Harness In-Context Operator Learning with Chain of Operators

Problem
The paper addresses the limitations of existing neural operators, particularly In-Context Operator Networks (ICON), which struggle with generalization to out-of-distribution (OOD) operator tasks and often require fine-tuning or retraining. This work is a preprint and has not undergone peer review, highlighting the need for a more robust framework that can leverage ICON’s capabilities without necessitating parameter updates.

Method
The authors propose the Chain of Operators (CHOP) framework, which utilizes a frozen ICON model to tackle OOD operator tasks. CHOP constructs a sequence of operators that includes explicit elementary transformations alongside the frozen ICON. This architecture allows the model to adapt to various operator tasks by leveraging the learned representations from ICON while maintaining interpretability and closed-form expressions for each operator in the chain. The training process involves using numerical context to prompt the model, enabling it to learn specific operators dynamically.

Results
Experiments conducted on a scalar conservation law and a mean-field control problem demonstrate that CHOP significantly reduces relative inference error compared to direct evaluations of ICON. The results indicate that CHOP achieves a reduction in error rates, although specific numerical values are not disclosed in the summary. Notably, the framework shows that a chain constructed for one family of partial differential equations (PDEs) can generalize effectively to a different family, suggesting the presence of shared mechanisms across different operator tasks.

Limitations
The authors acknowledge that while CHOP improves generalization capabilities, it may still be limited by the inherent constraints of the frozen ICON model. Additionally, the performance of CHOP on highly diverse or complex OOD tasks remains to be fully explored. The paper does not address potential computational overhead introduced by the chaining process or the scalability of the framework to larger operator families.

Why it matters
The introduction of CHOP has significant implications for the field of operator learning, particularly in scenarios where fine-tuning is impractical or infeasible. By enabling a frozen model to adapt to new tasks through a structured chain of operators, this work paves the way for more efficient and interpretable operator learning systems. The findings contribute to the ongoing discourse on enhancing the robustness of neural operators, as discussed in related literature on operator learning and generalization strategies, and are available on arXiv.