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

Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar

TL;DR

This work introduces a property-preserving kernel-based operator learning framework for incompressible flows governed by the Navier–Stokes equations. By decoupling the learning into a spatial, property-preserving kernel interpolant and an operator-valued kernel that maps inputs to expansion coefficients, the method analytically enforces incompressibility, periodicity, and turbulence power laws while achieving strong generalization. Across 2D and 3D laminar and turbulent problems, the approach yields orders-of-magnitude improvements in pointwise accuracy and significantly faster training than neural operators, with analytical guarantees on divergence and other properties. The framework also supports uncertainty quantification via operator-valued Gaussian processes and is designed to be meshfree, scalable, and interpretable, offering a practical surrogate for complex incompressible flows with potential extensions to compressible regimes and magnetohydrodynamics.

Abstract

We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier-Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields analytically and simultaneously preserve the aforementioned physical properties. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative $\ell_2$ errors upon generalization and trains up to five orders of magnitude faster compared to neural operators. Moreover, while our method enforces incompressibility analytically, neural operators exhibit very large deviations. Our results show that our method provides an accurate and efficient surrogate for incompressible flows.

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

TL;DR

This work introduces a property-preserving kernel-based operator learning framework for incompressible flows governed by the Navier–Stokes equations. By decoupling the learning into a spatial, property-preserving kernel interpolant and an operator-valued kernel that maps inputs to expansion coefficients, the method analytically enforces incompressibility, periodicity, and turbulence power laws while achieving strong generalization. Across 2D and 3D laminar and turbulent problems, the approach yields orders-of-magnitude improvements in pointwise accuracy and significantly faster training than neural operators, with analytical guarantees on divergence and other properties. The framework also supports uncertainty quantification via operator-valued Gaussian processes and is designed to be meshfree, scalable, and interpretable, offering a practical surrogate for complex incompressible flows with potential extensions to compressible regimes and magnetohydrodynamics.

Abstract

We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier-Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields analytically and simultaneously preserve the aforementioned physical properties. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative errors upon generalization and trains up to five orders of magnitude faster compared to neural operators. Moreover, while our method enforces incompressibility analytically, neural operators exhibit very large deviations. Our results show that our method provides an accurate and efficient surrogate for incompressible flows.
Paper Structure (43 sections, 8 theorems, 56 equations, 14 figures, 2 tables, 2 algorithms)

This paper contains 43 sections, 8 theorems, 56 equations, 14 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Results
  2. 2D flow past a cylinder
  3. 2D spacetime Taylor--Green vortices
  4. 3D species transport
  5. Uncertainty quantification
  6. 3D flow past an airfoil
  7. Discussion
  8. Methods
  9. Overview
  10. An overview of operator learning
  11. Our contribution
  12. The property-preserving kernel interpolant
  13. Incompressibility
  14. Periodicity
  15. Turbulence through power laws
  16. ...and 28 more sections

Key Result

Theorem 1

Let $X\subset\mathbb R^n$ be compact, fix $\nu>0$, let $k_\nu(x,x'):=\kappa_\nu(x-x')$ be the translation-invariant Matérn kernel on $\mathbb R^n$, let $B\in\mathbb R^{M\times M}$ be symmetric positive definite, and define the separable matrix-valued kernel ${K_X(x,x'):=k_\nu(x,x')\,B}$ with associa

Figures (14)

  • Figure 1: Schematic diagram of the proposed property-preserving kernel method.
  • Figure 2: The 2D laminar flow past a cylinder problem. (A) and (B) show examples of an input function (the initial velocity) and an output function (the final velocity), respectively, from the vortex shedding regime. (C) and (D) show the test relative $\ell_2$ errors and training runtimes as functions of $N$ for the vortex shedding regime. (E) and (F) show the same results for the regime without vortex shedding.
  • Figure 3: The 2D laminar Taylor--Green vortices problem for the spacetime operator map. (A) and (B) show examples of an input function (the initial velocity) and an output function (the final velocity) at time $T=1$, respectively. Here, the output functions are snapshots of the velocity field at four timesteps (see Section \ref{['sec:taylor_green_spacetime']} for details). (C) and (D) show the test relative $\ell_2$ errors and training runtimes as functions of $N$ for the operator map from the initial velocity to the final velocity at the four timesteps. (E) and (F) show the same results for the operator map from the flow parameters to the final velocity at the four timesteps.
  • Figure 4: The 3D turbulent species transport example. (A) and (B) show examples of an input function (the inlet velocity of the two gaseous species at the plane $z=0$) and an example output function (the final velocity), respectively. (C) shows a top-down yz view of the domain and its placement of the three blades and the inner walls. (D) and (E) show the test relative $\ell_2$ errors and training runtimes as functions of $N$.
  • Figure 5: The 3D turbulent flow past an airfoil. (A) and (B) show examples of an input function (the 2D set of points constituting an airfoil shape) and an example output function (the final velocity), respectively. (C) and (D) show the test relative $\ell_2$ errors and training runtimes as functions of $N$.
  • ...and 9 more figures

Theorems & Definitions (16)

  • Theorem 1: Universal Approximation
  • proof
  • Theorem 2: Coefficient interpolation rate for vector-valued $C^4$ Matérn kernels in $L^2(X)$
  • proof
  • Corollary 3: Ridge perturbation of coefficient interpolation in $L^2(X)$
  • proof
  • Theorem 4: Divergence-free Matérn reconstruction error in $L^2(\Omega)$
  • proof
  • Corollary 5: Ridge operator bound in $L^2(X;L^2(\Omega))$
  • proof
  • ...and 6 more