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arXiv AI

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

The paper introduces GeoDiv KAN, a geometry‑gated, divergence‑free Gaussian radial basis function Kolmogorov Arnold network designed to solve time fractional Navier–Stokes equations on complex domains. By incorporating geometry‑dependent feature gating and a streamfunction formulation, the architecture preserves incompressibility and achieves higher predictive accuracy than existing PINN, PIKAN, RBF KAN, and DF RBFKAN models on square, circular, and star‑shaped benchmarks. GeoDiv KAN is further validated on two data‑driven CFD cases: unsteady flow past a circular cylinder and turbulent RANS flow through a complex U‑bend.

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arXiv Machine Learning
Oct 8

Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows

The paper introduces a kernel‑based operator learning method that preserves key physical properties—such as incompressibility, periodicity, and turbulence—of the incompressible Navier–Stokes equations. By mapping input functions to expansion coefficients in a property‑preserving kernel basis, the method guarantees that predicted velocity fields analytically maintain these properties. The authors provide theoretical convergence guarantees, develop efficient computational techniques for large‑scale training, and demonstrate significant accuracy and speed improvements over neural operators on 2D and 3D flow benchmarks.

By Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar
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Hugging Face Trending Papers
Jun 17

Advances in Scientific Machine Learning for Coupled Fluid Flow and Transport

This chapter reviews recent advances in Scientific Machine Learning (SciML) for modeling coupled fluid flow and transport phenomena governed by the incompressible Navier-Stokes and scalar transport equations. Such systems, found in applications like turbidity currents and thermal convection, feature strong nonlinear coupling and multiscale behavior that make high-fidelity simulations computationally expensive.

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arXiv Machine Learning
Oct 22

Local and Sparse Kernel Operator Learning for Incompressible Flows: An Operator Lagrange Formulation

The paper introduces a local and sparse kernel method for learning solution operators of incompressible flows, replacing global output reconstruction with local divergence‑free interpolants and a multiscale kernel that approaches Kolmogorov statistics under refinement. A compactly supported, positive‑semidefinite kernel yields sparse operator regression, with support radius fitted to data and optional affine polynomials capturing global trends. The method employs Operator Lagrange weights to avoid forming local coefficients, achieving accuracy comparable to or better than previous dense methods and neural operators, while offering significant GPU speed‑ups and reduced storage.

By Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar
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arXiv Machine Learning
Oct 28

Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology

Neural Eigenfluid Bases (NEB) extend Laplacian Eigenfluids by providing a neural representation whose basis functions can be evaluated anywhere in space, enforcing incompressibility and impermeable boundaries via differential lifts and potential boundary conditions. Geometry‑local orthogonalization and Rayleigh minimization learn an ordered basis, while gauge fixing and a Dirichlet energy loss enable efficient learning of circulation fields around obstacles. With only 1024 modes, NEB achieves velocity errors below 1% in periodic vortex merger and outperforms numerical Laplacian bases on random obstacle flows, offering a thousand‑fold reduction in state size and millisecond‑scale updates through precomputed Galerkin operators.

By Baiming Zhang, Shiying Xiong, Zhecheng Wang
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