Data Science and Engineering Analytics Technical Section

I have read the paper behind this July '22 JPT paper summary:

Physics-Informed Neural Networks Help Predict Fluid Flow in Porous Media, page 52

The paper on which this summary is based is this:

SPE 203033,

Prediction of Fluid Flow in Porous Media Using Physics Informed Neural Networks, by Muhammad M. Almajid, SPE, and Moataz 0. Abu-Alsaud, SPE, Saudi Aramco, 2020

An important reference in this SPE paper 203033 that explains the mechanics behind the application is this:

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, by Raissia et al, 2019

https://www.sciencedirect.com/science/article/pii/S0021999118307125

This last paper explains that the informed physics part of the system is in two parts: a listing of input data space points or vectors, and a partial differential equation used to explain/predict the behavior of the process generating the input data space points or vectors:

u(t,x) + N[u] = 0

where u(t,x) are the data points or vectors generated by the true system "u", and N is a partial derivative function acting on an equation-of-state representation of "u". A neural network is used to "encode" or "learn" the u(t,x) data space, after which u(t,x) in the equation above is replace with the neural network.

This above development is essentially a special case of the hybrid system (not so named at the time) published in the 1989 SPWLA London LASER symposium:

Simultaneous Global Optimization of Unknowns by Simulated Annealing in Neural Networks

I say that the 2019 paper is essentially a special case of the 1989 paper because the function leaned by the neural network in the LASER paper was a workflow process and not a partial differential equation-of-state as was used in the 2019 paper. The system presented by the LASER paper can be applied to any function producing data used for neural network training.

A limitation of this sort of physics informed neural network is in how representative are the input data and the data-generating function at the time of neural net training, and then later at the time of application. I.e., (1) the function may be limited in structure or parameterization, and (2) the data space input to the neural network training process may not adequately cover the space to be encountered in the future. That the partial differential equation approach offers bounds on computed output values is again a special case of how the LASER paper handled bounds (e.g., all volumes must sum to 100%).

I say this just as a cautionary note.