physics.flu-dynOct 19, 2026

GeoDiv KAN Solver for Time Fractional Navier Stokes Equations on Complex Domains to Realistic CFD Reconstruction

Authors: Yatin Kumar, Shivani Saini, Ramesh Kumar Vats

Organizations: Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, Himachal Pradesh, 177005, India

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Abstract

Scientific machine learning provides a promising framework for solving complex problems in science and engineering. Notably, physics informed neural networks (PINNs) have shown considerable potential for solving differential equations and modelling engineering systems. Solving time fractional Navier Stokes equations (TFNSEs) on complex domains is challenging due to temporal nonlocality, nonlinear flow dynamics, geometric complexity, and incompressibility constraints. To address these challenges, we propose a geometry gated, divergence-free Gaussian radial basis function Kolmogorov Arnold network (GeoDiv KAN). The proposed architecture uses geometry dependent feature gating and a streamfunction formulation to preserve incompressibility by construction. GeoDiv KAN is evaluated for the time fractional incompressible Navier-Stokes equations on square, circular, and star-shaped domains. Numerical experiments show that GeoDiv KAN achieves higher predictive accuracy than PINN, PIKAN, RBF KAN, and DF RBFKAN across the considered benchmark problems. To establish practical relevance, GeoDiv KAN is applied to two data driven CFD cases: unsteady flow past a circular cylinder using the Zenodo dataset and turbulent RANS flow through a complex U bend using the U-bend dataset.

