The deep learning model is like a black box that can predict any function given enough data and enough computation power is available and it makes it more powerful in many problems where predicting mathematical equation is difficult. The more the data, the more accurate the prediction is. There are certain features of deep learning models that make them extremely powerful in many complex problems.
Deep learning can predict any function
The mathematical model generally tries to fit the data with some mathematical equation. It is excellent for a problem that has a precise mathematical form. But it becomes difficult to estimate the form of the mathematical equation when the problem becomes complex. Sometimes proper mathematical form does not exist and then the mathematical form is predicted only in a small range where a particular mathematical equation is valid. For the other region, another mathematical form is predicted. This makes mathematical model complex cumbersome. Deep learning model work as a black box that can predict any function that can define the output of the problem for a particular set of inputs. For a deep learning model to predict the robust function one needs to only have enough data and enough computation power. You may not know the functional form of the solution of the problem, but you have the solution is a black box and that black box will give the proper output for the given input.
Deep learning break complex features in term's of simple features
When one tries to detect an object mathematically in an image they try to find edges using an edge filter. There are more sophisticated mathematical models also that goes beyond detecting edges, but the more ore less try to identify the feature. Deep learning models on the other hand can build complex feature detections based on simple feature detection. For example, a deep convolutional neural network to detect an object (say human) consists of many layers. The initial layers try to find simple features such as edges from an input image. The next few layers try to detect more complex features such as a circle or rectangular or other slightly more complex features based on the previous simple features. The next few layers detect more complex features such as eyes nose etc in an image and that way the final layer is able to detect a human in an input image. It is very difficult on the other hand to represent a human image mathematically. The images below shows the similar things considering car as input.
Deep learning models not only see the near future but also infer based on the far future
For the problems where a model or agent need to take decision-based on the previous set of events, the machine learning models perform extremely well. Generally, this type of problem reinforcement learning is used. The mathematical model tries to take decision based on previous or previous few inputs and it tries to make a decision so that in the near future the model performs better. But sometimes the decision which might be costly in the near future can bring a good amount of benefit for the far future. The reinforcement learning models take the decision not only based on the near future benefits but also consider the far future effects of the decision taken at the present time. That's why reinforcement learning is extremely good at playing games and own over many games such as chess, Dota over human experts.
Where the mathematical model is more powerful?
Though the deep learning model is extremely powerful there are certain areas where mathematical models are more robust and perform better than a deep learning model
1. Where enough data is not available
2. The problems whose solution has precise mathematical form, for example, particle under gravity, an object in some central force field etc.