The Role of Data in Model Building and Prediction: A Survey Through Examples

Abstract

The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play-and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components. arXiv:1810.10446v1 [cond-mat.stat-mech] 24 Oct 2018The ideal formal model would be one which would cover the entire universe, which would agree with it in complexity, and which would have a one to one correspondence with it. Any one capable of elaborating and comprehending such a model in its entirety, would find the model unnecessary, because he could then grasp the universe directly as a whole. He would possess the third category of knowledge described by Spinoza. This ideal theoretical model cannot probably be achieved. Partial models, imperfect as they may be, are the only means for understanding the universe. This statement does not imply an attitude of defeatism but the recognition that the main tool of science is the human mind and that the human mind is finite.It is pretty impossible to detail the different procedures which had been followed to build the many models used in science, also because it can be safely claimed that there are not systematic protocols for model building. Possibly a lucky non-trivial exception is given by the framework of classical physics where a fairly clear approach exists. Once the forces are understood, one can write down proper differential equations that may be difficult to solve, but that always allow us to obtain reliable and useful results, for instance by means of qualitative or numerical analysis. Another example is when a reduced model is derived from a more general theory, for instance this the case of the Lorenz's model [4], which is obtained from a rather crude simplification of fluid dynamics equations. Such a model, in spite of its (apparent) simplicity, played a key role by allowing us to realize that many irregular motions encountered in nature can be due to chaos. This is a clear example of how even a "partial model" can help for understanding natural phenomena.The above approach is decidedly not applicable t...

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