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

Lennon J. Shikhman

cs.LGMar 2, 2026

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

Authors: Lennon J. Shikhman

Organizations: College of Computing, Georgia Institute of Technology · Department of Mathematics and Systems Engineering, Florida Institute of Technology

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.

Figures & tables

Panel of Figure 1 — 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.

Model	Test on $\mu_{B_{0}}$	Test on $\mu_{B_{1}}$
FNO trained on $\mu_{B_{0}}$	$0.078\pm 0.005$	$0.489\pm 0.022$
FNO trained on $\mu_{B_{1}}$	$0.601\pm 0.036$	$0.102\pm 0.003$
FNO (no BC channels)	$0.999\pm 0.001$	$1.001\pm 0.001$

Table 1: Cross-distribution generalization under boundary-condition shift. Relative

L^{2}

errors are reported as mean

\pm

standard deviation over evaluation batches.

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.

PDE Operator Learning

Partial differential equation (PDE) operator learning builds models that map inputs such as coefficients and initial conditions to solution fields.

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Developing foundational neural simulators for Partial Differential Equations (PDEs) requires robust generalization across diverse physical parameters and boundary conditions. However, current deep learning approaches largely face a structural trade-off between condition-agnostic deployment and physical fidelity. Purely data-driven operators infer the underlying physics implicitly and thus lack the explicit constraints needed to ensure physically valid solutions across varying domains, rendering the learning problem ill-posed. On the other hand, Physics-Informed Neural Networks (PINNs) enforce rigorous physical constraints but necessitate costly, instance-specific optimization. Furthermore, the massive scale of emerging foundational operators has severely degraded their inference speeds, making them computationally uncompetitive with traditional numerical solvers. To address this bottleneck between condition-agnostic deployment, physical rigor, and inference efficiency, we propose a Generalized Neural Operator. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining strict inference efficiency comparable to conventional numerical baselines.

Abstract

PDE Operator Learning

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