基于规范等变内在神经算子的几何一致椭圆PDE映射学习
Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps
AI总结：
本文提出GINO，一种通过内在谱乘子和规范等变非线性结构学习椭圆PDE映射的神经算子，实现了几何一致性和对度量扰动的鲁棒性。
摘要：
学习偏微分方程（PDE）的解算子从数据中已出现为多查询科学工作流中快速替代模型的有前途的途径。然而，对于几何PDEs，其输入和输出在局部框架变换（规范）下变换，现有算子学习架构仍依赖于表示、对度量扰动脆弱且对离散化敏感。我们提出Gauge-Equivariant Intrinsic Neural Operators（GINO），一种通过内在谱乘子作用于几何依赖谱来参数化椭圆解映射的神经算子类，结合规范等变非线性。此设计解耦了几何与可学习函数依赖，并在框架变换下保持一致性。我们验证了GINO在平坦扭体（T²）上的受控问题，其中真实解算子和正则化Helmholtz-Hodge分解允许闭式傅里叶表示，从而实现理论一致的诊断。在E1-E6实验中，GINO实现了低算子近似误差、接近机器精度的规范等变性、对结构化度量扰动的鲁棒性、强跨分辨率泛化性和结构保持性能。消融进一步将学习的谱乘子的平滑性与几何扰动下的稳定性联系起来。这些结果表明，强制内在结构和规范等变性可产生更几何一致且对离散化鲁棒的椭圆PDE算子替代模型。
英文摘要：
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.
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