[论文解读] One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers

Lennon Shikhman|arXiv (Cornell University)|Mar 2, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

论文主张神经PDE求解器学习的是一个按边界索引划分的算子族，而非一个对边界不敏感的单一算子；在边界条件发生变化时，存在不可识别性和泛化差的问题。文中给出理论与 Poisson 方程实验，以说明基础模型 PDE 求解器对边界感知的需求。

ABSTRACT

Neural PDE solvers are often described as learning solution operators that map problem data to PDE solutions. In this work, we argue that this interpretation is generally incorrect when boundary conditions vary. We show that standard neural operator training implicitly learns a boundary-indexed family of operators, rather than a single boundary-agnostic operator, with the learned mapping fundamentally conditioned on the boundary-condition distribution seen during training. We formalize this perspective by framing operator learning as conditional risk minimization over boundary conditions, which leads to a non-identifiability result outside the support of the training boundary distribution. As a consequence, generalization in forcing terms or resolution does not imply generalization across boundary conditions. We support our theoretical analysis with controlled experiments on the Poisson equation, demonstrating sharp degradation under boundary-condition shifts, cross-distribution failures between distinct boundary ensembles, and convergence to conditional expectations when boundary information is removed. Our results clarify a core limitation of current neural PDE solvers and highlight the need for explicit boundary-aware modeling in the pursuit of foundation models for PDEs.

研究动机与目标

• Motivate why neural PDE solvers with varying boundary conditions cannot be treated as learning a single operator.
• Frame operator learning as conditional risk minimization with respect to boundary conditions.
• Show non-identifiability outside the training boundary distribution and empirical evidence of boundary-conditioned degradation.
• Demonstrate through Poisson equation experiments how boundary shifts degrade performance and how boundary-ablated models converge to conditional expectations.

提出的方法

• Formalize neural operator learning as conditional risk minimization over boundary conditions.
• Use a controlled Poisson equation setup with mixed Dirichlet and Neumann boundaries and Fourier-based boundary distributions.
• Train Fourier Neural Operator models with and without explicit boundary condition channels.
• Evaluate cross-distribution generalization, boundary extrapolation, and boundary ablation effects.

实验结果

研究问题

• RQ1 Do neural PDE solvers learn a single boundary-agnostic operator or a boundary-indexed family of operators?
• RQ2 How does changing the boundary-condition distribution affect generalization and identifiability?
• RQ3 What happens when boundary information is omitted or poorly represented during training?
• RQ4 Can boundary shifts lead to convergence to conditional expectations rather than true solution operators?

主要发现

Model	Test on μB0 (relative L2)	Test on μB1 (relative L2)	Notes
FNO trained on μB0	0.078±0.005	0.489±0.022	Boundary-aware, trained on μB0
FNO trained on μB1	0.601±0.036	0.102±0.003	Boundary-aware, trained on μB1
FNO (no BC channels)	0.999±0.001	1.001±0.001	Boundary-ablated, no boundary input

• Training learns a boundary-indexed family of operators rather than a fixed operator invariant to boundary conditions.
• Models trained on one boundary distribution degrade sharply under boundary-condition shifts, while cross-distribution performance remains poor.
• Boundary extrapolation (shifts in boundary mean or higher-frequency components) causes monotonic error growth.
• Boundary-ablated models (no explicit BC channels) perform like conditional expectations over boundary conditions, not true solution operators.
• The results highlight non-identifiability under boundary distribution shift and the need for explicit boundary-aware modeling.
• Empirical results are demonstrated with Poisson equation experiments showing these effects.