Mechanistic Interpretability (MI) proposes a path toward fully understanding how neural networks make their predictions. Prior work demonstrates that even when trained to perform simple arithmetic, models can implement a variety of algorithms (sometimes concurrently) depending on initialization and hyperparameters. Does this mean neuron-level interpretability techniques have limited applicability? Here, we argue that high-dimensional neural networks can learn useful low-dimensional representations of the data they were trained on, going beyond simply making good predictions: Such representations can be understood with the MI lens and provide insights that are surprisingly faithful to human-derived domain knowledge. This indicates that such approaches to interpretability can be useful for deriving a new understanding of a problem from models trained to solve it. As a case study, we extract nuclear physics concepts by studying models trained to reproduce nuclear data.

Today's best language models still struggle with "hallucinations", factually incorrect generations, which impede their ability to reliably retrieve information seen during training. The *reversal curse*, where models cannot recall information when probed in a different order than was encountered during training, exemplifies limitations in information retrieval. To better understand these limitations, we reframe the reversal curse as a *factorization curse* --- a failure of models to learn the same joint distribution under different factorizations. We more closely simulate finetuning workflows which train pretrained models on specialized knowledge by introducing *WikiReversal*, a realistic testbed based on Wikipedia knowledge graphs. Through a series of controlled experiments with increasing levels of realism, including non-reciprocal relations, we find that reliable information retrieval is an inherent failure of the next-token prediction objective used in popular large language models. Moreover, we demonstrate reliable information retrieval cannot be solved with scale, reversed tokens, or even naive bidirectional-attention training. Consequently, various approaches to finetuning on specialized data would necessarily provide mixed results on downstream tasks, unless the model has already seen the right sequence of tokens. Across five tasks of varying levels of complexity, our results uncover a promising path forward: factorization-agnostic objectives can significantly mitigate the reversal curse and hint at improved knowledge storage and planning capabilities.

We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations, whose training dynamics and dependence on training set size can be predicted by our effective theory (in a toy setting). We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a "Goldilocks zone" (including comprehension and grokking) between memorization and confusion. Compared to the comprehension phase, the grokking phase stays closer to the memorization phase, leading to delayed generalization. The Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning.