# Learning Generalizable Neural Operators for Inverse Problems

arXiv: https://arxiv.org/abs/2512.18120 · HTML: https://mlsanity.com/paper/2512.18120/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 36 references that are in the map.
- First posted on arXiv 19 Dec 2025.

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems](https://mlsanity.com/paper/2001.01258/) (2020)
- [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Function Encoders: A Principled Approach to Transfer Learning in Hilbert Spaces](https://mlsanity.com/paper/2501.18373/) (2025)
- [To be or not to be stable, that is the question: understanding neural networks for inverse problems](https://mlsanity.com/paper/2211.13692/) (2022)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [A Frequency-Velocity CNN for Developing Near-Surface 2D Vs Images from Linear-Array, Active-Source Wavefield Measurements](https://mlsanity.com/paper/2207.09580/) (2022)
- [SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling](https://mlsanity.com/paper/2505.04738/) (2025)

## Abstract (arXiv)

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.