🤖 AI Summary
This work addresses the limited generalization of neural PDE solvers under varying boundary conditions, which arises because these models learn a family of operators conditioned on the training boundary distribution rather than a universal operator. The authors formulate operator learning as a conditional risk minimization problem with respect to boundary conditions and introduce the concept of a “boundary-indexed operator family.” They theoretically prove that standard neural operators suffer from non-identifiability outside the training boundary distribution, revealing the root cause of their generalization bottleneck. Through controlled experiments, numerical simulations of Poisson’s equation, and boundary perturbation analyses, they demonstrate significant performance degradation under boundary shifts and show that removing boundary information reduces the model to a conditional expectation. This study is the first to systematically elucidate the critical role of boundary conditions in the generalization of neural operators.
📝 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.