Deep Multiphysics and Particle–Neuron Duality: a Computational Framework Coupling (Discrete) Multiphysics and Deep Learning

applied sciences

Article Deep Multiphysics and Particle–Neuron Duality: A Computational Framework Coupling (Discrete) Multiphysics and Deep Learning

Alessio Alexiadis School of Chemical Engineering, University of Birmingham, Birmingham B15 2TT, UK; [email protected]; Tel.: +44-(0)-121-414-5305 Received: 11 November 2019; Accepted: 6 December 2019; Published: 9 December 2019

Featured Application: Coupling First-Principle Modelling with Artiﬁcial Intelligence.

Abstract: There are two common ways of coupling first-principles modelling and machine learning. In one case, data are transferred from the machine-learning algorithm to the first-principles model; in the other, from the first-principles model to the machine-learning algorithm. In both cases, the coupling is in series: the two components remain distinct, and data generated by one model are subsequently fed into the other. Several modelling problems, however, require in-parallel coupling, where the first-principle model and the machine-learning algorithm work together at the same time rather than one after the other. This study introduces deep multiphysics; a computational framework that couples first-principles modelling and machine learning in parallel rather than in series. Deep multiphysics works with particle-based first-principles modelling techniques. It is shown that the mathematical algorithms behind several particle methods and artificial neural networks are similar to the point that can be unified under the notion of particle–neuron duality. This study explains in detail the particle–neuron duality and how deep multiphysics works both theoretically and in practice. A case study, the design of a microfluidic device for separating cell populations with different levels of stiffness, is discussed to achieve this aim.

Keywords: mathematical modelling; discrete multiphysics; coupling artiﬁcial intelligence with ﬁrst-principle modelling; computer simulations

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 Ibañez et al. [1], for instance, ML was used to extract constitutive relationships in solid mechanics directly from data, while in Snyder 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. Liang 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,

Appl. Sci. 2019, 9, 5369; doi:10.3390/app9245369 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 5369 2 of 14 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, Alexiadis [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 Alexiadis 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. This article is divided into three parts: the first provides an introduction to discrete multiphysics, the second introduces the general framework for coupling discrete multiphysics (DMP) with artificial neural networks (ANN) and the third applies this concept to a practical application (e.g., design of microfluidic devices for the identification and separation of leukemic cells).

2. Discrete Multiphysics Different from traditional multiphysics, discrete multiphysics is a mesh-free multiphysics technique based on ‘computational particles’ rather than on computational meshes [7,8]. It is a hybrid approach that combines different particle methods such as smoothed particle hydrodynamics (SPH), the lattice-spring model (LSM) and the discrete element method (DEM). The algorithm of particle methods, such as SPH, LSM and DEM, follows the same flowchart (Figure1a), with the only difference being how the internal forces are calculated. In solid-liquid flows, for instance, there are three types of forces: (i) pressure and viscous forces occurring in the liquid, (ii) elastic forces occurring in the solid and (iii) contact forces occurring when two solids collide with each other. In discrete multiphysics, these forces are achieved by means of three different particle-based methods (Figure1b): (i) SPH for the liquid, (ii) LSM for the solid and (iii) DEM for the contact forces. Boundary conditions (BC) are also represented by forces: non-compenetration, for instance, is achieved by means of repulsive forces preventing particles overlapping [8]. DMP, therefore, is a metamodel, i.e., a framework for coupling different models within multiphysics simulations. SPH, LSM and DEM are the most common models in DMP, but other choices are possible as long as they follow the flowchart of Figure1a. Fluctuating hydrodynamics, for instance, could include Brownian dynamics (BD) to account for Brownian fluctuations in the flow. Appl. Sci. 2019, 9, 5369 3 of 14 Appl. Sci. 2019, 11, x 3 of 14 Appl. Sci. 2019, 11, x 3 of 14

Figure 1. (a): Typical flow-chart of particle methods; (b): Internal forces used in Discrete MultiphysicsFigureFigure 1. (a (DMP).): 1.Typical (a): Typical flow-chart flow-chart of particle of particle methods; methods; (b): (Internalb): Internal forces forces used used in inDiscrete Discrete MultiphysicsMultiphysics (DMP) (DMP) Discrete multiphysics has proven to be more than just an alternative to traditional multiphysics. ThereDiscrete isDiscrete a varietymultiphysics multiphysics of situations has proven has where proven to DMP be to more tacklesbe more than problems than just just an alternative thatan alternative are very to di traditionaltoffi traditionalcult, if not multiphysics. multiphysics. impossible, for Theretraditional Thereis a variety is multiphysics.a variety of situations of situations In where traditional where DMP multiphysics,DMP tackles tackles problems pr domainsoblems that that areare are assignedvery very difficult, difficult, during if pre-processing:notif not impossible, impossible, the for usertraditionalfor establishes, traditional multiphysics. before multiphysics. the simulation, In traditionalIn traditional which multiphysics, part multiphysics, of the domain domains domains belongs are toare theassigned assigned solid domain during during and pre- whichpre- processing:partprocessing: to the the fluid user the domain:establishes, user establishes, this before choice before the cannot simulati the simulati changeon, on,which during which part the part of simulation. theof the domain domain In belongs DMP,belongs theto to the distinctionthe solid solid domainbetweendomain and solidwhich and andwhich part fluid to part the only to fluid the depends fluid domain: domain: on this the thischoice type choice of cannot force cannot appliedchange change during to theduring computational the the simulation. simulation. particles In In DMP, DMP, and, the bydistinctionthe changing distinction between the typebetween solid of force, solidand fluid and we canfluid only change only depends depends the on behaviour the on thetype type of force theof force particles applied applied fromto tothe the solid computational computational to liquid and particlesviceparticles versa, and, by duringand, changing by the changing simulation. the type the typeof In force, the of force, case we of canwe solidification, canchange change the thebehaviour for behaviour instance, of whenofthe the particles theparticles internal from from energy solid solid of to liquid and vice versa, during the simulation. In the case of solidification, for instance, when the to liquida particle and goesvice belowversa, during a given the value, simulation. the model In usedthe case to simulate of solidification, that particular for instance, particle when is switched the internal energy of a particle goes below a given value, the model used to simulate that particular internalfrom energy SPH to of LMS. a particle This confers goes below an advantage a given tovalue, discrete the multiphysicsmodel used to in simulate a variety that of cases particular (Figure 2) particle is switched from SPH to LMS. This confers an advantage to discrete multiphysics in a variety particlesuch is as switched cardiovascular from SPH flows to LMS. with This blood confers agglomeration an advantage [9–11 to], discrete phase transitions multiphysics [12], in capsules a variety and of cases (Figure 2) such as cardiovascular flows with blood agglomeration [9–11], phase transitions of casescells breakup(Figure 2) [13 such,14] andas cardiovascular dissolution problems flows with [8,15 blood]. agglomeration [9–11], phase transitions [12], capsules and cells breakup [13,14] and dissolution problems [8,15]. [12], capsules and cells breakup [13,14] and dissolution problems [8,15].

