NeuralFVM: Neural-physics-based Finite Volume Method for Turbulent Flows Using the $k$-$ω$ Model

Abstract

In this work, we develop a neural-physics solver based on finite volume method (FVM), namely NeuralFVM, for turbulent flows by implementing the standard $k$-$ω$ model designed for efficient Graphics Processing Unit (GPU) execution. The governing equations for fluid flow and heat transfer are reformulated as local tensor operations using convolution-based stencil operators, which enables compatibility with deep learning libraries while preserving the conservative properties of the FVM. A key challenge in implementing the turbulent model within such a framework is the treatment of the stiff destruction terms in the $k$ and $ω$ transport equations. To address this issue, an operator-splitting strategy is introduced in which the stiff destruction terms are handled semi-implicitly while the remaining terms are advanced explicitly. This formulation avoids global matrix assembly and allows the entire solver to be implemented using local tensor operations. In addition, the pressure-velocity coupling is solved using a convolution-based geometric multigrid algorithm embedded within a neural network architecture. The resulting NeuralFVM solver is validated through comparison with simulations conducted using the commercial CFD software ANSYS Fluent for several channel-flow configurations and an indoor airflow scenario. The results demonstrate close agreement in velocity, temperature, and turbulence quantities, confirming the accuracy of the proposed approach. The developed GPU framework achieves a speedup of around 19-46 times compared with its Central Processing Unit (CPU) counterpart under different meshes. Moreover, the proposed solver naturally integrates with machine learning workflows, providing a promising foundation for future data-driven turbulence modeling and optimization.

Figures (16)

• Figure 1: Illustrations of the neural-physics to be applied in CFD (a) convolutional kernels used for simulation of face-centered values from cell-centered values, (b) linear interploation and spatial derivative kernels.
• Figure 2: Illustration of convolutional operations implemented through simple addition or subtraction of shifted tensors.
• Figure 3: Pressure correction using the Jacobi method coupled with a multigrid approach. Red and blue arrows indicate restriction and prolongation respectively. $\otimes$ indicate the left multiplication of a matrix, which can be implemented using convolutional layer.
• Figure 4: Flow chart showing the sequence of computation in one time advancement step
• Figure 5: External BCs are implemented by padding the tensor with values that enforce the specified boundary conditions.