Learning Generalizable Neural Operators for Inverse Problems

Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B${}^{-1}$, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple function representation from the inverse map. We learn neural basis functions for the input and output spaces, then train inverse models that operate on the resulting coefficient space. This structure allows us to learn deterministic, invertible, and probabilistic models within a single framework, and to choose models based on the degree of ill-posedness. We evaluate our approach on six inverse PDE benchmarks, including two novel datasets, and compare against existing invertible neural operator baselines. We learn probabilistic models that capture uncertainty and input variability, and remain robust to measurement noise due to implicit denoising in the coefficient calculation. Our results show consistent re-simulation performance across varying levels of ill-posedness. By separating representation from inversion, our framework enables scalable surrogate models for inverse problems that generalize across instances, domains, and degrees of ill-posedness.

SetONet: A Deep Set-based Operator Network for Solving PDEs with permutation invariant variable input sampling

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 in scenarios with variable sensor configurations, missing data, or irregular grids. We introduce the Set Operator Network (SetONet), a novel architecture that integrates Deep Sets principles into the DeepONet framework to address this limitation. The core innovation lies in the SetONet branch network, which processes the input function as an unordered \emph{set} of location-value pairs. This design ensures permutation invariance with respect to the input points, making SetONet inherently robust to variations in the number and locations of sensors. SetONet learns richer, spatially-aware input representations by explicitly processing spatial coordinates and function values. We demonstrate SetONet's effectiveness on several benchmark problems, including derivative/anti-derivative operators, 1D Darcy flow, and 2D elasticity. Results show that SetONet successfully learns operators under variable input sampling conditions where standard DeepONet fails. Furthermore, SetONet is architecturally robust to sensor drop-off; unlike standard DeepONet, which requires methods like interpolation to function with missing data. Notably, SetONet can achieve comparable or improved accuracy over DeepONet on fixed grids, particularly for nonlinear problems, likely due to its enhanced input representation. SetONet provides a flexible and robust extension to the neural operator toolkit, significantly broadening the applicability of operator learning to problems with variable or incomplete input data.