Deep Collocation Method: A Numerical Algorithm Combining Reversible Neural Network and Spectral Differentiation for Solving the Incompressible Navier–Stokes Equations.
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| Title: | Deep Collocation Method: A Numerical Algorithm Combining Reversible Neural Network and Spectral Differentiation for Solving the Incompressible Navier–Stokes Equations. |
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| Authors: | Chen, Ruilin1,2,3,4 (AUTHOR) ruilinchen@hrbeu.edu.cn, Ma, Guihui1,2,3,4 (AUTHOR), Fang, Ming1,2,3,4 (AUTHOR) |
| Source: | International Journal for Numerical Methods in Engineering. 4/15/2026, Vol. 127 Issue 7, p1-24. 24p. |
| Subjects: | Reversible computing, Numerical differentiation, Discretization methods, Computational fluid dynamics, Artificial neural networks, Iterative methods (Mathematics), Incompressible flow |
| Abstract: | Neural networks have the potential in approximating the solutions of the Navier–Stokes equations. To address the high computational complexity of high‐order derivatives based on automatic differentiation and the high memory requirements of deep neural network training, a novel Deep Collocation Method (DCM) that combines reversible neural network (RevNet) and spectral differentiation is proposed. The memory‐efficient RevNet is employed to approximate the solutions, while Chebyshev differentiation matrices are derived to rapidly and accurately compute the spatiotemporal derivatives of the approximate solutions, making the computational complexity of high‐order derivatives comparable to that of the forward pass, and significantly reducing memory requirements. A unified spectral discretization scheme is applied in both temporal and spatial dimensions, enabling seamless integration of the neural network and the spatiotemporal Chebyshev differentiation matrices. It effectively incorporates global spatiotemporal information into the residual computation of the governing equations at each collocation point, enabling the neural network to actively account for nonlocal effects of the flow field. Validation results demonstrate that DCM performs well various benchmark flow problems and enables efficient training of deep models even on memory‐constrained devices. [ABSTRACT FROM AUTHOR] |
| Copyright of International Journal for Numerical Methods in Engineering is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 192938203 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Deep Collocation Method: A Numerical Algorithm Combining Reversible Neural Network and Spectral Differentiation for Solving the Incompressible Navier–Stokes Equations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Chen%2C+Ruilin%22">Chen, Ruilin</searchLink><relatesTo>1,2,3,4</relatesTo> (AUTHOR)<i> ruilinchen@hrbeu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Ma%2C+Guihui%22">Ma, Guihui</searchLink><relatesTo>1,2,3,4</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Fang%2C+Ming%22">Fang, Ming</searchLink><relatesTo>1,2,3,4</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22International+Journal+for+Numerical+Methods+in+Engineering%22">International Journal for Numerical Methods in Engineering</searchLink>. 4/15/2026, Vol. 127 Issue 7, p1-24. 24p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Reversible+computing%22">Reversible computing</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+differentiation%22">Numerical differentiation</searchLink><br /><searchLink fieldCode="DE" term="%22Discretization+methods%22">Discretization methods</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+fluid+dynamics%22">Computational fluid dynamics</searchLink><br /><searchLink fieldCode="DE" term="%22Artificial+neural+networks%22">Artificial neural networks</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Incompressible+flow%22">Incompressible flow</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Neural networks have the potential in approximating the solutions of the Navier–Stokes equations. To address the high computational complexity of high‐order derivatives based on automatic differentiation and the high memory requirements of deep neural network training, a novel Deep Collocation Method (DCM) that combines reversible neural network (RevNet) and spectral differentiation is proposed. The memory‐efficient RevNet is employed to approximate the solutions, while Chebyshev differentiation matrices are derived to rapidly and accurately compute the spatiotemporal derivatives of the approximate solutions, making the computational complexity of high‐order derivatives comparable to that of the forward pass, and significantly reducing memory requirements. A unified spectral discretization scheme is applied in both temporal and spatial dimensions, enabling seamless integration of the neural network and the spatiotemporal Chebyshev differentiation matrices. It effectively incorporates global spatiotemporal information into the residual computation of the governing equations at each collocation point, enabling the neural network to actively account for nonlocal effects of the flow field. Validation results demonstrate that DCM performs well various benchmark flow problems and enables efficient training of deep models even on memory‐constrained devices. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of International Journal for Numerical Methods in Engineering is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1002/nme.70317 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 24 StartPage: 1 Subjects: – SubjectFull: Reversible computing Type: general – SubjectFull: Numerical differentiation Type: general – SubjectFull: Discretization methods Type: general – SubjectFull: Computational fluid dynamics Type: general – SubjectFull: Artificial neural networks Type: general – SubjectFull: Iterative methods (Mathematics) Type: general – SubjectFull: Incompressible flow Type: general Titles: – TitleFull: Deep Collocation Method: A Numerical Algorithm Combining Reversible Neural Network and Spectral Differentiation for Solving the Incompressible Navier–Stokes Equations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Chen, Ruilin – PersonEntity: Name: NameFull: Ma, Guihui – PersonEntity: Name: NameFull: Fang, Ming IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 04 Text: 4/15/2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00295981 Numbering: – Type: volume Value: 127 – Type: issue Value: 7 Titles: – TitleFull: International Journal for Numerical Methods in Engineering Type: main |
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