Enforcing Dirichlet Boundary Conditions in Operator Learning

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.

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitly into the neural operator architecture, so that the operator structure matches the kernel-expansion form used in boundary-type kernel-expansion methods. Unlike traditional neural operators such as DeepONet, which learn basis functions implicitly through deep networks, KernelOnet replaces the trunk network with explicit kernels and offers three complementary kernels: a data-driven learnable kernel, in which a neural network parameterizes a radial basis function learned from data, and which for constant-coefficient linear problems can be regarded as a non-singular fundamental solution; a physics-informed kernel, which embeds physical information such as analytic fundamental solutions into the network structure, so that the expansion satisfies the governing equation automatically and can be trained without supervision on boundary conditions alone, with no interior solution data; and a hybrid kernel, which splits the solution, according to the linear principal part of the governing equation, into a homogeneous part spanned by analytic fundamental solutions and a source part carried by low-rank learned correction kernels, thereby balancing physical priors against data fitting on nonlinear problems lacking an analytic fundamental solution. On three benchmarks and one engineering problem in a shallow-water waveguide, KernelOnet attains high accuracy; where comparable with DeepONet, it is more accurate with fewer learnable parameters. Its unsupervised configuration needs no interior solution labels, and its per-query inference cost is far below that of per-instance solvers, offering an effective route to acoustic propagation in unbounded exterior domains that general-purpose neural operators struggle to handle.