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

Pengcheng Cheng

cs.AIMar 16, 2026

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

Authors: Pengcheng Cheng

Organizations: School of Mathematics, Jilin University, Changchun, 130012, China

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.

Figures & tables

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.

Panel of Figure 4 — 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.

Panel of Figure 5 — 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.

Figure 6: Discretization consistency diagnostic (E4). GINO achieves negligible commutation errors ( $<0.01$ ), while CoordCNN errors exceed 0.6, revealing strong grid dependence.

Figure 7: Summary of E5 performance. Final relative errors for exact/coexact components are $\approx 3.4\times 10^{-2}$ , with decomposition residual $\approx 1.45\times 10^{-2}$ .

Figure 8: Training dynamics for E5. Left: gauge equivariance error stays below $4\times 10^{-6}$ . Middle/right: projection errors and decomposition residual decrease rapidly during training.

Figure 9: Ablation on spectral truncation (E6A). Error metrics (RelL2, RelEnergy) are minimized at intermediate $\Lambda$ , while roughness increases monotonically with $\Lambda$ .

Panel of Figure 10 — Figure 10: Effect of smoothness on geometric stability (E6B). Left: stronger regularization yields smoother multipliers. Right: unregularized models amplify error by up to 3.4× under metric perturbation, while smoothed models remain stable.

Explore similar work

Jun 2, 2026

EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

Sungwon Kim, Juho Song, Seungmin Shin +3

Graduate School of Data Science, KAIST, Daejeon, Republic of Korea · Mathematical Sciences, KAIST, Daejeon, Republic of Korea · LG Electronics, Pyeong-taek, Republic of Korea +1

Fourier Neural Operator PDE Surrogate Modeling PDE Operator Learning Neural Operators

Ask PDF

Aug 30, 2026

Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains

Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the Graph Spectral Neural Operator (GSNO), a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.

Abdolmehdi Behroozi, Chaopeng Shen

Department of Civil and Environmental Engineering, Penn State University, University Park, PA 16802, USA

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Neural Operators

Ask PDF

Jul 1, 2026

GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative

L^2

error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

Meenakshi Krishnan, Pranav Pulijala, Ke Chen +2

University of Maryland, College Park · University of Delaware, Newark

PDE Solving Image Reconstruction PDE Operator Learning PDE Inverse Problems Neural Operators

Ask PDF