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

Yatin Kumar, Shivani Saini, Ramesh Kumar Vats

arXiv:2610.30179v1physics.flu-dyncs.AI

TL;DR

Time-fractional Navier–Stokes equations on complex domains present challenges from history dependence, nonlinear flow, geometry, and incompressibility. GeoDiv-KAN combines geometry-dependent gating and a streamfunction velocity representation, and reports higher accuracy than the compared baselines across benchmark domains, alongside CFD reconstructions.

  • Problem

    Solving TFNSEs on complex domains must address temporal memory, nonlinear flow, geometry, and incompressibility, while conventional long-time simulations are computationally and memory intensive.

    Source: §1 · pp. 2–3

  • Method

    GeoDiv-KAN combines Gaussian RBF edge functions and geometry-dependent feature gating with a streamfunction-derived velocity that preserves incompressibility by construction.

    Source: §1 · p. 3, §3 · p. 5

  • Results

    GeoDiv-KAN showed steady improvement over PINN, PIKAN, RBF-KAN, and DF-RBFKAN on square, circular, and star-shaped domains, and reconstructed the reported cylinder and U-bend CFD fields.

    Source: §1 · p. 4, §6 · p. 31

  • Takeaways & Limitations

    The reported tests demonstrate GeoDiv-KAN’s applicability to two-dimensional benchmark domains and data-driven CFD reconstruction cases.

    Source: §1 · p. 4, §6 · p. 31

    The study is restricted to CFD reconstruction testing and prescribed two-dimensional geometries; unobserved geometries, sparse or noisy data, and uncertainty quantification remain future work.

    Source: §6 · p. 31

Abstract

from arXiv · show

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 \href{https://doi.org/10.5281/zenodo.5039610}{Zenodo dataset} and turbulent RANS flow through a complex U bend using the \href{https://huggingface.co/datasets/JDecke/ubend-cfd}{U-bend dataset}.

1. Introduction

Time-fractional Navier–Stokes equations combine flow dynamics with history dependence, making long simulations costly and complex-domain solutions difficult to represent. GeoDiv-KAN combines geometry-dependent gating with a streamfunction formulation and is evaluated on several geometries and CFD reconstruction cases.

1. Introduction

  • TFNSEs add flow-history dependence to the nonlinear convection, viscous diffusion, pressure, and incompressibility features of classical Navier–Stokes equations.The Caputo derivative has order 0 < α ≤ 1; α = 1 recovers the classical system.Source: §1 · pp. 1–2
  • Conventional TFNSE solvers retain information from previous time levels, making long-time simulations computationally expensive and memory intensive.They approximate the Caputo derivative using methods such as the L1 scheme or convolution quadrature.Source: §1 · p. 2
  • Neural solvers represent solution fields from spatial and temporal coordinates, while PINNs incorporate governing equations and conditions into their loss functions.Prior work also includes fractional PINNs, geometry-aware PINNs, and physics-informed KANs.Source: §1 · pp. 2–3
  • A unified solver for TFNSEs on complex domains remains underexplored because geometry representation, fractional memory, nonlinear coupling, and exact incompressibility must be addressed together.Soft continuity-loss penalties do not guarantee an exactly divergence-free velocity field.Source: §1 · p. 3
  • GeoDiv-KAN uses Gaussian RBF edge functions and a learnable geometry-dependent feature gate, with a streamfunction-derived velocity that is divergence-free by construction.Pressure is predicted through a separate network output, and no additional divergence penalty is required.Source: §1 · pp. 3–4
  • The study compares GeoDiv-KAN with PINN, PIKAN, RBF-KAN, and DF-RBFKAN on square, circular, and star-shaped domains, then tests two data-driven CFD reconstructions.The CFD cases are cylinder flow at Re = 100 and turbulent RANS flow through a U-bend at Re = 40,000.Source: §1 · p. 4

2. Preliminary

This section introduces fractional-calculus and KAN preliminaries, then describes the functional KAN layer and Gaussian radial basis parameterization used by the model.

