# Solving multiphysics-based inverse problems with learned surrogates and constraints

arXiv: https://arxiv.org/abs/2307.11099 · HTML: https://mlsanity.com/paper/2307.11099/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 113 references that are in the map.
- First posted on arXiv 18 Jul 2023.

## Builds on (most cited first, 30 of 34)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Tackling Climate Change with Machine Learning](https://mlsanity.com/paper/1906.05433/) (2019)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Real-time high-resolution CO2 geological storage prediction using nested Fourier neural operators](https://mlsanity.com/paper/2210.17051/) (2022)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Active learning of deep surrogates for PDEs: Application to metasurface design](https://mlsanity.com/paper/2008.12649/) (2020)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Total-variation regularization strategies in full-waveform inversion](https://mlsanity.com/paper/1608.06159/) (2016)
- [Fully Hyperbolic Convolutional Neural Networks](https://mlsanity.com/paper/1905.10484/) (2019)
- [Model-Parallel Fourier Neural Operators as Learned Surrogates for Large-Scale Parametric PDEs](https://mlsanity.com/paper/2204.01205/) (2022)
- [Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks](https://mlsanity.com/paper/2101.03906/) (2021)
- [Learning by example: fast reliability-aware seismic imaging with normalizing flows](https://mlsanity.com/paper/2104.06255/) (2021)
- [A Conditional Normalizing Flow for Accelerated Multi-Coil MR Imaging](https://mlsanity.com/paper/2306.01630/) (2023)
- [Amortized Normalizing Flows for Transcranial Ultrasound with Uncertainty Quantification](https://mlsanity.com/paper/2303.03478/) (2023)
- [Learned coupled inversion for carbon sequestration monitoring and forecasting with Fourier neural operators](https://mlsanity.com/paper/2203.14396/) (2022)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [De-risking geological carbon storage from high resolution time-lapse seismic to explainable leakage detection](https://mlsanity.com/paper/2211.03527/) (2022)
- [DeepFlow: History Matching in the Space of Deep Generative Models](https://mlsanity.com/paper/1905.05749/) (2019)

## Built on by (most cited first, 9 of 9)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Uncertainty Quantification of Surrogate Models using Conformal Prediction](https://mlsanity.com/paper/2408.09881/) (2024)
- [Inverse Design in Nanophotonics via Representation Learning](https://mlsanity.com/paper/2507.00546/) (2025)
- [InvertibleNetworks.jl: A Julia package for scalable normalizing flows](https://mlsanity.com/paper/2312.13480/) (2023)
- [Inference of CO2 flow patterns -- a feasibility study](https://mlsanity.com/paper/2311.00290/) (2023)
- [Physics-informed Deep Learning to Solve Three-dimensional Terzaghi Consolidation Equation: Forward and Inverse Problems](https://mlsanity.com/paper/2401.05439/) (2024)
- [A reduced-order derivative-informed neural operator for subsurface fluid-flow](https://mlsanity.com/paper/2509.13620/) (2025)
- [Neural surrogates for designing gravitational wave detectors](https://mlsanity.com/paper/2511.19364/) (2025)

## Abstract (arXiv)

Solving multiphysics-based inverse problems for geological carbon storage monitoring can be challenging when multimodal time-lapse data are expensive to collect and costly to simulate numerically. We overcome these challenges by combining computationally cheap learned surrogates with learned constraints. Not only does this combination lead to vastly improved inversions for the important fluid-flow property, permeability, it also provides a natural platform for inverting multimodal data including well measurements and active-source time-lapse seismic data. By adding a learned constraint, we arrive at a computationally feasible inversion approach that remains accurate. This is accomplished by including a trained deep neural network, known as a normalizing flow, which forces the model iterates to remain in-distribution, thereby safeguarding the accuracy of trained Fourier neural operators that act as surrogates for the computationally expensive multiphase flow simulations involving partial differential equation solves. By means of carefully selected experiments, centered around the problem of geological carbon storage, we demonstrate the efficacy of the proposed constrained optimization method on two different data modalities, namely time-lapse well and time-lapse seismic data. While permeability inversions from both these two modalities have their pluses and minuses, their joint inversion benefits from either, yielding valuable superior permeability inversions and CO2 plume predictions near, and far away, from the monitoring wells.