Modern numerical techniques, typicalized by finite difference methods and neural networks, only require the equations of motion and initial conditions to iteratively solve most of the physical problems. No analytical analysis is required in this whole process. These numerical methods discretalize the whole space and determine each grid value iteratively using simple mathematical operations such as “plus”, “minus”, “multiplication”, and “division”. This is made possible due to the drastic boost of computer performance in recent decades or so.
The traditional analytical methods, that well-known functions are used to derive and express the solution of a physical question, seem dying. While analytical methods are powerful in terms of computation resources (you only need to determine some well-known function values), the problems can be solved by them are generally limited, because of limited known-function. In fact, most complex problems cannot be solved analytically.
There is, however, an incompensible disadvantage in numerical methods. That is there are infinite many physical problems of a single kind. For example, just change one parameter of the problem, numerical methods have to redo every calculation. This set the parameter search in a discreate and infinite design space, thus it’s very hard to find the global optimized solution. This is not a problem in analytical analysis, because we just need to find the zero point of the derivative.
To overcome the limitations of analytical methods, we call on that new analytical functions should be created by exploring numerical methods. This uses the numerical methods as a tool for the purpose of expanding the current scope of analytical methods, and the analytical functions are like permanent memory.
Rui & Photonics 