Figures & tables

Figure 1: Architecture and training workflow of the proposed GeoDiv-KAN solver.
ModelKAN edgesGaussian RBFsDivergence-freeGeometry gate
PINNNoNoNoNo
Base PIKANYesNoNoNo
RBF-KANYesYesNoNo
DF-RBFKANYesYesYesNo
GeoDiv-KANYesYesYesYes
Table 1: Architectural features of the models considered in this study.
ModelLayers / WidthBasis / ActivationParametersOutputs
PINN5×355\times 35Tanh5,288(u,v,p)(u,v,p)
Base PIKAN4×104\times 10Spline grid = 105,400(u,v,p)(u,v,p)
RBF-KAN3×153\times 158 RBF bases5,400(u,v,p)(u,v,p)
DF-RBFKAN3×153\times 158 RBF bases5,250(ψ,p)(\psi,p)
GeoDiv-KANBranched width = 128 RBF bases5,400(ψ,p)(\psi,p)
Table 2: Final architectures and parameter counts of the comparison models for Square Case 1.
Figure 2: Distribution of interior and boundary collocation points on the square domain.
Figure 3: Comparison of the exact and GeoDiv-KAN predicted fields uu , together with absolute errors, for square domain at t=0.4t=0.4 .
Figure 4: Comparison of the exact and GeoDiv-KAN predicted fields vv , together with absolute errors, for square domain at t=0.4t=0.4 .
Figure 5: Comparison of the exact and GeoDiv-KAN predicted fields pp , together with absolute errors, for square domain at t=0.4t=0.4 .
Figure 6: GeoDiv-KAN predicted flow field diagnostics for square domain at t=1t=1 , first two panels show uu and vv with the velocity vectors, and third presents pp with the pressure-force vectors.
Figure 7: GeoDiv-KAN predicted temporal evolution of the solution fields uu on the square domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 8: GeoDiv-KAN predicted temporal evolution of the solution fields vv on the square domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 9: GeoDiv-KAN predicted temporal evolution of the solution fields pp on the square domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 10: Domain-wide distributions of the predicted velocity pairs (u,v)(u,v) for square case at different t, coloured by the corresponding predicted pressure.
Figure 11: Physical diagnostics of the GeoDiv-KAN solution for square domain at t=1t=1 : velocity magnitude, kinetic-energy density, vorticity, pressure-gradient magnitude, divergence magnitude, and velocity-error magnitude.
ModelComponentRMSEMax. Absolute ErrorRelative L2L_{2} Error
GeoDiv-KANuu8.9488×10−4\mathbf{8.9488\times 10^{-4}}3.4069×10−3\mathbf{3.4069\times 10^{-3}}4.1196×10−2\mathbf{4.1196\times 10^{-2}}
vv9.8067×10−4\mathbf{9.8067\times 10^{-4}}3.2151×10−3\mathbf{3.2151\times 10^{-3}}4.5145×10−2\mathbf{4.5145\times 10^{-2}}
pp2.9098×10−3\mathbf{2.9098\times 10^{-3}}2.6783×10−2\mathbf{2.6783\times 10^{-2}}2.3385×10−2\mathbf{2.3385\times 10^{-2}}
DF-RBFKANuu1.4090×10−11.4090\times 10^{-1}3.7635×10−13.7635\times 10^{-1}6.4863×1006.4863\times 10^{0}
vv1.3969×10−11.3969\times 10^{-1}6.1511×10−16.1511\times 10^{-1}6.4304×1006.4304\times 10^{0}
pp1.6893×10−11.6893\times 10^{-1}7.6173×10−17.6173\times 10^{-1}1.3576×1001.3576\times 10^{0}
Table 3: Comparison of different models for the square case at t=1.0t=1.0 .
Figure 12: Distribution of interior and boundary collocation points on the circular domain.
Figure 13: Comparison of the exact and GeoDiv-KAN predicted fields uu , together with absolute errors, for circle domain at t=0.4t=0.4 .
Figure 14: Comparison of the exact and GeoDiv-KAN predicted fields vv , together with absolute errors, for circle domain at t=0.4t=0.4 .
Figure 15: Comparison of the exact and GeoDiv-KAN predicted fields pp , together with absolute errors, for circle domain at t=0.4t=0.4 .
Figure 16: GeoDiv-KAN predicted flow field diagnostics for circle domain at t=1t=1 , first two panels show uu and vv with the velocity vectors, and third presents pp with the pressure-force vectors.
Figure 17: GeoDiv-KAN predicted temporal evolution of the solution fields uu on circle domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 18: GeoDiv-KAN predicted temporal evolution of the solution fields vv on circle domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 19: GeoDiv-KAN predicted temporal evolution of the solution fields pp on the circle domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 20: Domain-wide distributions of the predicted velocity pairs (u,v)(u,v) for circular case at different t, coloured by the corresponding predicted pressure.
