In-context operator learning with data prompts for differential equation problems.

Area of Science:

• Scientific Machine Learning
• Numerical Analysis
• Differential Equations

Background:

• Traditional neural networks require retraining for new problems, limiting their adaptability.
• Existing methods approximate specific solutions or operators, lacking generalization.
• Switching between different equations necessitates extensive model re-computation.

Purpose of the Study:

• Introduce the novel paradigm of in-context operator learning.
• Present In-Context Operator Networks (ICON) for simultaneous operator learning and application.
• Enable few-shot learning of operators without weight updates.

Main Methods:

• Training a single neural network as a general operator learner.
• Utilizing prompted data for in-context learning during inference.
• Leveraging commonalities across operators for efficient learning.

Main Results:

• Demonstrated capability of ICON for diverse differential equation problems.
• Successful application to forward and inverse problems (ODEs, PDEs, mean-field control).
• Generalization of learning to operators outside the training distribution.

Conclusions:

• In-context operator learning offers a powerful, adaptable approach to solving differential equations.
• ICON eliminates the need for retraining, significantly improving efficiency.
• The model shows strong generalization and few-shot learning capabilities.