One-shot learning for solution operators of partial differential equations

Anran Jiao, Haiyang He, Rishikesh Ranade, Jay Pathak and Lu Lu ()
Additional contact information
Anran Jiao: Yale University
Haiyang He: Ansys Inc.
Rishikesh Ranade: NVIDIA
Jay Pathak: Ansys Inc.
Lu Lu: Yale University

Nature Communications, 2025, vol. 16, issue 1, 1-18

Abstract: Abstract Learning and solving governing equations of a physical system, represented by partial differential equations (PDEs), from data is a central challenge in many areas of science and engineering. Traditional numerical methods can be computationally expensive for complex systems and require complete governing equations. Existing data-driven machine learning methods require large datasets to learn a surrogate solution operator, which could be impractical. Here, we propose a solution operator learning method that requires only one PDE solution, i.e., one-shot learning, along with suitable initial and boundary conditions. Leveraging the locality of derivatives, we define a local solution operator in small local domains, train it using a neural network, and use it to predict solutions of new input functions via mesh-based fixed-point iteration or meshfree neural-network based approaches. We test our method on various PDEs, complex geometries, and a practical spatial infection spread application, demonstrating its effectiveness and generalization capabilities.

Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.nature.com/articles/s41467-025-63076-z Abstract (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:nat:natcom:v:16:y:2025:i:1:d:10.1038_s41467-025-63076-z

Ordering information: This journal article can be ordered from
https://www.nature.com/ncomms/

DOI: 10.1038/s41467-025-63076-z

Access Statistics for this article

Nature Communications is currently edited by Nathalie Le Bot, Enda Bergin and Fiona Gillespie

More articles in Nature Communications from Nature
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().