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At work there is an idea of solving a problem with machine learning. I was assigned the task to have a look at this, since I'm quite good at both mathematics and programming. But I'm new to machine learning.

In the problem a box would be discretized into smaller boxes (e.g. $100 \times 100 \times 100$ or even more), which I will call 'cells'. Input data would then be a boolean for each cell, and output data would be a float for each cell. Thus both input and output have dimensions of order $10^6$ to $10^9$.

Do you have any recommendations about how to do this? I guess that it should be done with a ConvNet since the output depends on relations between close cells.

I have concerns about the huge dimensions, especially as our training data is not at all that large, but at most contains a few thousands of samples.


Motivation

It can be a bit sensitive to reveal information from a company, but since this is a common problem in computational fluid dynamics (CFD) and we already have a good solution, it might not be that sensitive.

The big boxes are virtual wind tunnels, the small boxes ('cells' or voxels) are a discretization of the tunnel. The input tells where a model is located and the output would give information about where the cells of a volume mesh need to be smaller.

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    $\begingroup$ Hello. Welcome to Artificial Intelligence Stack Exchange. It might be a good idea to describe your problem at a higher level. What are these boxes? What do the cells represent? What do the floating-point values associated with each cell (I suppose these are your "labels") represent? $\endgroup$ – nbro Mar 25 at 10:49
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    $\begingroup$ Thanks, @nbro. It can be a bit sensitive to reveal information from a company, but since this is a common problem in computational fluid dynamics and we already have a good solution, it might not that sensitive. The big boxes are virtual wind tunnels, the small boxes ('cells' or voxels) are a discretization of the tunnel. The input tells where a model is located and the output would give information about where the cells of a volume mesh need to be smaller. $\endgroup$ – md2perpe Mar 25 at 12:05
  • $\begingroup$ @md2perpe if the model exists, is it confined exclusively to one cell? If not maybe you could do some generalisation of the inputs/outputs based on whether the model exists within an area of space. Its approximately what the CNN attempts to do (smoothing + sub-sampling) but by doing it manually instead imposing your external knowledge of the system it could help reduce dimensionality exponentially whilst also speeding up learning $\endgroup$ – quest ions Mar 28 at 12:49
  • $\begingroup$ @questions. The model will exist; without it the task is meaningless. The model will span a lot of cells, but we're looking for numbers in the cells outside of the model. The model will probably be placed in the middle of the space, with some margin around (like this but in 3 dimensions and with higher resolution). $\endgroup$ – md2perpe Mar 28 at 15:42
  • $\begingroup$ @md2perpe by "numbers in the cells outside of the model" i assume you mean outputs where the input is 0. If the number of cells the model occupies is deterministic (w/ the margin) maybe there exists some parameterisation that can approximately describe the number of cells which are 1. A motivating idea is how one describes 1D gaussians by their mean and standard deviation instead of trying to find how each individual input in $\mathbb{R}$ is mapped. $\endgroup$ – quest ions Mar 28 at 22:14

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