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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
When I skimmed through the paper, I had the following queries:1. Is integration really a tree to tree translation? Because, neural network is predicting the resulting expression tree for the input equation. However, integration is not a predictive operation. Moreover, how can we define that whether the model is over-fitting or not on the training data? What was the size of the training data set?
2. Does the data set of equations contains noise? Is there any uncertainty in the data set? For example, while translating from one language to another there are chances that one word can be mapped to different ones having the same meaning. However, here, it is not the case, there may be multiple results but we can check whether they are correct or not with 100% surety.
3. Is this model able to generalise over any mathematical expression. The way they generated data sets is algorithmic and deterministic. It is not random(random number generators are itself deterministic). So, how can we say that this model is the one that outperforms any CAS?
Neural Networks don't learn, the way human beings do. They just imitate the underlying distribution of the data. But, I don't think that mathematical expressions have any such underlying distribution. Well, we can take a subset of those expressions which can have a distribution and I think that's the case with their model.
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