# A reduced-order derivative-informed neural operator for subsurface fluid-flow

arXiv: https://arxiv.org/abs/2509.13620 · HTML: https://mlsanity.com/paper/2509.13620/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 27 references that are in the map.
- First posted on arXiv 17 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [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 networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs](https://mlsanity.com/paper/2011.15110/) (2020)
- [Large-scale Bayesian optimal experimental design with derivative-informed projected neural network](https://mlsanity.com/paper/2201.07925/) (2022)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Solving multiphysics-based inverse problems with learned surrogates and constraints](https://mlsanity.com/paper/2307.11099/) (2023)
- [Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems](https://mlsanity.com/paper/2102.13245/) (2021)
- [Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution](https://mlsanity.com/paper/2402.08383/) (2024)
- [Learned coupled inversion for carbon sequestration monitoring and forecasting with Fourier neural operators](https://mlsanity.com/paper/2203.14396/) (2022)
- [Accelerating PDE-constrained Inverse Solutions with Deep Learning and Reduced Order Models](https://mlsanity.com/paper/1912.08864/) (2019)
- [An uncertainty-aware Digital Shadow for underground multimodal CO2 storage monitoring](https://mlsanity.com/paper/2410.01218/) (2024)

## Built on by (most cited first, 1 of 1)

- [SAGE: Subsurface AI-driven Geostatistical Extraction with proxy posterior](https://mlsanity.com/paper/2604.00307/) (2026)

## Abstract (arXiv)

Neural operators have emerged as cost-effective surrogates for expensive fluid-flow simulators, particularly in computationally intensive tasks such as permeability inversion from time-lapse seismic data, and uncertainty quantification. In these applications, the fidelity of the surrogate's gradients with respect to system parameters is crucial, as the accuracy of downstream tasks, such as optimization and Bayesian inference, relies directly on the quality of the derivative information. Recent advances in physics-informed methods have leveraged derivative information to improve surrogate accuracy. However, incorporating explicit Jacobians can become computationally prohibitive, as the complexity typically scales quadratically with the number of input parameters. To address this limitation, we propose DeFINO (Derivative-based Fisher-score Informed Neural Operator), a reduced-order, derivative-informed training framework. DeFINO integrates Fourier neural operators (FNOs) with a novel derivative-based training strategy guided by the Fisher Information Matrix (FIM). By projecting Jacobians onto dominant eigen-directions identified by the FIM, DeFINO captures critical sensitivity information directly informed by observational data, significantly reducing computational expense. We validate DeFINO through synthetic experiments in the context of subsurface multi-phase fluid-flow, demonstrating improvements in gradient accuracy while maintaining robust forward predictions of underlying fluid dynamics. These results highlight DeFINO's potential to offer practical, scalable solutions for inversion problems in complex real-world scenarios, all at substantially reduced computational cost.