## 1 Introduction  
Today, machine learning is used in a variety of fields. However, it normally produces black-box models that are not based on the underlying physics, chemistry or biology of the system under investigation. To alleviate this issue, a variety of approaches combine first-principle modelling (FPM) with machine learning (ML). Examples are data-driven modelling and clustering. In [PERSON] et al. [1], for instance, ML was used to extract constitutive relationships in solid mechanics directly from data, while in [PERSON] et al. [2], ML was used to learn density functionals. Other methodologies include reduce-order modelling and field-reconstruction. The idea is to use ML to learn from data generated by expensive computational models and produce a black-box model capable of imitating the output of the FPM with lower computational costs. This approach is used in turbulence [3], where the ML algorithm is trained with data generated by direct numerical simulations (DNS) and is used as an additional term in less expensive Reynolds-averaged Navier-Stokes (RANS) models. [PERSON] et al. [4] used a similar idea in solid mechanics: stress distribution data generated by finite element analysis were fed into neural networks to train the network to quickly estimate stress distributions. In all the examples above, the coupling between FPM and ML is _in series_: the two components remain distinct, and data generated by one model are subsequently fed into the other. However,as explained in [5], in-series coupling is not satisfactory for problems such as in-silico modelling of human physiology. Computer simulations of human organs, for instance, should account for the active intervention of the autonomous nervous system (ANS) that responds dynamically to environmental stimuli, ensuring the correct functioning of the body. Since it is not currently possible to model the ANS by first-principles, [PERSON] [5] used Artificial Intelligence (AI)to replicate the activity of the ANS in the case of peristalsis in the oesophagus. This approach requires a constant and localized exchange of information between the first-principles (FP) model and the ML algorithm. Biological neurons, in fact, are dispersed all over the surface of the oesophagus, and their activation is temporarily and spatially dependent on the physical interaction between the food and the oesophagus, which are calculated by the FP model and evolve continuously during the simulation. For this reason, in this case we cannot adopt in-series coupling, where the FP simulation and the ML algorithm occur separately (i.e., in series (one after the other)); we need a different approach, where the two models work together (i.e., in parallel (at the same time)). In this study, this new approach is called 'in parallel' as opposite to the traditional approach defined as 'in series'; the reader should not confuse this terminology with the parallel or serial programming paradigms that can be used to numerically implement both FPM and ANNs. Deep multiphysics is a computational framework that allows for this type of parallel coupling between FPM and ML. It is based on the so-called particle-neuron duality and has discrete multiphysics (DMP) and artificial neural networks (ANNs) as special cases. In [PERSON] 2019 [5], deep multiphysics was applied to human physiology, but since that study was based on reinforcement learning [6] rather than ANNs, the particle-neuron duality stayed in the background and was not discussed in detail. The present paper fills this gap by focusing on the framework rather than the application. A practical example (cell sorting based on the rigidity of the external membrane) is presented as a mean to facilitate the understanding of how particle duality works.