High-Dimensional Neural Network Potentials for Atomistic Simulations

Abstract

The interaction between syntax (formal language) and its semantics (meanings of language) is one which has been well studied in categorical logic. The results of this particular study are employed to understand how the brain is able to create meanings. To emphasize the toy character of the proposed model, we prefer to speak of the homunculus brain rather than the brain per se. The homunculus brain consists of neurons, each of which is modeled by a category, and axons between neurons, which are modeled by functors between the corresponding neuron-categories. Each neuron (category) has its own program enabling its working, i.e. a theory of this neuron. In analogy to what is known from categorical logic, we postulate the existence of a pair of adjoint functors, called Lang and Syn, from a category, now called BRAIN, of categories, to a category, now called MIND, of theories. Our homunculus is a kind of ``mathematical robot'', the neuronal architecture of which is not important. Its only aim is to provide us with the opportunity to study how such a simple brain-like structure could ``create meanings'' and perform abstraction operations out of its purely syntactic program. The pair of adjoint functors Lang and Syn model the mutual dependencies between the syntactical structure of a given theory of MIND and the internal logic of its semantics given by a category of BRAIN. In this way, a formal language (syntax) and its meanings (semantics) are interwoven with each other in a manner corresponding to the adjointness of the functors Lang and Syn. Higher cognitive functions of abstraction and realization of concepts are also modelled by a corresponding pair of adjoint functors. The categories BRAIN and MIND interact with each other with their entire structures and, at the same time, these very structures are shaped by this interaction.