Mathematical models can be analytically solved, though some cannot be solved and use certain iteration algorithms as Taylor series or Runge Kutta Algorithm. It is not always possible to solve a problem strictly under mathematical formulation. Sometimes, algorithms- sequence of logical steps- may solve more efficiently and more accurate a problem. One should not look down on algorithms. Hilbert Huang Transform Algorithm. The algorithm is able to separate signals with difference phase, frequency, and amplitude. It has revolutionised several sciences as meteorology, hydrology and signal processing as it achieves what conventional Fourier series cannot.

Ultimately what you want is a good explanation (or rather an explanation that is better than the current ones). Sometimes this will be achieved via an algorithm, sometimes via a mathematical model. The criterion here is that the better model or algorithm better "replicates" the real processes. In other words a good model or algorithm is not only one that allows you to predict the end state of your system knowing the initial state, it is one that can be applied as well as possible to the whole process and its mechanisms.

It is not completely clear what you mean by a mathematical model. Mathematicians usually have no problems with infinite structures or non-constructive proofs (that show that something exists, bit do not show you how to find/construct it). Thus a model can be quite useless for algorithms but useful for explaining things. On the other hand, an algorithm always is some kind of model. So the "model" concept is probably wider.

Further, an algorithm might get lost in many technical details, whose implementation is tedious but basically straight-forward. In such a case, an abstract model of the key elements is probably more useful and adequate for a research article.

Of course, model concept is wider even a model can be verbal explanation/description or diagrammatic pointing out interconnections (both +/-) between components of a system. To the best of belief, Pradeep intends to ask whether mathematical formulation of a problem is better or an algorithm in a research paper. Obviously, analytical solution is possible, one should go for mathematical formulation of the problem. But, one should not look down an algorithm as sometimes algorithms are most powerful way to solve a problem where mathematical solution is not possible or as precise as an algorithm can give. It depends on the nature of the problem.

However, "Mathematicians usually have no problems with infinite structures"is underestimation of power of mathematics to model infinite structures.

Hi Mohammad!

In answer to your first paragraph I would say that also an algorithm is a mathematical solution - if its correctness is proven, but otherwise it is not useful anyway.

I do not see how my statement that "mathematicians usually have no problems with infinite structures" underestimates the power of mathematics. i am saying that often it has NO problems with infinite structures, while an algorithm normally cannot accommodate this type of thing.

let me quote the C. O'Neil and R. Schutt book - Doing Data Science: 'In the purest sense, an algorithm is a set of rules or steps to follow to accomplish some task, and a model is an attempt to describe or capture the world'.

and I agree with that:-)

A model and an algorithm are two different things. A model is how we formulate a problem and its instances (in the algorithmic sense), and that can be analyzed for properties. An algorithm is a step-by-step process of well-defined steps that need not be correct unless proven to be correct via a proof of correctness. An algorithm takes in an input instance of a problem instance (of a problem), and produces output. These would be well defined by the model. It depends on the research I find for when papers do this. For example, if this is an Operations Research paper, more emphasis is usually given to the modelling process than the algorithm for solving the problem unless there is a novel approach to solving the problem. Say the paper involves a Linear Program. Experts will know there is a polynomial-time algorithm for that problem, so all they need is the LP for the problem (then just push the method into a statement with citation).

Hi Peter,

You might be knowing well the difference between often and usual.

The problem here is more with the notion of model than with that of algorithm. If a model is "an attempt to describe or capture the world", as Jerzy says, a vague sentence like "Y is a function of X" is a model or at least the program of one. For me, a model has -- or should have -- an explanatory ambition, but i have to admit that it is not always the case. "Model", in empirical science, is a somewhat vague notion that goes from a full-fledged hypothetico-deductive system to a set of laws or rules concerning a system. An example of the former is the "keynesian model", whereas the model of a farm is an example of the latter. Obviously, in order to be precise, a model has to be couched in mathematical terms, hence the notion of "mathematical model". (The models occurring in model theory are altogether different and irrelevant to factual science.)