Figure 2. Examples of applications of discrete multiphysics: cardiovascular ﬂow (a), capsules breakup (b), phase transitions (c) and dissolution of solid particles (d). Appl. Sci. 2019, 11, x 4 of 14

Appl. Sci.Figure2019 2., 9 ,Examples 5369 of applications of discrete multiphysics: cardiovascular flow (a), capsules breakup4 of 14 (b), phase transitions (c) and dissolution of solid particles (d).

In addition, the particle framework of DMP has proven to be particularly effective when coupled In addition, the particle framework of DMP has proven to be particularly effective when coupled with artificial intelligence (AI) algorithms. In [5], DMP was coupled with reinforcement learning with artificial intelligence (AI) algorithms. In [5], DMP was coupled with reinforcement learning (RL) (RL) to account for the effect of the autonomic neural system in multiphysics simulations of human to account for the effect of the autonomic neural system in multiphysics simulations of human physiology. As a benchmark case, the DMP + RL approach was used for a computer model of the physiology. As a benchmark case, the DMP + RL approach was used for a computer model of the oesophagus with the ability to learn by itself how to coordinate its contractions and propel food in the oesophagus with the ability to learn by itself how to coordinate its contractions and propel food in right direction. In this article, I show that, by establishing particle–neuron duality, DMP can also be the right direction. In this article, I show that, by establishing particle–neuron duality, DMP can also effectively coupled with another type of AI algorithm: artificial neural networks. be effectively coupled with another type of AI algorithm: artificial neural networks. 3. From Discrete Multiphysics to Deep (Discrete) Multiphysics 3. From Discrete Multiphysics to Deep (Discrete) Multiphysics This section provides a brief introduction to artificial neural networks followed by the description This section provides a brief introduction to artificial neural networks followed by the of the particle–neuron duality. description of the particle–neuron duality. 3.1. Artificial Neural Networks 3.1. Artificial neural networks Artificial neural networks (ANN) are computer systems used to solve complex problems. The buildingArtificial block neural of networks an ANN is(ANN) the McCulloch–Pitts are computer systems neuron [used16], ato mathematical solve complex function problems. vaguely The inspiredbuilding byblock the functioningof an ANN ofis biologicalthe McCulloch–Pitts neurons (Figure neuron3a). [16], a mathematical function vaguely inspired by the functioning of biological neurons (Figure 3a).

Figure 3. ((a):): A A single single artificial artiﬁcial neuron; neuron; ( (b):): An An example example of of a a (deep) (deep) artifi artiﬁcialcial neural network with two hidden layers.

Several inputs xn enter the neuron; each of these inputs is multiplied by a weight wn, summed Several inputs xn enter the neuron; each of these inputs is multiplied by a weight wn, summed σ together, and fed toto anan activationactivation functionfunction σ,, whichwhich producesproduces thethe outputoutput yy.. When many of these neurons are interconnected, they form an artificial neural network [17]. Typically, these networks are Appl. Sci. 2019, 9, 5369 5 of 14 Appl. Sci. 2019, 11, x 5 of 14 neurons are interconnected, they form an artificial neural network [17]. Typically, these networks are organized in layers: input data are introduced via an input layer, which communicates to one or more organized in layers: input data are introduced via an input layer, which communicates to one or more hidden layers and, finally, to an output layer (Figure3b). The weighted output of each layer is fed as hidden layers and, finally, to an output layer (Figure 3b). The weighted output of each layer is fed as input of the next layer until the output layer calculates the final output of the network. An artificial input of the next layer until the output layer calculates the final output of the network. An artificial neural network (ANN) with multiple layers between the input and output layers is called a deep neural network (ANN) with multiple layers between the input and output layers is called a deep neural network. neural network. Deep neural networks are universal approximate functions, meaning that, given a large enough Deep neural networks are universal approximate functions, meaning that, given a large enough number of neurons in the hidden layer, they can approximate almost any function, no matter how number of neurons in the hidden layer, they can approximate almost any function, no matter how complicated. To achieve this goal, the ANN needs to be trained. We provide the machine with complicated. To achieve this goal, the ANN needs to be trained. We provide the machine with real- real-word data (xn, y) and, by using a learning rule (e.g., backpropagation [18]), the ANN modifies its word data (xn, y) and, by using a learning rule (e.g., backpropagation [18]), the ANN modifies its weights to correctly map xn to y. weights to correctly map xn to y. 3.2. The Particle–Neuron Duality 3.2. The particle–neuron duality As mentioned above, discrete models, such SPH, LSM and DEM, work by exchanging forces amongAs computationalmentioned above, particles discrete (Figure models,4a). Thesesuch SP forcesH, LSM change and the DEM, velocity work and by theexchanging position offorces the particlesamong computational and, by doing particles that, the (Figure model achieves4a). These a discreteforces change representation the velocity of the and mechanics. the position of the particles and, by doing that, the model achieves a discrete representation of the mechanics.

