From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning
Zhentao Tan, Ruijie Quan, Yi Yang
cs.AI, cs.CV
Submitted: 2026-08-07
Updated: 2026-08-10
Code: https://github.com/Tanpig-X/ESO
License: http://creativecommons.org/licenses/by/4.0/
The gist: Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations.
Terminology
Abstract
Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations. However, many PDEs contain physics-sensitive local structures that are critical to the underlying physical behavior. For example, in Darcy flow, local material interfaces are often reflected by sharp changes in the permeability field and can strongly influence the solution. Existing spectral operators primarily adapt modal mixing based on center-point representations, making them insufficiently responsive to such localized structural variations. We propose the Edge-Conditioned Spectral Operator (ESO), a novel spectral operator framework that modulates global spectral mixing using local edge-wise variations. By incorporating the Pairwise-Variation Modal Mixer (PVMM) to inject local edge information into spectral mode selection, ESO preserves the global approximation capability of spectral neural operators while enabling the learned kernel to adapt to physics-sensitive local structures. Furthermore, we introduce a task-adaptive Physics-Aware Reweighting (PAR) that emphasizes physically important regions, identified by taskspecific physical quantities. Across nine PDE benchmarks, ESO consistently achieves state-of-the-art performance. Visual and region-wise analyses further demonstrate that ESO reduces solution errors near coefficient jumps, high-gradient flow structures, and other physically sensitive regions. The code is available at https://github.com/Tanpig-X/ESO.
Sources
- Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning
- Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers
- Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations
- DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training
- Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations
- Reformer: The Efficient Transformer
- Fourier Neural Operator for Parametric Partial Differential Equations
- Transformer for Partial Differential Equations' Operator Learning
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
- Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects
- Factorized Fourier Neural Operators
- PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems
- CViT: Continuous Vision Transformer for Operator Learning
- Transolver: A Fast Transformer Solver for PDEs on General Geometries
- Improved Operator Learning by Orthogonal Attention
- Simple yet Effective: Low-Rank Spatial Attention for Neural Operators
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