identifiability issues arise, in which the parameters minimizing the loss are not uniquely determined because of both the high dimensionality of the parameter space and the symmetries inherent in the problem. These identifiability issues often destabilize the estimation and hinder the interpretation of the estimated results. This proposal aims to clarify what kinds of identifiability issues exist in geophysical inverse problems, based on the theoretical frameworks of Lie groups and singular Bayesian learning theory. Specifically, identifiability in inverse problems will be formulated as a Lie group induced by the observation equations and the physical model, and its structure will be characterized both through mathematical derivation and data analysis. When the loss function or likelihood function is invariant under the action of a Lie group, the optimal solution is not unique but instead appears as a group orbit, resulting in non-uniqueness in estimation. This study will clarify what kinds of Lie groups underlie the indeterminacies inherent in fault slip inversion. Furthermore, within the framework of singular Bayesian learning theory developed by Sumio Watanabe, it is known that algebraic invariants such as the real log canonical threshold and singular fluctuation can be used to quantify the asymptotic behavior of singular models and the difficulty of learning. This study will investigate how the degeneracy structure of the parameter space arising from Lie groups is reflected in these theoretical quantities. The results of this research are expected to provide a common theoretical foundation for a wide range of geophysical inverse problems in which observational constraints are limited. In addition, the proposed framework is highly compatible with rapidly developing approaches such as physics-informed machine learning, and is expected to contribute to more advanced uncertainty quantification and model interpretation in geophysics.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	矢野 恵佑（統計数理研究所・准教授）