# Reservoir History Matching of the Norne field with generative exotic priors and a coupled Mixture of Experts -- Physics Informed Neural Operator Forward Model

arXiv: https://arxiv.org/abs/2406.00889 · HTML: https://mlsanity.com/paper/2406.00889/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 18 of its 64 references that are in the map.
- First posted on arXiv 2 Jun 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Toward Multimodal Image-to-Image Translation](https://mlsanity.com/paper/1711.11586/) (2017)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Understanding Probabilistic Sparse Gaussian Process Approximations](https://mlsanity.com/paper/1606.04820/) (2016)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [Deep Structured Mixtures of Gaussian Processes](https://mlsanity.com/paper/1910.04536/) (2019)
- [Cluster, Classify, Regress: A General Method For Learning Discountinous Functions](https://mlsanity.com/paper/1905.06220/) (2019)
- [A Unifying Framework for Gaussian Process Pseudo-Point Approximations using Power Expectation Propagation](https://mlsanity.com/paper/1605.07066/) (2016)
- [Prediction of Porosity and Permeability Alteration based on Machine Learning Algorithms](https://mlsanity.com/paper/1902.06525/) (2019)
- [The regularizing Levenberg-Marquardt scheme for history matching of petroleum reservoirs,](https://mlsanity.com/paper/1302.3501/) (2013)
- [4D Seismic History Matching Incorporating Unsupervised Learning](https://mlsanity.com/paper/1905.07469/) (2019)
- [A Novel A.I Enhanced Reservoir Characterization with a Combined Mixture of Experts -- NVIDIA Modulus based Physics Informed Neural Operator Forward Model](https://mlsanity.com/paper/2404.14447/) (2024)

## Abstract (arXiv)

We developed a novel reservoir characterization workflow that addresses reservoir history matching by coupling a physics-informed neural operator (PINO) forward model with a mixture of experts' approach, termed cluster classify regress (CCR). The inverse modelling is achieved via an adaptive Regularized Ensemble Kalman inversion (aREKI) method, ideal for rapid inverse uncertainty quantification during history matching. We parametrize unknown permeability and porosity fields for non-Gaussian posterior measures using a variational convolution autoencoder and a denoising diffusion implicit model (DDIM) exotic priors. The CCR works as a supervised model with the PINO surrogate to replicate nonlinear Peaceman well equations. The CCR's flexibility allows any independent machine-learning algorithm for each stage. The PINO reservoir surrogate's loss function is derived from supervised data loss and losses from the initial conditions and residual of the governing black oil PDE. The PINO-CCR surrogate outputs pressure, water, and gas saturations, along with oil, water, and gas production rates. The methodology was compared to a standard numerical black oil simulator for a waterflooding case on the Norne field, showing similar outputs. This PINO-CCR surrogate was then used in the aREKI history matching workflow, successfully recovering the unknown permeability, porosity and fault multiplier, with simulations up to 6000 times faster than conventional methods. Training the PINO-CCR surrogate on an NVIDIA H100 with 80G memory takes about 5 hours for 100 samples of the Norne field. This workflow is suitable for ensemble-based approaches, where posterior density sampling, given an expensive likelihood evaluation, is desirable for uncertainty quantification.