Enforcing Dirichlet Boundary Conditions in Operator Learning

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arXiv:2608.27256v1 Announce Type: cross Abstract: Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators …

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