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As the comments to your question point out, there are a lot of people working on finding something better. I would though like to answer this question by expanding the comment left by @josh

All models are wrong but some are useful (Wiki)

The above statement is a general truth used to describe the nature of statistical models. Using data that we have available, we can create models that let us do useful things such as approximate a predicted value.

Take for example Linear Regression

Using a number of observations, we can fit a model to give us an approximate value for a dependent variable given any value(s) for the independent variable(s).

Burnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel > Inference: A Practical Information-Theoretic Approach (2nd ed.): "A model is a simplification or approximation of reality and hence will not reflect all of reality. ... Box noted that “all models are wrong, but some are useful.” While a model can never be “truth,” a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless."

Deviations from our model (as can be seen in the image above) appear random, some observations are below the line and some are above, but our regression line shows a general correlation. Whilst deviations in our model appear random, in realistic scenarios there will be other factors at play which cause this deviation. For example, imagine watching cars as they drove through a junction where they must turn either left or right to continue, the cars turn in no particular pattern. Whilst we could say that the direction the cars turn is completely random, does every driver reach the junction and at that point make a random decision of which way to turn? In reality they are probably heading somewhere specific for a specific reason, and without attempting to stop each car to ask them about their reasoning, we can only describe their actions as random.

Where we are able to fit a model with minimal deviation, how certain can we be that an unknown, unnoticed or immeasurable variable wont at some point throw our model? Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

The problem with using the Linear and SVN models you mention alone is that we are somewhat required to manually observe our variables and how they each affect each other. We then need to decide what variables are important and write a task-specific algorithm. This can be straight forward if we only have a few variables, but what if we had thousands? What if we wanted to create a generalised image recognition model, could this realistically be achieved with this approach?

Deep Learning and Artificial Neural Networks (ANNs) can help us create useful models for huge data sets containing huge amounts of variables (e.g. image libraries). As you mention, there's an incomprehensible number of solutions which could fit the data using ANNs, but is this number really any different to the amount of solutions we would need to develop ourselves through trial and error?

The application of ANNs do much of the work for us, we can specify our inputs and our desired outputs (and tweak them later to make improvements) and leave it up to the ANN to figure out the solution. This is why ANNs are often described as "black boxes". From a given input they output an approximation, however (in general terms) these approximations don't include details on how they were approximated.

And so it really comes down to what problem you are trying to solve, as the problem will dictate what model approach is more useful. Models are not absolutely accurate and so there is always an element of being 'wrong', however the more accurate your results the more useful they are. Having more detail in the results on how the approximation was made may also be useful, depending on the problem it may even be more useful than increased accuracy.

If for example you are calculating a persons credit score, using regression and SVMs provides calculations that can be better explored. Being able to both tweak the model directly and explain to customers the effect separate independent variables have on their overall score is very useful. An ANN may aid in processing larger amounts of variables to achieve a more accurate score, but would this accuracy be more useful?