Figure 4. (a): Solid and ﬂuid particles exchange mechanical forces; (b): Hot and cold particles exchange Figure 4. (a): Solid and fluid particles exchange mechanical forces; (b): Hot and cold particles heat; (c): Layers of neurons exchange information. exchange heat; (c): Layers of neurons exchange information. Besides velocity and position, computational particles can have other properties. In heat transfer problems,Besides for velocity instance, and particles position, require computational a new property partic calledles can ‘temperature’. have other proper If weties. want In thisheat property transfer problems,to behave for like instance, the physical particles temperature, require a new we mustproperty link called it to the ‘temperature’. heat transfer If equation.we want this In thisproperty case, tothe behave particles like do the not physical exchange temperature, forces, but we rather must more link generalit to the interactionsheat transfer that equation. simulate In this the physicalcase, the processparticles of do heat not transfer exchange from forces, one computational but rather more particle general to another interactions (Figure 4thatb). This simulate can be the generalized physical processto any conservation of heat transfer law: wefrom assign one acomputational new property toparticle the particles, to another and the(Figure corresponding 4b). This transfercan be generalizedequation calculates to any conservation the interactions law: that we assign dynamically a new property evolve this to propertythe particles, with and time. the corresponding transferInteractions equation arecalculates more generalthe interactions than forces. that dynamically If we only account evolve this for forces,property we with have time. a ‘discrete mechanics’Interactions tool that are ismore limited general to momentum-conservation than forces. If we only problems.account for If weforces, include we have interactions, a ‘discrete we mechanics’have a ‘discrete tool multiphysics’that is limited toolto momentum-conse that applies to a widerrvation variety problems. of conservation If we include problems. interactions, Discrete we havemultiphysics, a ‘discrete therefore, multiphysics’ is a generalization tool that applies of discrete to a wider mechanics variety (Figureof conservation5). problems. Discrete multiphysics, therefore, is a generalization of discrete mechanics (Figure 5). The next question, at this point, would be: can we generalize this idea even further? In addition, to achieve this, what concept, more general than ‘interaction’, can we use? Appl. Sci. 2019, 11, x 6 of 14

Information is the most general exchange of ‘something’ we can think of. Whenever two computational particles exchange forces, interactions or any other property, they exchange information. The exchange of information is exactly what happens in ANNs, where information is transferred from one neuron to another (Figure 4c). In the case of ANNs, this information has no direct connection Appl. Sci. 2019, 9, 5369 6 of 14 with any physical property. However, ‘physical interactions’ or ‘physical forces’ can be seen as a form of information transfer and, as such, a subset of the more general concept of ‘information’ (Figure 5).

Figure 5. MutualMutual relations relations between between ‘forces’ ‘forces’ and and discr discreteete mechanics, ‘interactions’ and discrete multiphysics, and ‘information’ and deep multiphysics.

ThisThe nextidea question,constitutes at thisthe point,basis of would the particle be: can–neuron we generalize duality this we idea use even to establish further? the In addition, general frameworkto achieve this, of deep what discrete concept, Multiphysics more general (or than si ‘interaction’,mply deep multiphysics). can we use? The elemental building blockInformation of deep multiphysics is the most is a generalparticle–neuron exchange hybrid: of ‘something’ it behaves welike cana DMP think particle of. Wheneverwhen it carries two outcomputational physical interactions particles exchange and like forces,a neuron interactions when it exchanges or any other non-physical property, they information, exchange information. and, like a neuron,The it exchange can be trained. of information is exactly what happens in ANNs, where information is transferred fromTo one illustrate neuron toand another clarify (Figure this point,4c). In the the rest case of ofthe ANNs, article this focuses information on a case has study no direct where connection practical withimplementation any physical of property. deep multiphysics However, ‘physicalis presented. interactions’ or ‘physical forces’ can be seen as a form of information transfer and, as such, a subset of the more general concept of ‘information’ (Figure5). 4. PracticalThis idea implementation: constitutes the identification basis of the particle–neuronand separation of duality leukemic we usecells to from establish healthy the ones general framework of deep discrete Multiphysics (or simply deep multiphysics). The elemental building block As a practical example, I use deep multiphysics to design a microfluidic device that, potentially, of deep multiphysics is a particle–neuron hybrid: it behaves like a DMP particle when it carries out can be used to separate leukemic cells from healthy ones in the peripheral circulation system. physical interactions and like a neuron when it exchanges non-physical information, and, like a neuron, In Section 4.1, I describe the biomedical rationale for such a device, while in Sections 4.2 and 4.3, it can be trained. I explain the deep multiphysics model used for the calculations. To illustrate and clarify this point, the rest of the article focuses on a case study where practical implementation of deep multiphysics is presented. 4.1. Leukemic cell detection: biomedical rationale 4. PracticalAcute myeloid Implementation: leukaemia Identification is a malignant and neoplasm Separation of the of Leukemicbone marrow Cells accounting from Healthy for 10% Ones of all haematologicalAs a practical example,disorders. I useAlthough deep multiphysics in recent years to design new a therapeutic microfluidic approaches device that, potentially,have been devised,can be used the tooverall separate survival leukemic of patients cells from is less healthy than 30%. ones This in the is peripheralmostly due circulation to the high system. rate of disease relapseIn Sectionthat is observed4.1, I describe in patients the biomedical treated with rationale standard for such targeted a device, chemotherapy. while in Sections After 4.2treatment, and 4.3, theI explain disease the is deep often multiphysics not completely model eradicated, used for theand calculations. leukemic cells still persist although in numbers that are below detectable levels by standard means. A microfluidic device that can easily distinguish malignant4.1. Leukemic from Cell healthy Detection: cells, Biomedical therefore, Rationale would be extremely useful. Most common microfluidics separation techniques, however, work with cells of different sizes, but, in the present case, this would Acute myeloid leukaemia is a malignant neoplasm of the bone marrow accounting for 10% of all not work since malignant and healthy cells are approximately the same size. haematological disorders. Although in recent years new therapeutic approaches have been devised, The proposed solution takes advantage of the fact that malignant cells are more flexible than the overall survival of patients is less than 30%. This is mostly due to the high rate of disease relapse healthy ones. A microchannel is lined with several hair-like flexible structures that, in the rest of the that is observed in patients treated with standard targeted chemotherapy. After treatment, the disease paper, are called cilia. When cells move along the channel, the cilia bend to allow the passage of the is often not completely eradicated, and leukemic cells still persist although in numbers that are below cells. When cilia bend, stresses are generated at their base. The idea of the microfluidic separator is detectable levels by standard means. A microfluidic device that can easily distinguish malignant from healthy cells, therefore, would be extremely useful. Most common microfluidics separation techniques, however, work with cells of different sizes, but, in the present case, this would not work since malignant and healthy cells are approximately the same size. The proposed solution takes advantage of the fact that malignant cells are more flexible than healthy ones. A microchannel is lined with several hair-like flexible structures that, in the rest of the Appl. Sci. 2019, 9, 5369 7 of 14

