# Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems

arXiv: https://arxiv.org/abs/2412.03161 · HTML: https://mlsanity.com/paper/2412.03161/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2024.

## Built on by (most cited first, 11 of 11)

- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines](https://mlsanity.com/paper/2501.09395/) (2025)
- [Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery](https://mlsanity.com/paper/2505.23106/) (2025)
- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems](https://mlsanity.com/paper/2605.25353/) (2026)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [Operator learning on domain boundary through combining fundamental solution-based artificial data and boundary integral techniques](https://mlsanity.com/paper/2601.11222/) (2026)
- [Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs](https://mlsanity.com/paper/2603.03922/) (2026)
- [Neural Constitutive Learning for Generalized Reaction-Diffusion Systems](https://mlsanity.com/paper/2609.37113/) (2026)
- [SBNO : Schr"odinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information](https://mlsanity.com/paper/2610.13317/) (2026)

## Abstract (arXiv)

Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.