# In-Context Learning of Linear Systems: Generalization Theory and Applications to Operator Learning

arXiv: https://arxiv.org/abs/2409.12293 · HTML: https://mlsanity.com/paper/2409.12293/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 51 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 30 of 34)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Transformers in Vision: A Survey](https://mlsanity.com/paper/2101.01169/) (2021)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention](https://mlsanity.com/paper/2307.03576/) (2023)
- [Pretraining task diversity and the emergence of non-Bayesian in-context learning for regression](https://mlsanity.com/paper/2306.15063/) (2023)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [The Evolution of Statistical Induction Heads: In-Context Learning Markov Chains](https://mlsanity.com/paper/2402.11004/) (2024)
- [How Transformers Learn Causal Structure with Gradient Descent](https://mlsanity.com/paper/2402.14735/) (2024)
- [In-Context Convergence of Transformers](https://mlsanity.com/paper/2310.05249/) (2023)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Linear attention is (maybe) all you need (to understand transformer optimization)](https://mlsanity.com/paper/2310.01082/) (2023)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [How Many Pretraining Tasks Are Needed for In-Context Learning of Linear Regression?](https://mlsanity.com/paper/2310.08391/) (2023)
- [How Do Transformers Learn In-Context Beyond Simple Functions? A Case Study on Learning with Representations](https://mlsanity.com/paper/2310.10616/) (2023)
- [Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality](https://mlsanity.com/paper/2402.19442/) (2024)
- [Asymptotic theory of in-context learning by linear attention](https://mlsanity.com/paper/2405.11751/) (2024)
- [Unveiling Induction Heads: Provable Training Dynamics and Feature Learning in Transformers](https://mlsanity.com/paper/2409.10559/) (2024)
- [Transformers Learn Nonlinear Features In Context: Nonconvex Mean-field Dynamics on the Attention Landscape](https://mlsanity.com/paper/2402.01258/) (2024)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [In-Context Learning with Representations: Contextual Generalization of Trained Transformers](https://mlsanity.com/paper/2408.10147/) (2024)
- [Transformers are Minimax Optimal Nonparametric In-Context Learners](https://mlsanity.com/paper/2408.12186/) (2024)

## Built on by (most cited first, 6 of 6)

- [From Sparse Dependence to Sparse Attention: Unveiling How Chain-of-Thought Enhances Transformer Sample Efficiency](https://mlsanity.com/paper/2410.05459/) (2024)
- [Chain of Operators: An Inference-Time Harness for In-Context Operator Learning](https://mlsanity.com/paper/2606.12318/) (2026)
- [A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schr"odinger Equations](https://mlsanity.com/paper/2601.12587/) (2026)
- [Taming Visual Neglect: A Variational Information Bottleneck Framework for Adaptive Attention in Multimodal In-Context Learning](https://mlsanity.com/paper/2608.23570/) (2026)
- [Theory on Attention Dynamics for Out-of-Distribution In-Context Learning](https://mlsanity.com/paper/2609.36448/) (2026)
- [Transformer-based In-Context Score Operator Learning for Diffusion Models](https://mlsanity.com/paper/2610.13690/) (2026)

## Abstract (arXiv)

We study theoretical guarantees for solving linear systems in-context using a linear transformer architecture. For in-domain generalization, we provide neural scaling laws that bound the generalization error in terms of the number of tasks and sizes of samples used in training and inference. For out-of-domain generalization, we find that the behavior of trained transformers under task distribution shifts depends crucially on the distribution of the tasks seen during training. We introduce a novel notion of task diversity and show that it defines a necessary and sufficient condition for pre-trained transformers generalize under task distribution shifts. We also explore applications of learning linear systems in-context, such as to in-context operator learning for PDEs. Finally, we provide some numerical experiments to validate the established theory.