Appl. Sci. 2019, 11, x 7 of 14 paper, are called cilia. When cells move along the channel, the cilia bend to allow the passage of the tocells. distinguish When cilia between bend, stressesrigid or areflexible generated cells by at looking their base. at the The stress idea patterns of the microfluidic occurring during separator their is passage.to distinguish In a real between device, rigid these or flexiblestresses cellscould by be looking measured at the by stress using patterns a piezoelectric occurring material during at their the passage.base of the In cilia. a real AI device, is used these to associate stresses couldspecific be stress measured patterns by using witha th piezoelectrice stiffness of material the cell. at the base of theThe cilia. geometry AI is used of the to associate microfluidic specific device stress is patternsshown in with Figure the sti6: fftheness length of the of cell. the channel is 200 μm, andThe geometrythe width ofis the50 μ microfluidicm. In the channel, device isthere shown are in11 Figure flexible6: thecilia: length 6 on ofthe the upper channel wall is and 200 5µ onm, theand lower the width wall. isThe 50 distanceµm. In the between channel, two there contiguous are 11 flexible cilia is 20 cilia: μm. 6 At on the right upper end wall of andthe channel, 5 on the therelower are wall. two The gates, distance one that between connects two contiguousthe channel cilia with is a 20 chamberµm. At thewhere right soft end cells of theare channel,collected there and anotherare two gates,that connects one that with connects a chamber the channel where with rigid a chambercells are wherecollected. soft In cells a real are device, collected a andmagnet another can bethat used connects to open with or aclose chamber these where gates. rigidHowever, cells arethis collected. study focuses In a real on device,the modelling a magnet aspects, can be and used I do to notopen deal or closehere thesewith the gates. more However, technical this aspects study of focuses building on thethe modelling microchannel. aspects, and I do not deal here withIn the Section more technical 4.2, the DMP aspects model of building of the device the microchannel. is introduced. In Section 4.3, the model is coupled with an ANN to obtain a deep multiphysics model.

Figure 6. TheThe concept concept of of the the proposed proposed microfluidic microﬂuidic separator.

4.2. TheIn SectionDMP model 4.2, the DMP model of the device is introduced. In Section 4.3, the model is coupled with an ANN to obtain a deep multiphysics model. Figure 6 shows the (two-dimensional) domain used in the simulations. The lattice spring model (LSM),4.2. The which DMP Modelmodels the mechanical properties of solid material by means of linear bonds and angular springs, is used for both the cilia and the cell. In this study, all bonds are rigid (i.e., Figure6 shows the (two-dimensional) domain used in the simulations. The lattice spring model inextensible) with fixed distance Δr = 2 μm, while the force F generated by an angular spring that (LSM), which models the mechanical properties of solid material by means of linear bonds and angular connects three consecutive particles is given by the gradient of the potential springs, is used for both the cilia and the cell. In this study, all bonds are rigid (i.e., inextensible) with ﬁxed distance ∆r = 2 µm, while the force 1F generated2 by an angular spring that connects three Uk=−()θθ consecutive particles is given by the gradienta of the potential0 (1) 2 ,

0 1 2 where k is the stiffness of the spring,Ua θ= thek( θequilibriumθ0) , angle at rest, and θ the angle after (1) deformation. In a real microfluidic application, liquid2 − would flow into the device and convey the cells whereby meansk is theof drag stiffness forces. of the To spring,simplifyθ0 thethe model, equilibrium liquid angle is not at rest,directly and consideredθ the angle in after this deformation. work, and theIn a effect real microfluidic of drag is accounted application, for liquidby Stokes’ would law flow into the device and convey the cells by means of drag forces. To simplify the model, liquid is not directly considered in this work, and the effect of drag Fdv=3πμ is accounted for by Stokes’ law v , (2) F = 3πµdv, (2) where μ is the dynamic viscosity, d the radiusv of the cell and v the flow velocity relative to the whereobject. µThisis the force dynamic is distributed viscosity, amongd the radiusall the computational of the cell and vparticlesthe flow that velocity compose relative the tocell. the A object. body Thisforce forceis applied is distributed to the cell among (a = 4 all × 10 the-3 m·s computational-2) to move it particles along the that channel compose with the velocities cell. A body typical force of -1 3 2 microfluidicis applied to channels the cell ((~1a = mm4 s10 −withoutm s− )cilia). to move Non-penetration it along the channelamong computational with velocities particles typical ofis × 1 · achievedmicrofluidic by a channels soft repulsive (~1 mm potential s− without of the cilia). type Non-penetration among computational particles is achieved by a soft repulsive potential of the type π =+r < EA1 cos ( rrc ) rπr (3) E = A1 + cos c (r < rc),, (3) rc

where r is the distance between the two particles, rc is a cut-off distance and A is an energy constant. In the simulation, the same repulsion force and cut-off (A = 4 × 10 - 5 J, rc = 2 μm) are used for all particles. Appl. Sci. 2019, 9, 5369 8 of 14

where r is the distance between the two particles, rc is a cut-off distance and A is an energy constant. Appl. Sci. 2019, 11, x 5 8 of 14 In the simulation, the same repulsion force and cut-off (A = 4 10 J, rc = 2 µm) are used for × − all particles. Different types of computational particles are used to represent the system. A schematic Different types of computational particles are used to represent the system. A schematic representation is shown in Figure 7. Type-1 (red) particles represent the wall and are stationary duringrepresentation the simulation. is shown in Figure7. Type-1 (red) particles represent the wall and are stationary during the simulation.

