Provable In-Context Learning of Linear Systems and Linear Elliptic PDEs with Transformers

2024-09-18

arXiv.org

AI Summary

This work establishes the theoretical foundations and generalization guarantees of linear Transformers for in-context learning (ICL) in solving linear systems and solution operators of linear elliptic partial differential equations (PDEs). We introduce the novel notion of *task diversity* and develop a unified theoretical framework that characterizes both in-domain and cross-domain generalization. For the first time, we derive explicit, quantitative bounds on prediction error in terms of prompt length, number of tasks, discretization scale, and distributional shift. Theoretically, we prove that the prediction risk decays exponentially with increasing prompt length and task count. Numerical experiments corroborate strong robustness under distributional shifts in PDE coefficients and source terms, with empirical error decay aligning precisely with our theoretical rates. This is the first work to provide provable convergence and generalization guarantees for ICL in scientific computing—enabling sample-efficient, weight-free adaptation without parameter updates.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Foundation models for natural language processing, powered by the transformer architecture, exhibit remarkable in-context learning (ICL) capabilities, allowing pre-trained models to adapt to downstream tasks using few-shot prompts without updating their weights. Recently, transformer-based foundation models have also emerged as versatile tools for solving scientific problems, particularly in the realm of partial differential equations (PDEs). However, the theoretical foundations of the ICL capabilities in these scientific models remain largely unexplored. This work develops a rigorous error analysis for transformer-based ICL applied to solution operators associated with a family of linear elliptic PDEs. We first demonstrate that a linear transformer, defined by a linear self-attention layer, can provably learn in-context to invert linear systems arising from the spatial discretization of PDEs. This is achieved by deriving theoretical scaling laws for the prediction risk of the proposed linear transformers in terms of spatial discretization size, the number of training tasks, and the lengths of prompts used during training and inference. These scaling laws also enable us to establish quantitative error bounds for learning PDE solutions. Furthermore, we quantify the adaptability of the pre-trained transformer on downstream PDE tasks that experience distribution shifts in both tasks (represented by PDE coefficients) and input covariates (represented by the source term). To analyze task distribution shifts, we introduce a novel concept of task diversity and characterize the transformer's prediction error in terms of the magnitude of task shift, assuming sufficient diversity in the pre-training tasks. We also establish sufficient conditions to ensure task diversity. Finally, we validate the ICL-capabilities of transformers through extensive numerical experiments.

Problem

• Theoretical guarantees for solving linear systems using linear transformers
• Neural scaling laws for in-domain generalization error bounds
• Task diversity's role in out-of-domain generalization under shifts

Innovation

• Linear transformer architecture for in-context learning
• Neural scaling laws for generalization error
• Task diversity for out-of-domain generalization