2. Preliminary

  • The preliminary definitions introduce the Riemann–Liouville fractional integral and Caputo fractional derivative, with the derivative order constrained by (k −1) ≤α < k.Source: §2 · p. 4

2.1. Kolmogorov–Arnold Representation

  • The Kolmogorov–Arnold representation theorem expresses continuous multivariate mappings through univariate functions, motivating KAN’s learnable univariate edge functions.Source: §2.1 · p. 4

2.2. Kolmogorov–Arnold Network Layer

  • A KAN layer maps its input to an output using a functional matrix whose entries are learnable univariate functions, rather than scalar weights.Source: §2.2 · pp. 4–5

2.3. Gaussian Radial Basis Edge Functions

  • The proposed model parameterizes its learnable KAN edge functions with Gaussian radial basis functions.Source: §2.3 · p. 5

3. Methodology

GeoDiv-KAN combines Gaussian RBF-based KAN layers, geometry-dependent feature gating, and a streamfunction representation for two-dimensional incompressible TFNSEs. Its training procedure imposes initial conditions, evaluates the fractional-time residuals, and optimizes a combined objective.

3. Methodology

  • GeoDiv-KAN combines Gaussian RBF-based KAN layers, geometry-dependent feature gating, and a streamfunction representation for the two-dimensional incompressible TFNSEs.Source: §3 · p. 5
  • The geometry encoder uses domain-specific representations, including a normalized star-boundary descriptor and nonsingular Cartesian geometry features.The encoded geometry features are fused with flow features through a learned gate.Source: §3 · pp. 6–7
  • The initial-condition construction multiplies the network correction by time, making it vanish at t = 0 and preserving the prescribed initial velocities.Source: §3 · p. 7
  • The streamfunction-derived velocities satisfy incompressibility exactly, eliminating the need for a continuity penalty and independent vorticity output.The method trains two momentum residuals and uses the L1 formula for the time-fractional derivative.Source: §3 · pp. 5, 8
  • Exact fields provide initial values, boundary values, and evaluation references, but are not used as interior observations.Source: §3 · p. 8
  • The training objective combines momentum-residual, boundary, pressure-anchor, and smoothness terms, and L-BFGS optimizes the trainable parameters.For the square, raw residuals are used; circle and star residual scales are fixed from the initial network.Source: §3 · p. 9

4. Numerical Experiments

The numerical experiments evaluate GeoDiv-KAN for time-fractional incompressible Navier–Stokes equations on square, circular, and star-shaped domains, examining quantitative accuracy, field behavior, and incompressibility. For the non-convex star-shaped domain, the reported results include the lowest errors across solution components and divergence below 4.44 × 10−15 at t = 1.

4. Numerical Experiments

  • The square, circular, and star-shaped cases used α = 0.8 and Re = 400, and prediction accuracy was evaluated with absolute error, RMSE, maximum absolute error, and relative L2 error.For the square case, initial conditions were embedded in the trial solution; the circle and star cases also embedded initial conditions directly or imposed them from the exact fields.Source: §4 · p. 10, §4.1 · p. 11, §4.2 · p. 17

4.1. Case 1: Square Domain

  • On the square domain, divergence magnitude was of order 10^-18, and velocity-error magnitude remained below approximately 3.6 × 10^-3, primarily near the boundary.The passage attributes machine-precision incompressibility to the streamfunction formulation.Source: §4.1 · p. 15, Figure 11 · p. 15
  • GeoDiv-KAN achieved the lowest RMSE, maximum absolute error, and relative L2 error for all three solution components on the square domain.The comparison attributes improved approximation to geometry-gated Gaussian RBF edge functions and the divergence-free streamfunction representation.Source: §4.1 · p. 15, Table 3 · p. 16

4.2. Case 2: Circular Domain

  • On the circular domain, divergence magnitude remained below 5.33 × 10^-15 and velocity error below approximately 1.3 × 10^-2.The predicted flow showed smooth counterclockwise circulation, with a central low-pressure region and higher pressure near the boundary.Source: §4.2 · pp. 17, 18, 20, Figure 16 · p. 18, Figure 21 · p. 21
  • GeoDiv-KAN achieved the lowest RMSE, maximum absolute error, and relative L2 error for all three solution components on the circular domain.Its relative L2 errors for u, v, and p were 3.4570 × 10^-3, 3.3167 × 10^-3, and 6.5927 × 10^-2, respectively.Source: §4.2 · pp. 20–21, Table 4 · p. 21

