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As a (perhaps contrived) example, let's say we want to discover from some empirical data Coulomb's law for an electric field:

$$F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{|q|}{r^2}$$

In this case we would have a matrix whose columns include force $F$, charge $q$, and radius $r$; as well as perhaps some columns containing irrelevant features. We also know in advance the values of $\epsilon_0$ and $\pi$.

Supplying that matrix as input to some out-of-box machine learning tools may indeed have us find that

$$F = c \cdot \frac{|q|}{r^2}$$

for some $c$ that is close to but probably not exactly $(4 \pi \epsilon_0)^{-1}$.

My question is: how do we inform such systems to make use of the known constants so that it may more closely approximate the true form of the equation? Would it useful to include, for example, an entire column containing only $\pi$ and another entire column containing only $\epsilon_0$?

Edit: It may be the case here that $\epsilon_0$ is the interesting quantity to be discovered (in addition to the form of the equation). However, the question still stands regarding how to make use of $\pi$.

rhombidodecahedron
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    There are many simple solutions. Consider what would occur when you create a matrix with columns of $F$, $|q|/(4\pi\epsilon_0)$, and $r^2$, for instance. But that begs a more important question: your machine learner is unlikely to find that $F=c|q|^1 r^2$; it might find that $F=c|q|^{0.97} r^{2.001},$ say. What then would it mean to force $c=1/(4\pi\epsilon_0)$? Not much, it would seem. – whuber Jun 17 '14 at 15:45

1 Answers1

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Putting in known constants won't help with the form of the equation at all.

Consider the underlying equation $F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{|q|}{r^2}$

and further assume that we know $\pi$ and $\epsilon_0$.

If we don't know the powers involved (i.e. we know the variables and guess they enter as powers,, but not the exact form of the equation), we might try to estimate

$F = k \frac{1}{\pi \epsilon_0} \cdot \frac{|q|^a}{r^b}$

or equivalently

$F^* =k \frac{|q|^a}{r^b}$

where $F^*= \pi \epsilon_0 \cdot F$

-- but this doesn't especially help us pin down the value $\frac{1}{4}$ if we haven't concluded that the estimates of $a$ and $b$ indicate we should use a model where $a= 1$ and $b=2$.

If we do know the functional form (via dimensional analysis say) and only need estimate the constant of proportionality, then we may apply the same kind of manipulation

$F^* =k \frac{|q|}{r^2}$

and so get an estimate of something that hopefully is close to $\frac{1}{4}$ (and for which presumably a CI should have a good chance of including that value)

But this hasn't really saved us anything; we could as readily fit:

$F = k_2 \frac{|q|}{r^2}$

and proceed to scale the estimate of the constant (and the bounds on the CI) by the known scale factor $\pi \epsilon_0$ and come to the same outcome.

In that sense, knowing all but one constant may help with finding the value of the remaining one, but it doesn't really impact the modelling.

Glen_b
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