One-shot learning for solution operators of partial differential equations.

Area of Science:

• Computational Science and Engineering
• Data-driven Scientific Discovery
• Applied Mathematics

Background:

• Solving partial differential equations (PDEs) is crucial for modeling physical systems.
• Traditional numerical methods are computationally intensive and require full equations.
• Existing machine learning methods need large datasets for surrogate modeling.

Purpose of the Study:

• To develop a data-driven method for learning and solving PDEs efficiently.
• To enable one-shot learning of PDE solution operators from minimal data.
• To address the computational cost and data requirements of existing approaches.

Main Methods:

• Proposed a local solution operator learning method using neural networks.
• Leveraged the locality of derivatives for localized operator definition.
• Employed mesh-based fixed-point iteration and meshfree neural network approaches for prediction.

Main Results:

• Demonstrated effective learning and solving of various PDEs.
• Validated the method on complex geometries and a real-world infection spread model.
• Showcased strong generalization capabilities of the proposed approach.

Conclusions:

• The one-shot learning method offers an efficient alternative for solving PDEs.
• The approach significantly reduces data requirements compared to existing methods.
• This technique holds promise for accelerating scientific discovery and engineering applications.