4.3. Case 3: Star-Shaped Domain

  • On the star-shaped domain, absolute errors were below 9.06 × 10^-3 for u, 6.84 × 10^-3 for v, and 1.20 × 10^-2 for p.The predicted fields reproduced the analytical patterns, symmetries, and amplitudes at t = 0.4.Source: §4.3 · p. 23
  • On the star-shaped domain, predicted fields maintained stable spatial patterns and symmetries over t ∈ [0, 1], while flow diagnostics showed counterclockwise circulation conforming to the non-convex boundary.Pressure was low near the center and higher near the outward lobes.Source: §4.3 · p. 23, Figure 26 · p. 23, Figure 27 · p. 24
  • Across time levels, predicted velocity amplitudes increase while the maintained pattern shows steady velocity–pressure coupling.The domain-wide velocity-pair distributions are colored by predicted pressure.Source: §4.3 · p. 25, Figure 30 · p. 25
  • At t = 1, divergence remains below 4.44 × 10−15 and localized velocity error below 2.6 × 10−2 on the non-convex star-shaped domain.The flow diagnostics are described as smooth and physically consistent, with incompressibility maintained to machine precision.Source: §4.3 · p. 25, Figure 31 · p. 26
  • GeoDiv-KAN achieves the lowest errors for all solution components among the compared models on the star-shaped domain.The comparison is at t = 1.0.Source: Table 5 · p. 26, §4.3 · p. 26

5. Application

GeoDiv-KAN is tested on unsteady cylinder flow and steady turbulent flow through a U-bend, with reconstructions closely matching reference CFD fields. Errors are concentrated mainly in high-gradient regions, separated shear layers, and wake structures.

5. Application

  • The applications assess reconstruction on periodic vortex shedding behind a fixed cylinder at Re = 100 and steady turbulent flow through a 180° U-bend at Re = 40,000.The cylinder reference data come from Boudina’s public dataset; the U-bend reference is the public Decke et al. dataset, sample_100, on a 240×60 fluid grid.Source: §5.1 · p. 27, §5 · p. 27, §5.2 · p. 30

5.1. Unsteady Flow Past a Circular Cylinder at Re = 100

  • For the cylinder case, global RMSEs were 3.83 × 10−3 for u, 2.65 × 10−3 for v, and 1.91 × 10−3 for p across the full time interval.At the unseen test time t = 419.9, errors were 6.49 × 10−3 for u, 1.60 × 10−2 for v, and 9.17 × 10−3 for gauge-aligned pressure.Source: Figure 34 · p. 28, §5.1 · p. 28
  • Cylinder time-averaged streamwise-velocity profiles at x/D = 1, 2, and 4 closely follow reference curves, including near-wake velocity deficit and downstream recovery.The selected cylinder region retains 40,036 of 58,205 spatial nodes, covering separated shear layers and the vortex-shedding wake.Source: Figure 35 · p. 28, §5.1 · pp. 28–29

5.2. Turbulent RANS Flow Reconstruction in a U-Bend

  • For the U-bend, reconstructed fields capture pressure distribution and velocity variation through the bend, with errors concentrated near high-gradient areas and the sharply curved inner wall.Velocity-vector fields capture flow direction and acceleration; signed errors are concentrated near zero, with no consistent over- or underprediction.Source: Figure 37 · p. 29, §5.2 · pp. 30–31, Figure 38 · p. 30

6. Conclusion

The conclusion presents GeoDiv-KAN as a geometry-gated, divergence-free Gaussian RBF-based KAN for TFNSEs, evaluated on benchmark domains and data-driven CFD cases.

6. Conclusion

  • GeoDiv-KAN combines Gaussian RBF edge functions, geometry-dependent feature gating, and a streamfunction velocity representation for incompressible TFNSE solutions on complex domains.Numerical tests on square, circular, and star-shaped domains report steady improvement over PINN, PIKAN, RBF-KAN, and DF-RBFKAN.Source: §6 · p. 31
  • The data-driven cases reconstruct the evolving cylinder wake and downstream mean velocity profiles, and the U-bend mean velocity and gauge-corrected pressure fields.The U-bend reconstruction includes curvature-induced flow variation through the bend.Source: §6 · p. 31
  • The study is restricted to CFD reconstruction testing and prescribed two-dimensional geometries; three-dimensional issues, unobserved geometries and circumstances, sparse or noisy data, and uncertainty quantification are deferred to future work.Source: §6 · p. 31

Funding Declaration

The authors report that they received no funding for the investigation.

Funding Declaration

Authors’ contributions

The contribution statement assigns investigation and visualization roles to Yatin Kumar and Shivani Saini, with additional methodology, analysis, validation, and supervision roles listed for the other authors.

Authors’ contributions

  • Yatin Kumar contributed investigation, methodology, conceptualization, visualization, software, and writing; Shivani Saini contributed investigation, visualization, and writing.Source: Authors’ contributions · p. 31
  • Ramesh Kumar Vats contributed supervision, formal analysis, investigation, validation, and writing; Takaharu Yaguchi contributed methodology, formal analysis, validation, and writing.Source: Authors’ contributions · p. 31
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