SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling

Most neural-operator surrogates for PDEs inherit from DeepONet-style formulations the requirement that the input function be sampled at a fixed, ordered set of sensors. This assumption limits applicability to problems with variable sensor layouts, missing data, point sources, and sample-based representations of densities. We propose SetONet, which addresses this gap by recasting the operator input as an unordered set of coordinate-value observations and encoding it with permutation-invariant aggregation inside a standard branch-trunk operator network while preserving the DeepONet synthesis mechanism and lightweight end-to-end training. A structured variant, SetONet-Key, aggregates sensor information through learnable query tokens and a position-only key pathway, thereby decoupling sampling geometry from sensor values. The method is assessed on four classical operator-learning benchmarks under fixed layouts, variable layouts, and evaluation-time sensor drop-off, and on four problems with inherently unstructured point-cloud inputs, including heat conduction with multiple point sources, advection-diffusion, phase-screen diffraction, and optimal transport problems. In parameter-matched studies, SetONet-Key achieves lower error than the DeepONet baseline on fixed-sensor benchmarks and remains reliable when layouts vary or sensors are dropped at evaluation. Comparisons across pooling rules show that attention-based aggregation is typically more robust than mean or sum pooling. On the point-cloud problems, SetONet operates directly on the native input representation, without rasterization or multi-stage preprocessing, and outperforms the larger VIDON baseline.