On 28/09/2019 14:27, Oscar Benjamin wrote:
Neural nets are trained for a particular statistical distribution of
inputs and in the paper they describe their method for generating a
particular ensemble of possibilities. There might be something
inherent about the examples they give that means they are all solved
using a particular approach. From their description I could imagine
writing a pattern-matching integrator that would explicitly try to
reverse the way the examples are generated.
Perhaps the examples from e.g. the SymPy test suite would in some ways
represent a more "natural" distribution since they are written by
humans and show the kinds of problems that humans wanted to solve. It
would be interesting to see how the accuracy of the net looks on that
distribution of inputs (although any comparison with SymPy on that
data would be unfair).
I suppose a fair test might be to take a set of integrals from
Abramowitz and Stegun, but of course, for indefinite integrals the
accuracy should be 100% because the output can be checked by
differentiation.
Sadly the paper mainly focuses on the generation of the test sets,
because the rest is inevitably completely opaque!
I noticed this sentence:
"Unfortunately, functions do not have an integral which can be expressed
with usual functions (e.g.f(x) = exp(x2) or f(x) = log(log(x))), and
solutions to arbitrary differential equations cannot always be expressed
with usual functions."
This suggests that their integrator will not handle anything that
resolves to a higher transcendental functions - e.g. elliptic integrals.
I guess this is a rather special case of a significant maths problem
that is hard in one direction but easy to check, and where pattern
matching can be used extensively. It would be sad if one day the whole
of SymPy were replaced with an opaque program like this - but I don't
think that is likely.
David
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