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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.
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]
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Database: Engineering Source
Description
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]
ISSN:00295981
DOI:10.1002/nme.70317