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

TL;DR

SetONet addresses the fixed-input limitation of DeepONet by encoding input functions as unordered sets of location–value pairs using Deep Sets, preserving parameter count. It achieves parity with DeepONet on fixed sensor layouts and remains accurate under variable sensor configurations and sensor drop-off, addressing practical data irregularities. It enables end-to-end learning for intrinsically unstructured inputs, such as point-source heat conduction, advection-diffusion plumes, and density transport maps, without rasterization. The approach is lightweight and acts as a drop-in replacement, expanding operator learning to variable, incomplete, or unstructured sensor data.

Abstract

Neural operators, particularly the Deep Operator Network (DeepONet), have shown promise in learning mappings between function spaces for solving differential equations. However, standard DeepONet requires input functions to be sampled at fixed locations, limiting its applicability when sensor configurations vary or inputs exist on irregular grids. We introduce the Set Operator Network (SetONet), which modifies DeepONet's branch network to process input functions as unordered sets of location-value pairs. By incorporating Deep Sets principles, SetONet ensures permutation invariance while maintaining the same parameter count as the baseline. On classical operator-learning benchmarks, SetONet achieves parity with DeepONet on fixed layouts while sustaining accuracy under variable sensor configurations or sensor drop-off - conditions for which standard DeepONet is not applicable. More significantly, SetONet natively handles problems where inputs are naturally represented as unstructured point clouds (such as point sources or density samples) rather than values on fixed grids, a capability standard DeepONet lacks. On heat conduction with point sources, advection-diffusion modeling chemical plumes, and optimal transport between density samples, SetONet learns operators end-to-end without rasterization or multi-stage pipelines. These problems feature inputs that are naturally discrete point sets (point sources or density samples) rather than functions on fixed grids. SetONet is a DeepONet-class architecture that addresses such problems with a lightweight design, significantly broadening the applicability of operator learning to problems with variable, incomplete, or unstructured input data.

Figures (12)

• Figure 1: Schematic of the proposed Set Operator Network (SetONet) architecture for solving PDEs with variable input sampling. The branch network (top) leverages Deep Sets principles zaheer2018deepsets to process the input function represented as an unordered set of $M$ location-value pairs $\{(\boldsymbol{x}_i, g(\boldsymbol{x}_i))\}_{i=1}^M$. This involves an element-wise transformation $\phi$ applied to each pair after positional encoding (see Sec. \ref{['sec:setonet_architecture']}), a permutation-invariant aggregation step (e.g., mean, sum, or attention pooling), and a final set-level transformation $\rho$ to produce latent branch coefficients $\boldsymbol{b} = [\boldsymbol{b}_1, \dots, \boldsymbol{b}_p]$. Concurrently, the trunk network (bottom) maps the query location $\boldsymbol{y}$ to corresponding basis vectors $\boldsymbol{t}(\boldsymbol{y}) = [\boldsymbol{t}_1(\boldsymbol{y}), \dots, \boldsymbol{t}_p(\boldsymbol{y})]$. The final operator approximation $\mathcal{T}_{\boldsymbol{\theta}}(g)(\boldsymbol{y})$ is computed by combining $\boldsymbol{b}$ and $\boldsymbol{t}(\boldsymbol{y})$ as detailed in Eq. \ref{['eq:setonet_output_simplified']}.
• Figure 2: Test performance (Mean Squared Error Loss vs. Steps) for DeepONet and SetONet under different sensor configurations. Left: Derivative benchmark example. Right: Integral benchmark example. SetONet achieves comparable performance under different settings. All results averaged over $5$ seeded runs.
• Figure 3: Example predictions for the Derivative approximation benchmark. Left (a): DeepONet (blue) and SetONet (red) predictions vs. ground truth (green) under the fixed sensor locations condition. Right (b): SetONet (red) predictions vs. ground truth (green) under the variable sensor locations condition.
• Figure 4: Example predictions for the Integral approximation benchmark. Left (a): DeepONet (blue) and SetONet (red) predictions vs. ground truth (green) under the fixed sensor locations condition. Right (b): SetONet (red) predictions vs. ground truth (green) under the sensor drop-off condition.
• Figure 5: Example predictions for the 1D Darcy flow benchmark. Left (a): DeepONet (blue) and SetONet (red) predictions vs. ground truth (green) under the fixed sensor locations condition. Right (b): SetONet (red) predictions vs. ground truth (green) under the variable sensor locations condition.