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Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps

Pengcheng Cheng

Abstract

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform under changes of local frame (gauge), many existing operator-learning architectures remain representation-dependent, brittle under metric perturbations, and sensitive to discretization changes. We propose Gauge-Equivariant Intrinsic Neural Operators (GINO), a class of neural operators that parameterize elliptic solution maps primarily through intrinsic spectral multipliers acting on geometry-dependent spectra, coupled with gauge-equivariant nonlinearities. This design decouples geometry from learnable functional dependence and enforces consistency under frame transformations. We validate GINO on controlled problems on the flat torus ($\mathbb{T}^2$), where ground-truth resolvent operators and regularized Helmholtz--Hodge decompositions admit closed-form Fourier representations, enabling theory-aligned diagnostics. Across experiments E1--E6, GINO achieves low operator-approximation error, near machine-precision gauge equivariance, robustness to structured metric perturbations, strong cross-resolution generalization with small commutation error under restriction/prolongation, and structure-preserving performance on a regularized exact/coexact decomposition task. Ablations further link the smoothness of the learned spectral multiplier to stability under geometric perturbations. These results suggest that enforcing intrinsic structure and gauge equivariance yields operator surrogates that are more geometry-consistent and discretization-robust for elliptic PDEs on form-valued fields.

Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps

Abstract

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform under changes of local frame (gauge), many existing operator-learning architectures remain representation-dependent, brittle under metric perturbations, and sensitive to discretization changes. We propose Gauge-Equivariant Intrinsic Neural Operators (GINO), a class of neural operators that parameterize elliptic solution maps primarily through intrinsic spectral multipliers acting on geometry-dependent spectra, coupled with gauge-equivariant nonlinearities. This design decouples geometry from learnable functional dependence and enforces consistency under frame transformations. We validate GINO on controlled problems on the flat torus (), where ground-truth resolvent operators and regularized Helmholtz--Hodge decompositions admit closed-form Fourier representations, enabling theory-aligned diagnostics. Across experiments E1--E6, GINO achieves low operator-approximation error, near machine-precision gauge equivariance, robustness to structured metric perturbations, strong cross-resolution generalization with small commutation error under restriction/prolongation, and structure-preserving performance on a regularized exact/coexact decomposition task. Ablations further link the smoothness of the learned spectral multiplier to stability under geometric perturbations. These results suggest that enforcing intrinsic structure and gauge equivariance yields operator surrogates that are more geometry-consistent and discretization-robust for elliptic PDEs on form-valued fields.
Paper Structure (75 sections, 13 theorems, 268 equations, 10 figures)

This paper contains 75 sections, 13 theorems, 268 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem Setup and Target Operator
  3. Geometric and Functional-Analytic Setting
  4. Elliptic PDE on 1-Forms and the Solution Operator
  5. Learning Objective and Data Model
  6. Gauge-Equivariant Intrinsic Neural Operators
  7. Gauge Structure on $1$ -Forms and an Equivariance Principle
  8. Intrinsic Linear Layers via Functional Calculus of $\Delta_H$
  9. Gauge-Equivariant Pointwise Nonlinearities on $1$ -Forms
  10. The Gauge-Equivariant Intrinsic Neural Operator Architecture
  11. Intrinsicness and Gauge Equivariance: Formal Guarantees
  12. Discrete Realization and Design for Consistency
  13. Discussion: Relation to the Target Elliptic Solution Operator
  14. Sobolev Operator Approximation
  15. Sobolev Norms via the Hodge Spectrum
  16. ...and 60 more sections

Key Result

Proposition 3.1

Let $(\mathcal{M}, g)$ be a compact smooth Riemannian manifold without boundary. Let $\varphi : \mathcal{M} \to \mathcal{M}$ be a diffeomorphism and set $g' := \varphi^* g$ . Consider a network operator $\widehat{\mathcal{S}}^{(g)}_{\theta,K}$ built by composing layers of the form where: Then the operator is intrinsic in the sense that

Figures (10)

  • Figure 1: Training convergence of GINO on the base geometry (E1). Evaluation metrics (MSE, RelL2, RelEnergy) rapidly decrease and stabilize, reaching near numerical precision.
  • Figure 2: Qualitative prediction example at step 4000 (E1). From left to right: $|f|$ , $|u|$ , $|\hat{u}|$ , and $|\hat{u} - u|$ . The model produces visually accurate predictions with small residual errors.
  • Figure 3: Gauge equivariance and sensitivity analysis (E2). Left: GINO's equivariance error remains below $2 \times 10^{-7}$ . Middle and right: CoordCNN shows strong gauge dependence with relative deviations near 1.
  • Figure 4: Metric perturbation stability (E3). GINO (blue) maintains low RelEnergy (left) and RelL2 (right) across $\|M - I\|_F \in [0, 0.4]$ , while CoordCNN (orange) shows large, invariant errors.
  • Figure 5: Cross-resolution generalization (E4). GINO generalizes across resolutions with low RelL2 (left) and RelEnergy (right); CoordCNN fails dramatically when tested on unseen grids.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 3.1: Gauge equivariance
  • Definition 3.2: Truncated spectral multiplier layer
  • Definition 3.3: Fiberwise gauge-equivariant nonlinearity
  • Proposition 3.1: Intrinsicness
  • Proposition 3.2: Gauge equivariance
  • Lemma 4.1: Sobolev boundedness of $\mathcal{S}_g$
  • Lemma 4.2: Truncation bias in Sobolev operator norm
  • Lemma 4.3: Uniform multiplier error implies Sobolev operator bound
  • Theorem 4.4: Sobolev operator approximation of the elliptic solution operator
  • Theorem 5.1: Metric stability of the shifted Hodge–Poisson resolvent
  • ...and 8 more