Figure 21: Physical diagnostics of the GeoDiv-KAN solution for circular domain at t=1t=1 : velocity magnitude, kinetic-energy density, vorticity, pressure-gradient magnitude, divergence magnitude, and velocity-error magnitude.
ModelComponentRMSEMax. Absolute ErrorRelative L2L_{2} Error
GeoDiv-KANuu3.3680×10−3\mathbf{3.3680\times 10^{-3}}1.3169×10−2\mathbf{1.3169\times 10^{-2}}3.4570×10−3\mathbf{3.4570\times 10^{-3}}
vv3.2314×10−3\mathbf{3.2314\times 10^{-3}}1.1034×10−2\mathbf{1.1034\times 10^{-2}}3.3167×10−3\mathbf{3.3167\times 10^{-3}}
pp7.7882×10−3\mathbf{7.7882\times 10^{-3}}2.2770×10−2\mathbf{2.2770\times 10^{-2}}6.5927×10−2\mathbf{6.5927\times 10^{-2}}
DF-RBFKANuu3.4127×10−13.4127\times 10^{-1}8.0141×10−18.0141\times 10^{-1}3.4997×10−13.4997\times 10^{-1}
vv3.5364×10−13.5364\times 10^{-1}8.8597×10−18.8597\times 10^{-1}3.6265×10−13.6265\times 10^{-1}
pp4.5295×10−14.5295\times 10^{-1}7.7488×10−17.7488\times 10^{-1}3.7897×1003.7897\times 10^{0}
Table 4: Comparison of different models for the circular domain at t=1.0t=1.0 .
Figure 22: Distribution of interior and boundary collocation points on the star-shaped domain.
Figure 23: Comparison of the exact and GeoDiv-KAN predicted fields uu , together with absolute errors, for star shape at t=0.4t=0.4 .
Figure 24: Comparison of the exact and GeoDiv-KAN predicted fields vv , together with absolute errors, for star shape at t=0.4t=0.4 .
Figure 25: Comparison of the exact and GeoDiv-KAN predicted fields pp , together with absolute errors, for star shape at t=0.4t=0.4 .
Figure 26: GeoDiv-KAN predicted flow field diagnostics for star-shaped domain at t=1t=1 , first two panels show uu and vv with the velocity vectors, and third presents pp with the pressure-force vectors.
Figure 27: GeoDiv-KAN predicted temporal evolution of the solution fields pp on the star-shaped domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 28: GeoDiv-KAN predicted temporal evolution of the solution fields uu on the star-shaped domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 29: GeoDiv-KAN predicted temporal evolution of the solution fields vv on the star-shaped domain at t=0, 0.2, 0.4, 0.6, 0.8,t=0,\,0.2,\,0.4,\,0.6,\,0.8, and 1.01.0 .
Figure 30: Domain-wide distributions of the predicted velocity pairs (u,v)(u,v) for star shape case at different t, coloured by the corresponding predicted pressure.
Figure 31: Physical diagnostics of the GeoDiv-KAN solution for star-shaped domain at t=1t=1 : velocity magnitude, kinetic-energy density, vorticity, pressure-gradient magnitude, divergence magnitude, and velocity-error magnitude.
ModelComponentRMSEMax. Absolute ErrorRelative L2L_{2} Error
GeoDiv-KANuu6.8132×10−3\mathbf{6.8132\times 10^{-3}}2.6302×10−2\mathbf{2.6302\times 10^{-2}}6.3748×10−3\mathbf{6.3748\times 10^{-3}}
vv6.0799×10−3\mathbf{6.0799\times 10^{-3}}2.0223×10−2\mathbf{2.0223\times 10^{-2}}5.6894×10−3\mathbf{5.6894\times 10^{-3}}
pp1.2668×10−2\mathbf{1.2668\times 10^{-2}}3.0300×10−2\mathbf{3.0300\times 10^{-2}}1.1350×10−1\mathbf{1.1350\times 10^{-1}}
DF-RBFKANuu3.7335×10−23.7335\times 10^{-2}9.0434×10−29.0434\times 10^{-2}3.5317×10−23.5317\times 10^{-2}
vv4.4447×10−24.4447\times 10^{-2}1.0828×10−11.0828\times 10^{-1}4.2060×10−24.2060\times 10^{-2}
pp1.7635×10−11.7635\times 10^{-1}2.9138×10−12.9138\times 10^{-1}1.5391×1001.5391\times 10^{0}
Table 5: Comparison of different models for the star-shaped domain at t=1.0t=1.0 .
Figure 32: Streamline topology of flow past a fixed circular cylinder at Re=100Re=100 , showing the inflow, cylinder, downstream vortex-shedding wake, and outflow.
Figure 33: Comparison of the reference CFD and GeoDiv-KAN reconstructed fields at the unseen test time t=419.9t=419.9 for uu , vv , pp , and pointwise absolute error.
Figure 34: Time-averaged spatial RMSE distributions of uu , vv , and gauge-aligned pp
Figure 35: Comparison of the time-averaged CFD and GeoDiv-KAN streamwise velocity profiles at x/D=1,2,x/D=1,2, and 44 .
Figure 36: Physical configuration of the selected U-bend flow problem, showing the inlet, outlet, inner and outer walls.
Figure 37: Reference RANS fields, GeoDiv-KAN reconstructions, and absolute errors for uu , vv , and gauge-corrected pressure pp .
Figure 38: Reference RANS and GeoDiv-KAN reconstructed velocity-vector fields, together with the vector-error magnitude.
Figure 39: Pointwise parity and signed-error distributions for reconstruction of uu , vv , and gauge corrected pressure pp .
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