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As a biologist with some background in mathematics I am really interested in mathematical modelling of biological processes, like tumor development or therapy responses. However, looking at recent publications especially machine-learning algorithms seem to very popular. As a matter of fact, I don't really understand the difference between machine learning and mathematic modelling. Which one is superior? As I see, machine learning algorithms can be used even by people without big knowledge in maths as long as they can code on a basic level. Mathematic modelling seems to be harder... What are the exact differences?

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    $\begingroup$ I would encourage you to edit the question by defining your notion of “superior”, which can help you to get an answer matching the question you intended. You seem to refer to ease of implementation, and the fact that ML can be used by people without an understanding of it. (So can a hacksaw, and I wouldn’t advise either…) $\endgroup$ May 11 at 14:37
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    $\begingroup$ Separately, you ask for definitions/boundaries of each. That’s a separate question that’s been asked before on this site, so hopefully the existing answers are illuminating. $\endgroup$ May 11 at 14:38
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    $\begingroup$ It all depends on what one wants to extract from the model. Explanation? Prediction? Validation? $\endgroup$
    – Xi'an
    May 11 at 14:54

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Which one is superior?

Neither is superior to the other. There are cases where one is preferable to another.

I will take what you mean by "mathematical modelling" to mean differential equations and the like, which are often used in the modelling of biological processes. ODE/PDE models are mechanistic in so far as we have an idea of how the system changes and we need only estimate the relevant parameters. Simple biological processes (e.g. population growth limited to some carrying capacity, the spread of an epidemic in a closed and homogeneous population, etc) are very well suited for mathematical modelling of this nature.

Machine learning is better suited when we don't have an idea of how the system evolves, but we do have relevant data about the system at the time of observation. Our approach here is really descriptive, rather than mechanistic, often opting to make $E(y \vert X)$ our prediction.

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Mathematical modelling appears in a lot different fields, especially statistics that are just a way of making models using empirical data or introduce randomness. The is no "superiority", since both field are intricated.

When it comes to machine learning (or statistics, this is the same), some models are more easily interpreted than others.

For example the simplest model: linear regression. It is easily interpreted because the physical meaning of the coefficients can be identified. However, this means that the observed phenomenon must be able to be modelled by a linear law, with one transformation, for example a power law is nothing other than a linear law in a log-log graph. If the phenomenon is "simple", you can probably come up with a theoretical model and then validate it on data using machine learning/statistics , good!

On the other hand, if you are studying an extremely complex phenomenon, the modelling task becomes much more difficult. This is where deep-learning models (or other very complex models that work like a black box) can outperform "classical" hand-made models if you have enough data, since the model itself will be learned. But in general, these models will be difficult to interpret. This can be a bit discouraging in a way...

Finally, the fact that machine learning is "mainstream" (which is not really the case, but there is a bit of a hype around it indeed) comes from the fact that there are libraries that have already implemented many algorithms for you. But if you're not able to understand what you're doing I don't think anyone will accept your work...

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