The Standard Interpretable Model: A general theory of interpretable machine learning to deductively design interpretable methods using Lagrangian mechanics

2026-06-10

AI Summary

This work addresses the lack of a unified theoretical foundation in interpretable machine learning, which has led to fragmented methodologies and inconsistent evaluation criteria. By introducing Lagrangian mechanics into this domain for the first time, the paper proposes a general theoretical framework grounded in user-oriented interpretability. Through systematic analysis of symmetries and constraints, the approach derives optimal interpretable models by minimizing a suitably defined Lagrangian. This deductive methodology not only unifies existing techniques under a coherent theoretical umbrella but also reveals novel research directions. It has successfully informed the design of core programming interfaces, mitigated limitations of current methods, and established a rigorous theoretical basis for interpretability education and interdisciplinary integration.

Abstract

As Artificial Intelligence models grow in complexity, interpretability has become an indispensable tool for understanding, debugging, and controlling their computations. However, interpretability lacks general theories to deductively design interpretable methods. This gap between theories and methods results in a fragmented literature and inconsistent evaluation protocols. To fill this gap, we introduce the Standard Interpretable Model (SIM), a general theory grounded in Lagrangian mechanics that enables the deductive design of interpretable methods. Specifically, the SIM summarises, in a set of premises, what interpretability is for a target user. From these premises, the SIM systematically derives interpretability symmetries and corresponding constraints, which shape the landscape of a Lagrangian whose minima correspond to optimal interpretable models. To reach the minima, one can either update the parameter values of an opaque model to make it more interpretable or compile constraints into an interpretable architecture. We empirically show that the SIM identifies and solves limitations of existing methods (including traditional, concept-based, and mechanistic interpretability), highlights underexplored research directions, and informs the design of core programming interfaces. Beyond being a research method, the deductive nature of the SIM offers pedagogical grounding for interpretability curricula and may shift the scientific community's perspective of a discipline that has long been fragmented.

Problem

• interpretable machine learning
• general theory
• deductive design
• interpretability
• Lagrangian mechanics

Innovation

• Standard Interpretable Model
• Lagrangian mechanics
• interpretable machine learning
• deductive design
• interpretability symmetries