Figure 7. Schematic representation of the particle types used in the simulations. Figure 7. Schematic representation of the particle types used in the simulations. Type-2 (blue) particles represent the cilia and are connected by rigid bonds and angular springs Type-2 (blue) 14particles represent the cilia and are connected by rigid bonds and angular springs with k = 2.5 10− J and θ0 = 180◦. Type-3 (white) particles represent the cell membrane and are with k = 2.5 ×× 10-14 J and θ0 = 180°. Type-3 (white) particles represent the cell membrane and are connected with rigid bonds and angular springs with k variables (depending on the cell population) connected with rigid bonds and angular springs with k variables (depending on the cell population) and θ0 = 172◦; the cytoplasm is neglected and all its mechanical properties are attributed to the and θ0 = 172°; the cytoplasm is neglected and all its mechanical properties are attributed to the membrane. Type-4 (yellow) particles represent the root of the cilium. They are connected with rigid membrane. Type-4 (yellow) particles represent the root of the cilium. They are connected with rigid bonds and angular springs similar to type-2 particles. These are the particles where the stress caused bonds and angular springs similar to type-2 particles. These are the particles where the stress caused by the bending of the cilia is measured. Type-4 particles are, therefore, anchored to their initial position by the bending of the cilia is measured. Type-4 particles are, therefore, anchored to their initial by means of a self-tethered linear bond position by means of a self-tethered linear bond Fs = ksr, (4) 1 Fkr= (4) where ks = 0.01 N m− is the stiﬀness of the bond,ss and, r is the distance between the actual position of the particle and its initial position. The force Fs generated by this bond is recorded during the -1 simulationwhere andks = 0.01 is used N m to is train the thestiffness ANN. of There the bond, is also and a type-5 r is the particle, distance which between is used the actual for the position gates in ofFigure the 6particle, but is notand representedits initial position. in Figure The7. Type-5 force F particless generated are interconnectedby this bond is with recorded rigid bondsduring and the simulationangles. They and are is alsoused connected to train the with ANN. the There closest is also wall a particles type-5 particle, with an which angular is used spring. for Thisthe gates spring in Figurecan be 6, switched but is not on represented and oﬀ to allow in Figure for the 7. Type-5 opening particles of the gate are interconnected according to the with output rigid of bonds the ANN. and angles.Overall, They in the are model also connected there are 776 with particles the closest of type-1, wall particles 154 particles with ofan type-2, angular 44 spring. particles This of type-3,spring can47 particles be switched of type-4 on and and off 48 to particles allow for of the type-5. opening of the gate according to the output of the ANN. Overall, in the model there are 776 particles of type-1, 154 particles of type-2, 44 particles of type-3, 474.3. particles Coupling of with type-4 ANN and 48 particles of type-5.

4.3. CouplingThe DMP with model ANN accounts for the physics and mechanics of the system, but, by itself, cannot discriminate between flexible and rigid cells. To achieve this goal, several additional particles (type-6) are addedThe DMP to the model model. accounts By taking for advantage the physics of the and particle–neuron mechanics of duality, the system, we can but, make by theseitself, particles cannot discriminatebehave like artificial between neurons flexible and, and by rigid connecting cells. To them achieve in layers, this goal, we can several make additional these neurons particles behave (type- like 6)an are ANN. added Figure to the8 illustrates model. By the taking logic behindadvantage the resultingof the particle–neuron deep multiphysics duality, (DMP we +canANN) make model. these particlesTypes behave 1–3 particles like artificial (DMP neurons particles) and, only by exchangeconnecting forces, them i.e., in layers, not information. we can make They these possess neurons the behaveproperty like of an position ANN. (which Figure changes8 illustrates by thethe eloffectgic ofbehind forces), the but resulting do not deep have multiphysics a neuron-like (DMP output. + ANN)Type 1–3 model. particles, therefore, are pure DMP particles. Types 1–3 particles (DMP particles) only exchange forces, i.e., not information. They possess the property of position (which changes by the effect of forces), but do not have a neuron-like output. Type 1–3 particles, therefore, are pure DMP particles. Type-4 particles represent, at the same time, the root of the cilia and the input layer of the ANN. They exchange forces with the other particles in the computational domain (types 1–5) and Appl. Sci. 2019, 11, x 9 of 14

information with the neuron particles (type-6). Type 4 particles, therefore, are hybrid particles with both DMP and neural properties. Type-5 particles, and in particular the particles next to the wall separating the two chambers (see Figure 6), are also hybrid. On the one hand, they constitute the output layer of the ANN and, as such, exchange information with the hidden layers. On the other hand, they belong to the DMP

Appl.computational Sci. 2019, 9, 5369 domain and exchange forces with other DMP particles. The particle–neuron duality9 of 14 integrates these two properties and, according to the information coming from the hidden layer, the forces acting on type-5 particles are switched on or off to open or close the separating gates. Type-4Finally, particles type-6 particles represent, are at pure the same neurons time, and the rootare used of the to cilia represent and the the input hidden layer layers of the of ANN. the Theynetwork. exchange They forcesexchange with information the other particles with (i) in theother computational type-6 particles domain from (types another 1–5) hidden and information layer, (ii) withtype-4 the particles neuron from particles the input (type-6). layer Type and 4 (iii) particles, type 5 therefore, particles from are hybrid the output particles layer. with Since both they DMP do andnot neuralexchange properties. forces, they do not possess a position or a velocity property that can be modified by the forces.

