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Extended physics-informed neural networks (Ex-PINNs): a machine-learning method for solving partial differential equations in domain with spatially heterogeneous parameters

Crossref DOI link: https://doi.org/10.1038/s41598-026-72553-4

Published Online: 2026-09-19

Update policy: https://doi.org/10.1007/springer_crossmark_policy

Authors

Yan, Wenda

Wang, Hongzhi

Zhang, Shuisheng

Zhang, Jian
Funding

Funding for this research was provided by:

The work was financially supported by the Heilongjiang Provincial Colleges and Universities Basic Scientific Research Business Funds Scientific Research Project (145309111)
License Information

Text and Data Mining valid from 2026-09-19

Accepted Manuscript valid from 2026-09-19
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Article History

Received: 17 September 2025

Accepted: 16 September 2026

First Online: 19 September 2026

Declarations

Competing interests: The authors declare no competing interests.

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