Imposing Boundary Conditions on Neural Operators via Learned Function Extensions

Neural operators have emerged as powerful surrogates for the solution of partial differential equations (PDEs), yet their ability to handle general, highly variable boundary conditions (BCs) remains limited. Existing approaches often fail when the solution operator exhibits strong sensitivity to boundary forcings. We propose a general framework for conditioning neural operators on complex non-homogeneous BCs through function extensions. Our key idea is to map boundary data to latent pseudo-extensions defined over the entire spatial domain, enabling any standard operator learning architecture to consume boundary information. The resulting operator, coupled with an arbitrary domain-to-domain neural operator, can learn rich dependencies on complex BCs and input domain functions at the same time. To benchmark this setting, we construct 18 challenging datasets spanning Poisson, linear elasticity, and hyperelasticity problems, with highly variable, mixed-type, component-wise, and multi-segment BCs on diverse geometries. Our approach achieves state-of-the-art accuracy, outperforming baselines by large margins, while requiring no hyperparameter tuning across datasets. Overall, our results demonstrate that learning boundary-to-domain extensions is an effective and practical strategy for imposing complex BCs in existing neural operator frameworks, enabling accurate and robust scientific machine learning models for a broader range of PDE-governed problems.

💡 Research Summary

The paper addresses a critical limitation of current neural operator (NO) models: their inability to robustly handle highly variable, non‑homogeneous boundary conditions (BCs). While NOs excel at learning mappings from interior coefficients or source terms to PDE solutions, they typically assume fixed or weakly varying BCs. This is problematic for many engineering and scientific applications where the BCs are the primary drivers of the solution, especially for elliptic problems where the solution at any interior point depends globally on the entire boundary.

Key Contributions

1. Unified BC Representation – The authors reformulate Dirichlet, Neumann, and Robin conditions into a single expression
   (B(q,u)=\alpha\odot B_D(u)+\beta\odot B_N(u)-\gamma)
   where (\alpha,\beta,\gamma) are scalar/vector fields defined on the whole boundary. They introduce a principled normalization scheme that respects physical units and prevents scale imbalance, ensuring stable training.

2. Extender Module (Ψ) – The central idea is to map the boundary data (q) to a pseudo‑extension (\psi) defined over the entire domain (\Omega). Three concrete instantiations are proposed:

   • Zero Extension (0X) – Simple padding with zeros away from the boundary.
   • Harmonic Extension (HX) – Solve a Laplace problem (\Delta\psi=0) with (\psi|_{\partial\Omega}=q), yielding the smoothest possible extension (minimal Dirichlet energy).
   • Learned Pseudo‑Extension (LX) – An attention‑based neural network that ingests raw boundary values and domain coordinates, repeatedly applies cross‑attention and feed‑forward layers, and outputs (\psi). This approach avoids the costly PDE solve required by HX while still embedding the full information of (q).

3. Integration with Existing Operators – Once (\psi) is obtained, it is concatenated with any standard domain‑to‑domain operator (\Phi) (e.g., Fourier Neural Operator, Graph Neural Operator, Transformer‑based operator). The overall learned operator becomes
   (G_\theta(a,q)=\Phi_\theta\big(a,\Psi_\theta(q)\big))
   where (a) denotes interior coefficients. Because both inputs to (\Phi) are full‑domain functions, no special boundary handling is required inside (\Phi).

4. Extensive Benchmark Suite – The authors construct 18 challenging datasets covering Poisson, linear elasticity, and hyperelasticity equations on six distinct geometries (circles, squares, boomerangs, etc.). Each sample features random boundary segment sizes, random locations, component‑wise BC types, and up to eight mixed‑type segments. Meshes are unstructured and locally refined, including domains with holes.

5. Empirical Results – Across all datasets, the LX‑based models consistently outperform baselines (BENO, PENN, BOON) by large margins. Error percentages drop from 30–70 % (baseline) to 5–10 % (LX). Even the simpler 0X and HX variants improve over baselines, confirming the value of any domain‑wide extension. Notably, the improvements hold without any dataset‑specific hyper‑parameter tuning, demonstrating the method’s robustness.

Technical Insights

• The unified BC formulation eliminates the need for branching logic in the network architecture, simplifying implementation and enabling batch processing of heterogeneous BCs.
• The pseudo‑extension concept leverages the fact that many PDE solution operators are affine in the boundary data for linear problems; the authors prove (Appendix A.3) that an affine mapping exists when the extension satisfies certain smoothness conditions.
• The attention‑based extender efficiently learns to propagate boundary information throughout the domain, effectively performing a learned “soft” PDE solve. Down‑sampling of coordinates before attention reduces computational cost while preserving accuracy.
• By treating the extension as a learned latent field, the method can be combined with any existing NO architecture, making it a plug‑and‑play upgrade for current scientific ML pipelines.

Impact and Future Directions

The proposed framework dramatically expands the applicability of neural operators to real‑world problems where boundary conditions are complex, spatially varying, and possibly component‑wise. It opens the door to fast surrogate models for design optimization, inverse problems, and uncertainty quantification in domains such as structural mechanics, fluid dynamics, and material science. Future work could explore (i) extensions to time‑dependent PDEs where boundary data evolves, (ii) multi‑physics coupling where different fields share boundary interfaces, and (iii) theoretical analysis of approximation guarantees for the learned extender. Overall, the paper delivers a practical, theoretically grounded, and empirically validated solution to a long‑standing bottleneck in operator learning.