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

arXiv: https://arxiv.org/abs/2505.04738 · HTML: https://mlsanity.com/paper/2505.04738/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 51 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Stabilizing Transformer Training by Preventing Attention Entropy Collapse](https://mlsanity.com/paper/2303.06296/) (2023)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs](https://mlsanity.com/paper/2403.12553/) (2024)
- [Super-Resolution Neural Operator](https://mlsanity.com/paper/2303.02584/) (2023)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Mitigating spectral bias for the multiscale operator learning](https://mlsanity.com/paper/2210.10890/) (2022)
- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
- [Variable-Input Deep Operator Networks](https://mlsanity.com/paper/2205.11404/) (2022)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Using neural networks to solve the 2D Poisson equation for electric field computation in plasma fluid simulations](https://mlsanity.com/paper/2109.13076/) (2021)
- [Gaussian Plane-Wave Neural Operator for Electron Density Estimation](https://mlsanity.com/paper/2402.04278/) (2024)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)

## Built on by (most cited first, 1 of 1)

- [Learning Generalizable Neural Operators for Inverse Problems](https://mlsanity.com/paper/2512.18120/) (2025)

## Abstract (arXiv)

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.