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Geometric Generalization of Neural Operators from Kernel Integral Perspective

Mingyu Han, Daniel Zhengyu Huang, Yuhan Wang, Yanshu Zhang, Jiayi Zhou

TL;DR

This work introduces a kernel-integral perspective on neural operators for PDEs on variable, nonparametric geometries, linking operator learning to geometry-dependent singular kernels from boundary integral methods. It develops the Multiscale Point Cloud Neural Operator (M-PCNO), an architecture that combines a Fourier-based long-range component with a local, geometry-aware short-range component to achieve near-linear evaluation cost on point clouds. The authors provide rigorous approximation guarantees via an Ewald-type decomposition, derive error bounds, and demonstrate geometric generalization across diverse 2D and 3D geometries, including a large-scale potential-flow aero example. The results indicate that a geometry-aware, multiscale neural operator can robustly generalize across topology changes and complex domains while offering practical speedups for large-scale simulations.

Abstract

Neural operators are neural network-based surrogate models for approximating solution operators of parametric partial differential equations, enabling efficient many-query computations in science and engineering. Many applications, including engineering design, involve variable and often nonparametric geometries, for which generalization to unseen geometries remains a central practical challenge. In this work, we adopt a kernel integral perspective motivated by classical boundary integral formulations and recast operator learning on variable geometries as the approximation of geometry-dependent kernel operators, potentially with singularities. This perspective clarifies a mechanism for geometric generalization and reveals a direct connection between operator learning and fast kernel summation methods. Leveraging this connection, we propose a multiscale neural operator inspired by Ewald summation for learning and efficiently evaluating unknown kernel integrals, and we provide theoretical accuracy guarantees for the resulting approximation. Numerical experiments demonstrate robust generalization across diverse geometries for several commonly used kernels and for a large-scale three-dimensional fluid dynamics example.

Geometric Generalization of Neural Operators from Kernel Integral Perspective

TL;DR

This work introduces a kernel-integral perspective on neural operators for PDEs on variable, nonparametric geometries, linking operator learning to geometry-dependent singular kernels from boundary integral methods. It develops the Multiscale Point Cloud Neural Operator (M-PCNO), an architecture that combines a Fourier-based long-range component with a local, geometry-aware short-range component to achieve near-linear evaluation cost on point clouds. The authors provide rigorous approximation guarantees via an Ewald-type decomposition, derive error bounds, and demonstrate geometric generalization across diverse 2D and 3D geometries, including a large-scale potential-flow aero example. The results indicate that a geometry-aware, multiscale neural operator can robustly generalize across topology changes and complex domains while offering practical speedups for large-scale simulations.

Abstract

Neural operators are neural network-based surrogate models for approximating solution operators of parametric partial differential equations, enabling efficient many-query computations in science and engineering. Many applications, including engineering design, involve variable and often nonparametric geometries, for which generalization to unseen geometries remains a central practical challenge. In this work, we adopt a kernel integral perspective motivated by classical boundary integral formulations and recast operator learning on variable geometries as the approximation of geometry-dependent kernel operators, potentially with singularities. This perspective clarifies a mechanism for geometric generalization and reveals a direct connection between operator learning and fast kernel summation methods. Leveraging this connection, we propose a multiscale neural operator inspired by Ewald summation for learning and efficiently evaluating unknown kernel integrals, and we provide theoretical accuracy guarantees for the resulting approximation. Numerical experiments demonstrate robust generalization across diverse geometries for several commonly used kernels and for a large-scale three-dimensional fluid dynamics example.
Paper Structure (23 sections, 2 theorems, 88 equations, 4 figures, 5 tables)

This paper contains 23 sections, 2 theorems, 88 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries and Literature Review
  4. Neural Operator Approaches
  5. Classical Fast Kernel Summation Methods
  6. Organization
  7. Approximation of Singular Kernels
  8. Neural Layers for Kernel Integrals
  9. Long-Range Approximation
  10. Short-Range Approximation
  11. Multiscale Point Cloud Neural Layer
  12. Multiscale Point Cloud Neural Operator
  13. Numerical Study
  14. Kernel Integral
  15. Exterior Laplace Equations
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\kappa$ be a periodic, translation-invariant kernel on $B_2= [-\frac{1}{2},\frac{1}{2}]^d$ with $d\geq 2$. Assume that $\kappa(x) \in L^1(B_2) \cap C^2(B_2\setminus\{0\})$ and that there exists a constant $C>0$ such that for all $x\in B_2\setminus\{0\}$. For $\delta\in(0,\tfrac{1}{2})$, define the Gaussian mollifier $\rho_\delta(y) = \frac{1}{(2\pi \delta^2)^ {d/2}} e^{-\frac{ \lVert y \rVer

Figures (4)

  • Figure 1: Representative results for learning the Laplacian single layer potential using a 5-layer M-PCNO with $p=32$ and $n=8000$. Each column shows the reference solution (top) and the prediction (bottom) for test cases with median and largest relative errors from the single-curve (left two) and two-curve (right two) test datasets.
  • Figure 2: Kernel integrals: relative test errors as functions of the truncated mode number $p$ with training dataset size $n=8000$ (top row), and as functions of the training dataset size $n$ with $p=32$ fixed (bottom row), for different models and test datasets. Each column corresponds to one kernel integral.
  • Figure 3: Neumann-to-Dirichlet map for the exterior Laplacian learned with a 5-layer M-PCNO: relative $L^2$ test errors as functions of the truncated mode number $p$ (left) and the training dataset size $n$ (right).
  • Figure 4: Representative results for the potential flow problem using a 5-layer M-PCNO with $p=16$ and $n=4000$. Each column shows the reference solution (top) and the prediction (bottom) for test cases with the largest relative error (leftmost, fighter jet), the median relative error (left-middle, turboprop), and three randomly selected samples.

Theorems & Definitions (7)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proof 1
  • Lemma A.1
  • Proof 2