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

Jeongjin Park, Grant Bruer, Huseyin Tuna Erdinc, Abhinav Prakash Gahlot, Felix J. Herrmann

TL;DR

DeFINO (Derivative-based Fisher-score Informed Neural Operator) is proposed, a reduced-order, derivative-informed training framework that captures critical sensitivity information directly informed by observational data, significantly reducing computational expense.

Abstract

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.

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

TL;DR

DeFINO (Derivative-based Fisher-score Informed Neural Operator) is proposed, a reduced-order, derivative-informed training framework that captures critical sensitivity information directly informed by observational data, significantly reducing computational expense.

Abstract

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.

Paper Structure

This paper contains 12 sections, 6 equations, 3 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Learning derivatives with reduced-order modeling
  3. Our contribution
  4. METHODOLOGY
  5. Problem description: empirical risk minimization
  6. Fourier Neural Operator li2020fourier
  7. Creating joint samples
  8. NUMERICAL EXPERIMENTS
  9. Training setup
  10. Results and evaluation metrics
  11. DISCUSSION AND CONCLUSION
  12. ACKNOWLEDGEMENT

Figures (3)

  • Figure 1: (Left) Example of a heterogeneous permeability model, $\mathbf{A}$, with injection location shown as red dot. Adapted from gahlot2024uncertainty (Right) CO$_2$ plume concentration after 1,825 days of injection.
  • Figure 2: (Left) First eigenvector $\mathbf{v}$ of the Fisher Information Matrix (FIM) for a sample permeability model after 1,095 days of CO$_2$ plume evolution. (Right) Corresponding ground-truth vector–Jacobian product, $\mathbf{v}^\top\mathbf{J}$, computed using the reservoir simulator $\mathcal{F}$.
  • Figure 3: Close-up view of the gradient $\mathbf{v}^\top\mathbf{J}$ after 365 days of CO$_2$ injection. The first column shows the ground-truth gradient, the second column shows the gradient predicted by DeFINO, and the third column shows the gradient predicted by the baseline FNO. The relative $\ell_2$-norm misfit of DeFINO's $\mathbf{v}^\top\mathbf{J}$ shown in the middle column is $0.0446$, compared to $0.1470$ for FNO shown in the right column.

Theorems & Definitions (1)

  • Definition 1: Fisher Information Matrix