Generalization Limits of In-Context Operator Networks for Higher-Order Partial Differential Equations

TL;DR

This paper examines the generalization capabilities of In-Context Operator Networks (ICONs) for solving higher-order partial differential equations, highlighting their strengths and limitations in extrapolating solution behaviors.

Contribution

It extends previous ICON work to higher-order PDEs, analyzing their generalization limits and demonstrating qualitative accuracy despite point-wise accuracy degradation.

Findings

ICONs retain qualitative solution dynamics for higher-order PDEs.

Point-wise accuracy decreases for complex, higher-order problems.

ICONs can extrapolate fundamental solution features beyond training data.

Abstract

We investigate the generalization capabilities of In-Context Operator Networks (ICONs), a new class of operator networks that build on the principles of in-context learning, for higher-order partial differential equations. We extend previous work by expanding the type and scope of differential equations handled by the foundation model. We demonstrate that while processing complex inputs requires some new computational methods, the underlying machine learning techniques are largely consistent with simpler cases. Our implementation shows that although point-wise accuracy degrades for higher-order problems like the heat equation, the model retains qualitative accuracy in capturing solution dynamics and overall behavior. This demonstrates the model's ability to extrapolate fundamental solution characteristics to problems outside its training regime.