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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

Scientific Reports, Published online: 19 September 2026; doi:10.1038/s41598-026-72553-4 Extended physics-informed neural networks (Ex-PINNs): a machine-learning method for solving partial differential equations in domain with spatially heterogeneous parameters

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