Solving partial differential equations (PDEs) for various initial and boundary conditions requires significant computational resources. We propose a neural operator $ G_\theta: \mathcal{A} \to \mathcal{U} $, mapping functional spaces, which combines dynamic mode decomposition (DMD) and deep learning for efficient modeling of spatiotemporal processes. The method automatically extracts key modes and system dynamics and uses them to construct predictions, reducing computational costs compared to traditional methods (FEM, FDM, FVM). The approach is demonstrated and compared with closest methods (DeepONet, FNO) on the heat equation and Laplace equation, where high accuracy of solution recovery is achieved.
Citation: Nikita Sakovich, Dmitry Aksenov, Ekaterina Pleshakova, Sergey Gataullin. A neural operator using dynamic mode decomposition analysis to approximate partial differential equations[J]. AIMS Mathematics, 2025, 10(9): 22432-22444. doi: 10.3934/math.2025999
Solving partial differential equations (PDEs) for various initial and boundary conditions requires significant computational resources. We propose a neural operator $ G_\theta: \mathcal{A} \to \mathcal{U} $, mapping functional spaces, which combines dynamic mode decomposition (DMD) and deep learning for efficient modeling of spatiotemporal processes. The method automatically extracts key modes and system dynamics and uses them to construct predictions, reducing computational costs compared to traditional methods (FEM, FDM, FVM). The approach is demonstrated and compared with closest methods (DeepONet, FNO) on the heat equation and Laplace equation, where high accuracy of solution recovery is achieved.
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Nikita Sakovich, Dmitry Aksenov, Ekaterina Pleshakova, Sergey Gataullin. A neural operator using dynamic mode decomposition analysis to approximate partial differential equations[J]. AIMS Mathematics, 2025, 10(9): 22432-22444. doi: 10.3934/math.2025999
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