Regarding the consistency of convolutional neural networks
I am currently building a 2-channel (also called double-channel) convolutional neural network in order to classify 2 binary images (containing binary objects) as 'similar' or 'different'. 

The problem I am having is that it seems as though the network doesn't always converge to the same solution. For example, I can use exactly the same ordering of training pairs and all the same parameters and so forth, and when I run the network multiple times, each time produces a different solution; sometimes converging to below 2% error rates, and other times I get 50% error rates. 

I have a feeling that it has something to do with the random initialization of the weights of the network, which results in different optimization paths each time the network is executed. This issue even occurs when I use SGD with momentum, so I don't really know how to 'force' the network to converge to the same solution (global optima) every time?

Can this have something to do with the fact that I am using binary images instead of grey-scale or color images, or is there something intrinsic to neural networks that is causing this issue?

There are several sources of randomness in training. 
Initialization is one. SGD itself is of course stochastic since the content of the minibatches is often random. Sometimes, layers like dropout are inherently random too. The only way to ensure getting identical results is to fix the random seed for all of them.

Given all these sources of randomness and a model with many millions of parameters, your quote 


  "I don't really know how to 'force' the network to converge to the same solution (global optima) every time?"


is something pretty much something anyone should say - no one knows how to find the same solution every time, or even a local optima, let alone the global optima.

Nevertheless, ideally, it is desirable to have the network perform similarly across training attempts (with fixed hyper-parameters and dataset). Anything else is going to cause problems in reproducibility, of course. 

Unfortunately, I suspect the problem is inherent to CNNs. 
You may be aware of the bias-variance tradeoff. For a powerful model like a CNN, the bias is likely to be low, but the variance very high. In other words, CNNs are sensitive to data noise, initialization, and hyper-parameters. Hence, it's not so surprising that training the same model multiple times yields very different results. (I also get this phenomenon, with performances changing between training runs by as much as 30% in one project I did.) My main suggestion to reduce this is stronger regularization.


  Can this have something to do with the fact that I am using binary images instead of grey-scale or color images, or is there something intrinsic to neural networks that is causing this issue?


As I mentioned, this problem is present inherently for deep models to an extent. However, your use of binary images may also be a factor, since the space of the data itself is rather discontinuous. Perhaps consider "softening" the input (e.g. filtering the inputs) and using data augmentation. A similar approach is known to help in label smoothing, for example. 
