New SIM theory uses Lagrangian mechanics to design interpretable AI models

As AI systems become more complex, interpretability tools are critical for debugging and trust, but the field lacks a unifying theory to systematically design them. Researchers from the University of Cambridge, Imperial College London, and other institutions present the Standard Interpretable Model (SIM) — a general theory grounded in Lagrangian mechanics. SIM starts from a set of premises defining what interpretability means for a specific user, then deductively derives symmetries and constraints that shape the loss landscape. The minima of this Lagrangian correspond to optimal interpretable models, which can be reached either by updating an opaque model’s parameters or by compiling constraints directly into the architecture.

The SIM framework is empirically shown to identify and fix shortcomings in existing interpretability approaches, including traditional feature-based, concept-based (like TCAV), and mechanistic interpretability (e.g., probing). It also highlights underexplored research directions and informs the design of core programming interfaces. Beyond research, the deductive nature of SIM offers a structured pedagogical method for teaching interpretability, potentially shifting the field from a fragmented collection of techniques to a principled science. The paper is available on arXiv (2606.12289).