# Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials

arXiv: https://arxiv.org/abs/2402.16726 · HTML: https://mlsanity.com/paper/2402.16726/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 44 references that are in the map.
- First posted on arXiv 26 Feb 2024.

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Faith and Fate: Limits of Transformers on Compositionality](https://mlsanity.com/paper/2305.18654/) (2023)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)
- [Explaining grokking through circuit efficiency](https://mlsanity.com/paper/2309.02390/) (2023)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworks](https://mlsanity.com/paper/2303.11873/) (2023)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Grokking of Hierarchical Structure in Vanilla Transformers](https://mlsanity.com/paper/2305.18741/) (2023)
- [Unifying Grokking and Double Descent](https://mlsanity.com/paper/2303.06173/) (2023)
- [Grokking as a First Order Phase Transition in Two Layer Networks](https://mlsanity.com/paper/2310.03789/) (2023)
- [Predicting Grokking Long Before it Happens: A look into the loss landscape of models which grok](https://mlsanity.com/paper/2306.13253/) (2023)
- [Benign Overfitting and Grokking in ReLU Networks for XOR Cluster Data](https://mlsanity.com/paper/2310.02541/) (2023)
- [Feature emergence via margin maximization: case studies in algebraic tasks](https://mlsanity.com/paper/2311.07568/) (2023)

## Built on by (most cited first, 14 of 14)

- [Locate, Steer, and Improve: A Practical Survey of Actionable Mechanistic Interpretability in Large Language Models](https://mlsanity.com/paper/2601.14004/) (2026)
- [Rethinking Evaluation of Sparse Autoencoders through the Representation of Polysemous Words](https://mlsanity.com/paper/2501.06254/) (2025)
- [Beyond Induction Heads: In-Context Meta Learning Induces Multi-Phase Circuit Emergence](https://mlsanity.com/paper/2505.16694/) (2025)
- [Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)](https://mlsanity.com/paper/2502.21009/) (2025)
- [Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representation](https://mlsanity.com/paper/2405.16658/) (2024)
- [A rationale from frequency perspective for grokking in training neural network](https://mlsanity.com/paper/2405.17479/) (2024)
- [Geometric Generality of Transformer-Based Gr"obner Basis Computation](https://mlsanity.com/paper/2504.12465/) (2025)
- [Emergence of Minimal Circuits for Indirect Object Identification in Attention-Only Transformers](https://mlsanity.com/paper/2510.25013/) (2025)
- [Is Grokking Worthwhile? Functional Analysis and Transferability of Generalization Circuits in Transformers](https://mlsanity.com/paper/2601.09049/) (2026)
- [Gradient Flow Through Diagram Expansions: Learning Regimes and Explicit Solutions](https://mlsanity.com/paper/2602.04548/) (2026)
- [Grokked Models are Better Unlearners](https://mlsanity.com/paper/2512.03437/) (2025)
- [Learning Large-Scale Modular Addition with an Auxiliary Modulus](https://mlsanity.com/paper/2605.07648/) (2026)
- [The Discrete-Log Clock: How a Transformer Learns Modular Multiplication](https://mlsanity.com/paper/2606.17399/) (2026)
- [Causal and Interpretable Structures in LLM Compositional Tasks](https://mlsanity.com/paper/2609.35970/) (2026)

## Abstract (arXiv)

Grokking has been actively explored to reveal the mystery of delayed generalization and identifying interpretable representations and algorithms inside the grokked models is a suggestive hint to understanding its mechanism. Grokking on modular addition has been known to implement Fourier representation and its calculation circuits with trigonometric identities in Transformers. Considering the periodicity in modular arithmetic, the natural question is to what extent these explanations and interpretations hold for the grokking on other modular operations beyond addition. For a closer look, we first hypothesize that any modular operations can be characterized with distinctive Fourier representation or internal circuits, grokked models obtain common features transferable among similar operations, and mixing datasets with similar operations promotes grokking. Then, we extensively examine them by learning Transformers on complex modular arithmetic tasks, including polynomials. Our Fourier analysis and novel progress measure for modular arithmetic, Fourier Frequency Density and Fourier Coefficient Ratio, characterize distinctive internal representations of grokked models per modular operation; for instance, polynomials often result in the superposition of the Fourier components seen in elementary arithmetic, but clear patterns do not emerge in challenging non-factorizable polynomials. In contrast, our ablation study on the pre-grokked models reveals that the transferability among the models grokked with each operation can be only limited to specific combinations, such as from elementary arithmetic to linear expressions. Moreover, some multi-task mixtures may lead to co-grokking -- where grokking simultaneously happens for all the tasks -- and accelerate generalization, while others may not find optimal solutions. We provide empirical steps towards the interpretability of internal circuits.