SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915904670924800 |
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| author | Tretiakov, Stepan Li, Xingjian Kumar, Krishna |
| author_facet | Tretiakov, Stepan Li, Xingjian Kumar, Krishna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling Tretiakov, Stepan Li, Xingjian Kumar, Krishna Machine Learning I.2; G.1.8 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. |
| title | SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling |
| topic | Machine Learning I.2; G.1.8 |
| url | https://arxiv.org/abs/2505.04738 |