Deriving Compact Laws Based on Algebraic Formulation of a Data Set

TL;DR

This paper introduces an algebraic method for discovering compact, meaningful laws from data sets by formulating the problem as a linear algebra model, enabling efficient and precise identification of governing equations.

Contribution

The paper presents a novel algebraic equation formulation and search algorithms for deriving compact laws from data, improving efficiency and mathematical rigor over existing methods.

Findings

The approach guarantees convergence for certain types of theories.

It enables efficient and precise discovery of governing equations.

Validated through rigorous proofs and algorithmic development.

Abstract

In various subjects, there exist compact and consistent relationships between input and output parameters. Discovering the relationships, or namely compact laws, in a data set is of great interest in many fields, such as physics, chemistry, and finance. While data discovery has made great progress in practice thanks to the success of machine learning in recent years, the development of analytical approaches in finding the theory behind the data is relatively slow. In this paper, we develop an innovative approach in discovering compact laws from a data set. By proposing a novel algebraic equation formulation, we convert the problem of deriving meaning from data into formulating a linear algebra model and searching for relationships that fit the data. Rigorous proof is presented in validating the approach. The algebraic formulation allows the search of equation candidates in an explicit mathematical manner. Searching algorithms are also proposed for finding the governing equations with improved efficiency. For a certain type of compact theory, our approach assures convergence and the discovery is computationally efficient and mathematically precise.