I do not agree. By a mathematical model one can calculate exactly all about the system, while numerical models are more or less numerical experiments, not necessarily exact. It is usually more difficult to create a correct mathematical model and to solve it analytically, while to do a program is easier. That is why analytical solutions are more valuable.

But, when analytical solution fails numerical analysis, approximation and algorithms are put in service to get an accurate not precise solution.

Yes, I agree. Numerical solution is at least a kind of solution.

Both algorithms and mathematical models deserve credit, as to what proportion the credits should be accorded; it very much depends on the context of the problem one is solving. If your inferences and conclusions very much depend on e.g, the efficiency, robustness or speed, then perhaps the algorithm should be given more weight. Otherwise, the model might deserve more credit. Bottom line? look closely at the message in the paper and judge for yourself which of the two (entire algorithm or specific mathematical model(s)) deserves the most credit. I hope this helps. Good luck!

In context of statistics, a model is usually a parametric explanation of the observations of interest, while an algorithm is usually used for inferring the optimal value of the model's parameter. If several different models are under consideration, an algorithm for model selection or comparison may be also needed.

To me Mathematical model of a system means we try to provide the same output from the given parameters input [27-28]- thus it aims for a best fit, in either subjective or objective approach [29]. Consequently it (the system) is not a close loop relation (no feed back nor feed forward in to the system block). On the other hand, generally algorithm of a system means the system can be a close loop. We can say that a math model of a system is a subset of an algorithm system solution.

[27] validating models, by Michigan State Univ, in ppt

[28] J. P. C. Kleijnen, “Verification and validation of simulation models,” Eur. J. Oper. Res., vol. 82, no. mimic, pp. 145–162, 1995.

[29] R. G. Sargent, “Verification and validation of simulation models,” in Proceedings of the 2010 Winter Simulation Conference, 2010, pp. 166–183.

Both the "algorithm" and the "mathematical model" are models. The first is the representative of the "numerical model" class and the second of the "analytical model" class. All models have 3 qualities (precision, realism, generality) which cannot be maximized simultaneously. Thus, numerical models maximize the precision and realism and sacrifice the generality, while analytical models maximize realism and generality, sacrificing the precision. So, a numerical model allows one to analyze the problem and make predictions only for precise initial and boundary conditions, while analytical models allow the same analysis, but in a more general way, not only for the chosen values of the initial and boundary conditions. Of course, usually an analytical model gives a nonlinear equation, the exact solution of which is most often impossible. In spite of this, there are some perturbation methods (regular or singular) which give an approximate solution of the problem (with a known and controllable error). I hope this answer will help.

Algorithm and mathematical model are different things. While algorithm is a sequence of pre-established steps, the model is a double mapping of the structure of a physical system into a mathematical structure, where the derivation of the results takes place. In the case of solving a problem, which seems to be your case, an algorithm (of solving) may be employed. However, a model is employed only if the problem is a problem of modeling. If it is about a pure-math problem, a mathematical model is not necessary.

What has been mentioned above are all OK but to prioritize, one would say the mathematical model has precedence over the algorithm that later will operate on it. For example, to design a controller for a plant, one would first define the mathematical model first, if it can be described by set of differential/difference equations. Else, the designer may use other tools such as ANN to model the plant using some lab test data. Once the model is well defined, the algorithm will be designed to come up with the best fit controller. It is very crucial to have a real "model" in the first place or else the controller that works fine on paper or in desk-top simulation may fail in actual working environment. Yes the algorithm, which may be called the "control strategy" may overcome some of the issues associated with the modeling such as "uncertainty" or "poorly modeling" of some components of the plant. However, having a good model saves a lot of efforts in the design of the controller.

In Flight Control System design, many of the reported works showed that a robust controller may demonstrate great performance and stability margin on desk-top simulations but may fail during flight test mainly because it has been based on "low fidelity" mathematical model of the aircraft. Others, showed successful results mainly as they have used "high fidelity" mathematical models which are normally nonlinear and takes into considerations all the components of the aircraft including the sensors, the actuators, and the operating environment.

mathematical model is use to have a proper solution to the proposed problem while algorithms is to have a sequence of steps that may include some mathematical model.