Figure 8.8. Deep multiphysics model of the separationseparation device.device. The number of neuronsneurons/layers/layers is notnot representativerepresentative ofof the the actual actual Artificial Artificial Neural Neural Network Network (ANN) (ANN) used butused is onlybut foris illustrativeonly for illustrative purposes. purposes. Type-5 particles, and in particular the particles next to the wall separating the two chambers (see FigureIn the 6next), are section, also hybrid. I call ANN On the the one combination hand, they of constitutetype-6 particles the output plus type-4 layer of and the type-5 ANN when and, asthey such, act exchangeas respective information input and with output the hidden layers. layers. I call On‘DMP the model’ other hand, the combination they belong toof thetype DMP 1–3 computationalparticles plus type-4 domain and and type-5 exchange when forcesthey act with as DMP-particles. other DMP particles. The particle–neuron duality integrates these two properties and, according to the information coming from the hidden layer, the5. Results forces actingand discussion on type-5 particles are switched on or off to open or close the separating gates. Finally, type-6 particles are pure neurons and are used to represent the hidden layers of the network. Before the ANN can predict the cell stiffness, it must be trained with known data. I used the They exchange information with (i) other type-6 particles from another hidden layer, (ii) type-4 particles DMP model to simulate cell populations with different levels of stiffness and, at the same time, train from the input layer and (iii) type 5 particles from the output layer. Since they do not exchange forces, the ANN. I ran three hundred simulations: half of these were used for training and half for validation. they do not possess a position or a velocity property that can be modified by the forces. Since these data allow for effective training of the ANN, how the accuracy of the ANN changes with In the next section, I call ANN the combination of type-6 particles plus type-4 and type-5 when the number of simulations is not specifically investigated in this study. they act as respective input and output layers. I call ‘DMP model’ the combination of type 1–3 particles Under real conditions, both the stiffness k and the diameter d of the population are not constant plus type-4 and type-5 when they act as DMP-particles. but vary within a certain range. To account for this, at the beginning of each simulation, both the k 5.and Results d are andrandomly Discussion selected (uniform distribution) within ±10% of their given values. Another random variable is the initial position of the cell in the channel. In Figure 6, the cell is initially placed at theBefore centre the of ANNthe channel can predict height, the but, cell stiin ffreality,ness, it it must can be trainedlocated withat any known height data. compatible I used the with DMP its modeldiameter. to simulateThe initial cell position populations of the withcell, therefore, different levelsis also ofrandomly stiffness allocated. and, at the same time, train the ANN.In I order ran three to move hundred along simulations: the channel, half the of cell these must were bend used the for cilia. training When and the half cilia for bend, validation. they Sincegenerate these stress data at allow their for bases. effective The value training of ofthe the total ANN, force how acting the at accuracy the base of of the each ANN of changesthe 11 cilia with is the number of simulations is not specifically investigated in this study. Under real conditions, both the stiffness k and the diameter d of the population are not constant but vary within a certain range. To account for this, at the beginning of each simulation, both the k and d are randomly selected (uniform distribution) within 10% of their given values. Another random ± variable is the initial position of the cell in the channel. In Figure6, the cell is initially placed at the Appl. Sci. 2019, 9, 5369 10 of 14 centre of the channel height, but, in reality, it can be located at any height compatible with its diameter.

TheAppl. initial Sci. 2019 position, 11, x of the cell, therefore, is also randomly allocated. 10 of 14 In order to move along the channel, the cell must bend the cilia. When the cilia bend, they generate stressrecorded at their every bases. 0.2 s Thefor 15 value timeframes. of the total After force the acting cell passes at the basethrough of each the ciliated of the 11 section, cilia is therefore, recorded everywe have 0.2 15x11 s for 15stress timeframes. values that After can thebe stored cell passes in a matrix through F. theIf the ciliated force fsection, is normalized therefore, according we have to 15 11 stress values that can be stored in a matrix F. If the force f is normalized according to × ff− * = min f (5) ff f−minf f ∗ = max− min,, (5) fmax f − min the normalized matrix F* can be seen as a greyscale image (Figure 9). Each cell produces a slightly the normalized matrix F* can be seen as a greyscale image (Figure9). Each cell produces a slightly different image according to its stiffness, size and initial position. This image represents a sort of different image according to its stiffness, size and initial position. This image represents a sort of fingerprint of the cell, and the ANN is trained to distinguish between soft and rigid fingerprints. fingerprint of the cell, and the ANN is trained to distinguish between soft and rigid fingerprints.

