The Peng-Robinson equation of state, for example, has no underlying physical meaning, and is just a model that was fit to data. Computer scientists have developed much, much better models for data than messing around with polynomials and exponentials, like neural networks. Why can't standards organizations take a bunch of data about the chemical, fit a neural network or whatever to it, and publish that as the "EOS", instead of coming up with a meaningless equation that almost certainly wouldn't do as well?

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    $\begingroup$ If you have a hammer, they say, every problem looks like a nail. All these equations of state were proposed long before the current progress in machine learning. Also, they can be written on paper, which was an advantage in those days. $\endgroup$ May 18, 2016 at 5:51
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    $\begingroup$ Furthermore, having an actual equation in front of you lets one do thought experiments that help you see what the equation is saying. Plopping down a USB drive and saying there is a neural network model on it is not revealing to a human at all. $\endgroup$
    – Jon Custer
    Jun 8, 2016 at 14:26
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    $\begingroup$ There is no qualitative difference between neural networks and "messing around with polynomials and exponentials". It is only a matter of taste and availability of data what kind of data model you put on your data points. $\endgroup$
    – Greg
    Oct 7, 2019 at 3:00

3 Answers 3


It is true that a high order polynomial can fit any training set. But that is not a strength - an unfalsifiable model overfits. In particular, a polynomial of order n is only likely to be predictive if the true function is n times differentiable. Since chemical space is discrete, and for many purposes some molecules are special cases, polynomial models are a poor choice in cheminformatics.

Neural nets can work, with an explicit regularizer or with dropout. It's true that the resulting model is a black box - after building it, there is the challenge of understanding the predictions. However, running the model to make predictions is cheaper than doing a lot of experiments. After gaining some understanding of the model you can do more targeted experiments to check it against reality.

With some problems, we have had good results with a fingerprint-based SVM.


For n data points there exists an n-1 order polynomial that perfectly fits the data.

Therefore there is no basis for "a neural network or whatever" being better.

Furthermore, it simply isn't true that the Peng-Robinson equation "has no underlying physical meaning".

The Peng-Robinson equation (like Van der Waals) recognizes that atoms/molecules occupy space and that attractive forces exist between them.

As Peng and Robinson say in their article:

Since two-constant equations have their inherent limitations, and the equation obtained in this study is no exception, the justification for the new equation is the compromise of its simplicity and accuracy.

That being said, neural networks have been used to find equations of states. See for example Equation of state and artificial neural network to predict the thermodynamic properties of pure and mixture of liquid alkali metals

Generally, ANN is powerful and successful method for complex non-linear systems due to unique advantages such as high speed, simplicity and large capacity which reduce engineering attempt. In recent years, ANN modeling has been successfully used for prediction of thermophysical properties of pure and mixture fluids [24], [25], [26] and [27].


I'm currently working with a R410a Equation of State and I found ML useful in the following cases:

  1. Getting a formula from a data set predicted by a simulator software. (Software companies knows this and are getting harder and harder to download large samples)

  2. Multilinear regressions when you have a table with data but you don't have a formula. (refrigerants , water, air, etc).

  3. Classification when you have to use different equations depending on the zone (gas, liquid, mixture, crystallization form, etc).

You will hear a lot about overfitting of neural networks, this is not a problem of neural networks, is a problem of low data density. When you have only a few of inputs in your model, there is not so hard to have a dataset big enough to easily avoid overfitting.


  • If you only have data and more than one input variables is the easy way.
  • You can perform several trainings changing the input parameters, so you don't have to iterate later.


  • Equations of state HAVE (a vague) underlying physical meaning.
  • It's harder to generalize for different gases and compositions. So, it's also hard to be seen in a chemical simulation software.
  • You can't write the trained model in a book.
  • It's harder to check if the model has a good behavior.
  • When there is a small zone with a big nonlinearity, your neural network will tend to ignore that zone. For example, it's a personal example, if you train a model to predict the Methane Number of natural gas without inerts. You will need to add A LOT of data of methane numbers near 100 to reach good predictions in that zone.

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