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

arXiv: https://arxiv.org/abs/2606.12289 · HTML: https://mlsanity.com/paper/2606.12289/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 133 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 23 of 23)

- [Qwen3-VL Technical Report](https://mlsanity.com/paper/2511.21631/) (2025)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [The (Un)reliability of saliency methods](https://mlsanity.com/paper/1711.00867/) (2017)
- [AxBench: Steering LLMs? Even Simple Baselines Outperform Sparse Autoencoders](https://mlsanity.com/paper/2501.17148/) (2025)
- [k-Sparse Autoencoders](https://mlsanity.com/paper/1312.5663/) (2013)
- [Concept-based Explainable Artificial Intelligence: A Survey](https://mlsanity.com/paper/2312.12936/) (2023)
- [A Closer Look at the Intervention Procedure of Concept Bottleneck Models](https://mlsanity.com/paper/2302.14260/) (2023)
- [Stochastic Concept Bottleneck Models](https://mlsanity.com/paper/2406.19272/) (2024)
- [Imposing Hard Constraints on Deep Networks: Promises and Limitations](https://mlsanity.com/paper/1706.02025/) (2017)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [Causal Concept Graph Models: Beyond Causal Opacity in Deep Learning](https://mlsanity.com/paper/2405.16507/) (2024)
- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [Towards Automatic Concept-based Explanations](https://mlsanity.com/paper/1902.03129/) (2019)
- [There Was Never a Bottleneck in Concept Bottleneck Models](https://mlsanity.com/paper/2506.04877/) (2025)
- [Interpretability Can Be Actionable](https://mlsanity.com/paper/2605.11161/) (2026)
- [Foundations of Interpretable Models](https://mlsanity.com/paper/2508.00545/) (2025)
- [Physics of Learning: A Lagrangian perspective to different learning paradigms](https://mlsanity.com/paper/2509.21049/) (2025)
- [Don't Lose Focus: Activation Steering via Key-Orthogonal Projections](https://mlsanity.com/paper/2605.06342/) (2026)
- [Prototype-Grounded Concept Models for Verifiable Concept Alignment](https://mlsanity.com/paper/2604.16076/) (2026)

## Built on by (most cited first, 1 of 1)

- [Graphical Design of Interpretable Architectures](https://mlsanity.com/paper/2608.18936/) (2026)

## Abstract (arXiv)

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.