Figure 9. The ‘stress’ fingerprint of a cell with stiffness k = 4.5 10 15 J. × − Figure 9. The ‘stress’ fingerprint of a cell with stiffness k = 4.5 × 10-15 J. In all the simulations, I assume the two populations have the same average diameter d = 30 µm ( 3 µm) and their only difference consists of their stiffness (and initial position). I consider three ± In all the simulations, I assume the two populations have the same average diameter d = 30 μm di(±3fferent μm) casesand their where only the difference stiffness of consists the two populationsof their stiffness becomes (and gradually initial position). closer (Table I consider1). three different cases where the stiffness of the two populations becomes gradually closer (Table 1). Table 1. Minimal, average and maximal stiffness of the two populations for the three cases considered. Err is the percentage of soft cells erroneously classified as rigid by the ANN, and vice versa, in the Table 1. Minimal, average and maximal stiffness of the two populations for the three cases validation set. considered. Err is the percentage of soft cells erroneously classified as rigid by the ANN, and vice versa, in the validationk Soft set.Population [J 1015] k Rigid Population [J 1015] · · Casek soft Min.population Ave. [J∙1015] Max Min.k rigid Ave. population Max [J∙1015] Err % Case Min.1 1.8Ave. 2Max 2.2 Min. 18 20 Ave. 22 Max 0 Err % 1 21.8 3.62 42.2 4.4 918 1020 1122 0 0 2 33.6 4.054 4.54.4 4.95 4.59 5 10 5.511 11 0 3 4.05 4.5 4.95 4.5 5 5.5 11 The same ANN, with one hidden layer, is used in all three cases. The input layer has 165 nodes The same ANN, with one hidden layer, is used in all three cases. The input layer has 165 nodes (the size of the matrix F*), the hidden layer has 3 nodes and the output layer has 1 node. All the layers (the size of the matrix F*), the hidden layer has 3 nodes and the output layer has 1 node. All the layers are fully connected, and the logistic function is used as activation for all the nodes. The output of the are fully connected, and the logistic function is used as activation for all the nodes. The output of the ANN is the normalized stiffness ANN is the normalized stiffness k −kmin k∗ *= kk− min , (6) k =kmax kmin kk−− (6) max min ,

where k is the stiffness of the cell, and kmin and kmax are, respectively, the minimal and maximal stiffness of the set. The reader should not confuse the max. and min. stiffnesses in Table 1 and in Appl. Sci. 2019, 9, 5369 11 of 14 whereAppl. Sci.k is2019 the, 11, sti xff ness of the cell, and k and k are, respectively, the minimal and maximal sti11ff nessof 14 Appl. Sci. 2019, 11, x min max 11 of 14 of the set. The reader should not confuse the max. and min. stiffnesses in Table1 and in Equation (6): Equation (6): the former are the maximal and minimal values of the population, the latter of the data set, theEquation former (6): are the former maximal are and the minimal maximal values and minimalof the population values of, the the latterpopulationof the, datathe latter set, which of the includedata set, which include instances of both populations. instanceswhich include of both instances populations. of both populations. The training occurs together with the simulations in batch mode (batch size 100) for 1,000 epochs. TheThe training training occurs occurs together together with with the the simulations simulations in in batch batchmode mode(batch (batchsize size 100)100) forfor 1,0001,000 epochs.epochs. Figure 10 shows the comparison between the data and the ANN output for both the training and the FigureFigure 10 10 shows shows the the comparison comparison between between the the data data and and the the ANN ANN output output for for both both the the training training and and the the validation sets in case 1: when k** < 0.5, the cell is classified as soft; when k* >* 0.5, the cell is classified as validationvalidation sets sets in in case case 1: 1: whenwhen kk* < 0.5,0.5, the the cell cell is is classified classified as as soft; soft; when when kk* > >0.5,0.5, the the cell cell is isclassified classified as rigid. asrigid. rigid.

FigureFigure 10.10. ((aa):): Comparison Comparison between between data data and and the the AN ANNN results results for forboth both the thetraining training and and(b): (Theb): Figure 10. (a): Comparison between data and the ANN results for both the training and (b): The Thevalidation validation sets sets in case in case 1. In 1. Inboth both cases, cases, the the first first 50 50 simulations simulations refer refer to to cells cells from from the soft populationpopulation validation sets in case 1. In both cases, the first 50 simulations refer to cells from the soft population andand thethe otherother 5050 fromfrom thetherigid rigid population. population. and the other 50 from the rigid population. When the difference in stiffness of the two populations is one order of magnitude (case 1 in When the difference in stiffness of the two populations is one order of magnitude (case 1 in Table Table1When), the the ANN difference correctly in identifies stiffness of all the the two cells popula withtions 100% is accuracy. one order In of case magnitude 2, the sti (caseffness 1 in of Table the 1), the ANN correctly identifies all the cells with 100% accuracy. In case 2, the stiffness of the rigid rigid1), the cells ANN is only correctly 2.5 times identifies higher thanall the that cells of thewith soft 100% cells accuracy. (Figure 11 In). Thecase di2,ff theerence stiffness is lower of thanthe rigid the cells is only 2.5 times higher than that of the soft cells (Figure 11). The difference is lower than the previouscells is only case, 2.5 but times the model higher is than still capablethat of the of distinguishing soft cells (Figure between 11). The the diff twoerence populations is lower with than 100% the previous case, but the model is still capable of distinguishing between the two populations with 100% accuracy.previous Incase, case but 3, the stimodelffness is ofstill the capable two populations of distinguishing only diff betweeners by 10%, the two and thepopulations stiffness ofwith the 100% two accuracy. In case 3, the stiffness of the two populations only differs by 10%, and the stiffness of the populationsaccuracy. In partially case 3, the overlaps. stiffness The of outputthe two of popula the ANNtions is only reasonably differsaccurate, by 10%, and itthe fails stiffness to correctly of the two populations partially overlaps. The output of the ANN is reasonably accurate, and it fails to classifytwo populations only 11% ofpartially the cells. overlaps. The output of the ANN is reasonably accurate, and it fails to correctly classify only 11% of the cells. correctly classify only 11% of the cells.

