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One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers

Lennon J. Shikhman

TL;DR

This work shows 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.

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.

One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers

TL;DR

This work shows 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.

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.
Paper Structure (39 sections, 8 equations, 4 figures, 1 table)

This paper contains 39 sections, 8 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Problem Setup and Notation
  4. Differential equation and domain.
  5. Boundary specification.
  6. Solution mappings.
  7. Data generation.
  8. Neural solver inputs.
  9. What Is Actually Being Learned
  10. Learning as conditional prediction.
  11. Dependence on the boundary distribution.
  12. ERM interpretation.
  13. Non-Identifiability Induced by Boundary Distribution Shift
  14. Implications for Generalization
  15. Forcing generalization versus boundary generalization.
  16. ...and 24 more sections

Figures (4)

  • Figure 1: Training dynamics for boundary-aware Fourier Neural Operators trained on $\mu_{B_0}$ (left) and $\mu_{B_1}$ (right). Relative $L^2$ error decreases smoothly and stabilizes in both cases, indicating that subsequent generalization failures are not due to optimization instability or lack of convergence.
  • Figure 2: Boundary extrapolation via Dirichlet mean shifts. Relative $L^2$ error increases smoothly and symmetrically as boundary conditions move away from the training distribution, despite unchanged forcing distribution, resolution, and PDE operator.
  • Figure 3: Boundary extrapolation via increased Dirichlet bandwidth. Performance degrades monotonically as higher-frequency boundary components are introduced beyond the training support.
  • Figure 4: Conditional expectation behavior under boundary ablation. For a fixed forcing $f^*$, the boundary-ablated model output (left) closely matches the Monte Carlo estimate of $\mathbb{E}[u \mid f^*]$ (center), with the absolute difference shown on the right. This provides empirical evidence that training via empirical risk minimization leads to conditional averaging when boundary information is unavailable.