# When is gray-box modeling advantageous for virtual flow metering?

arXiv: https://arxiv.org/abs/2110.05034 · HTML: https://mlsanity.com/paper/2110.05034/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 11 Oct 2021.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [Developing a Hybrid Data-Driven, Mechanistic Virtual Flow Meter -- a Case Study](https://mlsanity.com/paper/2002.02737/) (2020)
- [Bayesian Neural Networks for Virtual Flow Metering: An Empirical Study](https://mlsanity.com/paper/2102.01391/) (2021)
- [On gray-box modeling for virtual flow metering](https://mlsanity.com/paper/2103.12513/) (2021)
- [Identifiability and physical interpretability of hybrid, gray-box models -- a case study](https://mlsanity.com/paper/2010.13416/) (2020)

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

- [WISE-FM:Operation-Aware, Engineering-Informed Foundation Model for Multi-Task Well Design](https://mlsanity.com/paper/2604.23767/) (2026)

## Abstract (arXiv)

Integration of physics and machine learning in virtual flow metering applications is known as gray-box modeling. The combination is believed to enhance multiphase flow rate predictions. However, the superiority of gray-box models is yet to be demonstrated in the literature. This article examines scenarios where a gray-box model is expected to outperform physics-based and data-driven models. The experiments are conducted with synthetic data where properties of the underlying data generating process are known and controlled. The results show that a gray-box model yields increased prediction accuracy over a physics-based model in the presence of process-model mismatch. They also show improvements over a data-driven model when the amount of available data is small. On the other hand, gray-box and data-driven models are similarly influenced by noisy measurements. Lastly, the results indicate that a gray-box approach may be advantageous in nonstationary process conditions. Unfortunately, choosing the best model prior to training is challenging, and overhead on model development is unavoidable.