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I have a camera system with some special optics that warp the field of view of the camera, dependent on two variables, $\theta_1$ and $\theta_2$. Given a specific configuration of these two variables, each pixel on my camera (which is 500x600 resolution) will see a specific coordinate on a screen in front of the camera. I can calculate this for each pixel, but it requires too many computations and is too slow. So, I want to learn a model that fits this function, but computes much faster.

I have plenty of input/output data that I have generated, mapping the 500x600 input points to the 500x600 output points for different $\theta_i$ values, and I have already used some 2D polynomial least squares regression to learn these functions. They perform adequately, but I was wondering if a neural net could be used to learn a better function.

My question comes down to this: can a neural network learn what basically amounts to a regression problem trying to learn $f_{\theta_1,\theta_2}(px_1,px_2)=(a_1,b_1)$?

I know that neural nets excel in classification problems, which this is not, but I have also heard that a neural network can "learn arbitrary functions."

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Yes - neural networks can be used to predict regression problems. The output layer will be a set of nodes that make continuous numeric predictions.

In your problem, it would be two nodes that learn the weights for $\theta_1$ and $\theta_2$.

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Yes it is possible. You just need to define/select a neural network with two neurons(for θ1 and θ2) in the last layer and define a custom loss function which can include both of these neuron values.

The classification problems of neural networks tend to select the correct class by using something like argmax in the last layer.

Both the classification and regression neural networks will get trained in the same way.

For binary classification models we can use single neuron at the last layer just like regression problems.

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