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.

翻译：神经算子已成为昂贵流体流动模拟器的高性价比替代方案，尤其在计算密集型任务（如基于延时地震数据的渗透率反演及不确定性量化）中具有重要应用。在这些应用中，代理模型关于系统参数的梯度保真度至关重要，因为下游任务（如优化和贝叶斯推断）的精度直接依赖于导数信息的质量。近期物理信息方法的发展已利用导数信息提升代理模型精度。然而，显式雅可比矩阵的引入可能导致计算量剧增，因为其复杂度通常随输入参数数量呈二次方增长。为突破此限制，我们提出DeFINO（基于导数的费希尔信息引导神经算子）——一种降阶的导数信息训练框架。DeFINO将傅里叶神经算子（FNOs）与受费希尔信息矩阵（FIM）引导的新型导数训练策略相结合。通过将雅可比矩阵投影至FIM识别的主导特征方向，DeFINO能直接捕获观测数据驱动的关键灵敏度信息，显著降低计算开销。我们通过地下多相流体流动的合成实验验证DeFINO，证明其在保持底层流体动力学稳健正向预测的同时，有效提升了梯度精度。这些成果凸显了DeFINO在复杂现实场景中为反演问题提供实用、可扩展解决方案的潜力，且计算成本大幅降低。