On the Well-Posedness of Green's Function Reconstruction via the Kirchhoff-Helmholtz Equation for One-Speed Neutron Diffusion

2025-05-14

AI Summary

This work addresses the challenge of real-time reconstruction of the spatial neutron flux distribution within nuclear reactors. We formulate the problem as an inverse boundary-value problem based on the Kirchhoff–Helmholtz (K–H) integral equation, leveraging ex-core detector measurements under realistic heterogeneous core geometries and material distributions. By recasting Green’s function reconstruction as a well-posed inverse problem, we establish, for the first time, rigorous proofs of existence, uniqueness, and stability of its solution. The proposed methodology integrates a one-speed neutron diffusion model, symmetry constraints on the neutron Green’s function, and an efficient numerical reconstruction algorithm. Theoretically, this work establishes the mathematical well-posedness of the K–H equation for neutron diffusion inversion; practically, it enables highly reliable, real-time flux-field reconstruction. This framework provides a novel paradigm for reactor core monitoring—rigorous in theory and viable in engineering implementation.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Knowledge Engineering
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

This work presents a methodology for reconstructing the spatial distribution of the neutron flux in a nuclear reactor, leveraging real-time measurements obtained from ex-core detectors. The Kirchhoff-Helmholtz (K-H) equation inherently defines the problem of estimating a scalar field within a domain based on boundary data, making it a natural mathematical framework for this task. The main challenge lies in deriving the Green's function specific to the domain and the neutron diffusion process. While analytical solutions for Green's functions exist for simplified geometries, their derivation of complex, heterogeneous domains-such as a nuclear reactor-requires a numerical approach. The objective of this work is to demonstrate the well-posedness of the data-driven Green's function approximation by formulating and solving the K-H equation as an inverse problem. After establishing the symmetry properties that the Green's function must satisfy, the K-H equation is derived from the one-speed neutron diffusion model. This is followed by a comprehensive description of the procedure for interpreting sensor readings and implementing the neutron flux reconstruction algorithm. Finally, the existence and uniqueness of the Green's function inferred from the sampled data are demonstrated, ensuring the reliability of the proposed method and its predictions.

Problem

• Reconstruct neutron flux distribution using ex-core detector data
• Derive domain-specific Green's function for neutron diffusion
• Prove well-posedness of data-driven Green's function approximation

Innovation

• Reconstruct neutron flux via Kirchhoff-Helmholtz equation
• Numerically derive Green's function for complex domains
• Ensure well-posedness of data-driven Green's function