Provable In-Context Learning of Linear Systems and Linear Elliptic PDEs with Transformers (2409.12293v2)

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.

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

The research presented in this paper develops a rigorous error analysis of in-context learning (ICL) using transformers, specifically targeting linear systems and linear elliptic partial differential equations (PDEs). This paper addresses the gap in understanding the theoretical foundations of ICL capabilities in scientific modeling, leveraging transformers for PDE solutions.

Problem Context and Objectives

In recent advancements, foundation models (FMs) based on transformer architectures have demonstrated impressive adaptability in NLP through few-shot prompts without updating model weights. This adaptability is attributed to their in-context learning abilities. In the scientific domain, transformers have shown potential in solving complex PDEs, but theoretical underpinnings of such capabilities remain underexplored. This paper focuses on establishing the theoretical error bounds for transformers applied to a class of linear elliptic PDEs through ICL frameworks.

Core Contributions

1. In-context Learning of Linear Systems:
   • The authors reduce the infinite-dimensional PDE problem to a finite-dimensional linear system problem. They establish that transformers with linear self-attention layers can invert these linear systems in context.
   • They derive error bounds for the prediction risk of generalization from ICL, considering discretization size, number of training tasks, and prompt lengths.
2. Task Distribution Shifts:
   • The paper introduces the concept of task diversity to analyze shifts in task distribution. The authors demonstrate conditions ensuring sufficient task diversity, which allows for robust generalization even under significant distribution shifts.
   • They establish bounds quantifying the prediction error under distribution shifts, showcasing the robustness of transformers owing to their adaptability to task diversity.
3. Numerical Validation:
   • Extensive numerical experiments validate the theoretical results. These experiments highlight that the prediction risk decays predictably with increased sample sizes and prompt lengths during both training and inference phases.

Theoretical Results and Implications

In-Domain Generalization

The paper presents rigorous bounds for in-domain generalization. For transformers trained to learn linear systems from spatial discretization of PDEs, the generalization error scales as $O(1/m + 1/n^2 + 1/\sqrt{N})$, where $m$ is the prompt length during testing, $n$ the prompt length during training, and $N$ the number of pre-training tasks. This scaling indicates that longer prompts and more training tasks significantly reduce generalization error.

PDE Solution Error Estimation

Theoretical results extend to learning elliptic PDE solutions. The error bound on PDE solutions is expressed in terms of the discretization error and statistical error from learning linear systems. This yields an explicit error structure encompassing the trade-off between increasing discretization (basis functions) and the amount of training data, crucial for practical applications. An illustrative example using FEM discretization in one dimension exemplifies these bounds.

Out-of-Domain Generalization

Out-of-domain generalization is analyzed under two scenarios:

1. Task Distribution Shifts:
   • The paper shows that task diversity ensures OOD generalization by quantifying the prediction error again as $O(1/m)$.
   • Task distributions with centralizer conditions or simultaneously diagonalizable matrices are explored to ensure task diversity.
2. Covariate Distribution Shifts:
   • The error bounds under covariate shifts reveal that transformers are less robust when the data distribution shifts beyond the pre-trained distribution scope. The mismatch error scales with the operator norm difference of covariance matrices.

Future Directions

This paper sets a foundation for theoretically grounded ICL in scientific problems using transformers. Future research calls for exploring more complex settings, including nonlinear and time-dependent PDEs, where depth and nonlinearity play critical roles in approximating solutions. Moreover, a deeper understanding of task diversity in broader contexts beyond linear systems is essential to harness the full potential of transformers in diverse scientific applications.

This research paves the way for leveraging sophisticated in-context learning models in scientific computing, offering pathways to efficient problem-solving without extensive retraining. The blend of theoretical rigor and practical validation in this work underscores the transformative potential of transformer-based methods in scientific modeling and computation.