A Physics-informed Sheaf Model

TL;DR

This paper introduces a novel sheaf-theoretic framework for normal mode analysis in molecules, linking eigenmodes to sheaf Laplacians and proposing efficient graph construction methods.

Contribution

It establishes a new connection between normal mode analysis and sheaf theory, defining the anisotropic sheaf and relating it to the Hessian matrix as a sheaf Laplacian.

Findings

Demonstrates a one-to-one correspondence between zero-eigenvalue modes and global sections of the sheaf.

Analyzes the dimension of the harmonic signal space in normal mode analysis.

Proposes a systematic graph construction method preserving normal mode properties.

Abstract

Normal mode analysis (NMA) provides a mathematical framework for exploring the intrinsic global dynamics of molecules through the definition of an energy function, where normal modes correspond to the eigenvectors of the Hessian matrix derived from the second derivatives of this function. The energy required to 'trigger' each normal mode is proportional to the square of its eigenvalue, with six zero-eigenvalue modes representing universal translation and rotation, common to all molecular systems. In contrast, modes associated with small non-zero eigenvalues are more easily excited by external forces and are thus closely related to molecular functions. Inspired by the anisotropic network model (ANM), this work establishes a novel connection between normal mode analysis and sheaf theory by introducing a cellular sheaf structure, termed the anisotropic sheaf, defined on undirected, simple graphs, and identifying the conventional Hessian matrix as the sheaf Laplacian. By interpreting the global section space of the anisotropic sheaf as the kernel of the Laplacian matrix, we demonstrate a one-to-one correspondence between the zero-eigenvalue-related normal modes and a basis for the global section space. We further analyze the dimension of this global section space, representing the space of harmonic signals, under conditions typically considered in normal mode analysis. Additionally, we propose a systematic method to streamline the Delaunay triangulation-based construction for more efficient graph generation while preserving the ideal number of normal modes with zero eigenvalues in ANM analysis.