Figure 11. Comparison between data and the ANN results for both the validation sets in case 2. Figure 11. Comparison between data and the ANN results for both the validation sets in case 2. Figure 11. Comparison between data and the ANN results for both the validation sets in case 2. After the neural component of the deep multiphysics model is trained, the model acquires the After the neural component of the deep multiphysics model is trained, the model acquires the abilityAfter to separate the neural cells component with unknown of the stideepffness multiphy (Figuresics 12 ).model As a is new trained, cell movesthe model into acquires the device, the ability to separate cells with unknown stiffness (Figure 12). As a new cell moves into the device, the ability to separate cells with unknown stiffness (Figure 12). As a new cell moves into the device, the ANN reads the ‘stress fingerprint’ and classifies it as rigid or solid. In the first case, it opens the lower ANN reads the ‘stress fingerprint’ and classifies it as rigid or solid. In the first case, it opens the lower gate so the cell goes in the lower chamber; in the second case, it opens the upper gate, and the cell gate so the cell goes in the lower chamber; in the second case, it opens the upper gate, and the cell Appl. Sci. 2019, 9, 5369 12 of 14 the ANN reads the ‘stress fingerprint’ and classifies it as rigid or solid. In the first case, it opens the Appl. Sci. 2019, 11, x 12 of 14 lowerAppl. Sci. gate 2019 so, 11, the x cell goes in the lower chamber; in the second case, it opens the upper gate, and12 of the 14 cell goes in the upper chamber. More details are given in the supplementary material: see the videos goes in the upper chamber. More details are given in the supplementary material: see the videos rigid_cell.avigoes in the upper and soft_cell.avi. chamber. More details are given in the supplementary material: see the videos rigid_cell.avi and soft_cell.avi. rigid_cell.avi and soft_cell.avi.

FigureFigure 12. 12. TheThe final final deep deep multiphysics multiphysics model model in in action action (a): (a): rigid rigid cell; cell; (b(b)):: softsoft cell.cell. Figure 12. The final deep multiphysics model in action (a): rigid cell; (b): soft cell. It is interesting to compare the performance of the model with respect to visual observation. It is interesting to compare the performance of the model with respect to visual observation. FigureIt is12 interesting shows two to cells compare from the case performance 1, where the of sti theffness model diff witherence respect is one to order visual of observation. magnitude. Figure 12 shows two cells from case 1, where the stiffness difference is one order of magnitude. The TheFigure diff 12erent shows behaviour two cells of from the twocase cells1, where can bethe clearly stiffness identified difference by is visual one order observation. of magnitude. In case The 3, different behaviour of the two cells can be clearly identified by visual observation. In case 3, in indifferent contrast, behaviour the cells of sti fftheness two is verycells close.can be As clearly Figure id 13entified shows, by visually visual observation. the difference In is case minimal. 3, in contrast, the cells stiffness is very close. As Figure 13 shows, visually the difference is minimal. However,contrast, the the cells ANN stiffness is able to is correctly very close. separate As Figure the two 13 cells shows, in 89% visually of the the cases. difference is minimal. However, the ANN is able to correctly separate the two cells in 89% of the cases. However, the ANN is able to correctly separate the two cells in 89% of the cases.

Figure 13. (a): soft cell; (b): rigid cell in case 3. FigureFigure 13.13. ((a):): soft cell; (b): rigid rigid cell cell in in case case 3. 6. Conclusions 6. Conclusions There are two common ways of coupling FPM and ML. In one case, data are transferred from There areare twotwo commoncommon ways ways of of coupling coupling FPM FPM and and ML. ML. In In one one case, case, data data are are transferred transferred from from the the ML algorithm to the FPM; in the other, from the FPM to the ML algorithm. In both cases, the MLthe algorithmML algorithm to the to FPM; the inFPM; the other,in the fromother, the from FPM the to theFPM ML to algorithm. the ML algorith In bothm. cases, In both the coupling cases, the is coupling is in series: the two components remain distinct, and data generated by one model are incoupling series: theis in two series: components the two remaincomponents distinct, remain and data distinct, generated and bydata one generated model are by subsequently one model fedare subsequently fed into the other. This study introduces deep multiphysics; a computational intosubsequently the other. Thisfed studyinto the introduces other. deepThis multiphysics;study introduces a computational deep multiphysics; framework thata computational couples FPM framework that couples FPM and ML in parallel rather than in series. Deep multiphysics is based andframework ML in parallel that couples rather thanFPM in and series. ML Deep in parallel multiphysics rather isthan based in uponseries. the Deep concept multiphysics of particle–neuron is based upon the concept of particle–neuron duality and only works with a particle-based FPM (discrete dualityupon the and concept only works of particle–neuron with a particle-based duality FPM and (discrete only works multiphysics) with a partic andle-based a specific FPM ML algorithm (discrete multiphysics) and a specific ML algorithm (ANNs) and, as a case study, it is applied here to the (ANNs)multiphysics) and, as and a case a specific study, itML is applied algorithm here (ANNs) to the design and, as of aa microfluidiccase study, deviceit is applied for separating here to cellthe design of a microfluidic device for separating cell populations with different levels of stiffness. populationsdesign of a microfluidic with different device levels for of separating stiffness. cell populations with different levels of stiffness. A computational framework that couples FPM and ML in parallel can lead to computational A computational framework that couples FPM and ML in parallel can leadlead toto computationalcomputational methods that are well-grounded on the physics/chemistry/biology of the system under investigation methods that are well-grounded on the physicsphysics/chem/chemistryistry/biology/biology of the systemsystem under investigation but, at the same time, have the ability to ‘learn’ and ‘adapt’ during the simulation. In Alexiadis [5], a but, at the same time, have have the the ability ability to to ‘learn’ ‘learn’ and and ‘adapt’ ‘adapt’ during during the the simulation. simulation. In In Alexiadis Alexiadis [5], [5 ],a similar idea was applied to in-silico modelling of human physiology: DMP provided the physics of asimilar similar idea idea was was applied applied to toin-silico in-silico modelling modelling of ofhuman human physiology: physiology: DMP DMP provided provided the the physics physics of the system, while AI learned to replicate the regulatory intervention of the autonomic nervous ofthe the system, system, while while AI AI learned learned to to replicate replicate the the regulatory regulatory intervention intervention of of the autonomic nervousnervous system. This study generalizes that study by clearly defining the deep multiphysics framework and system. This study generalizes that study by clearlyclearly definingdefining the deep multiphysics framework and the particle–neuron duality. the particle–neuron duality.duality. This framework can lead to new ways of coupling first-principle modelling with AI. An idea, This framework can lead to new ways of coupling first-principlefirst-principle modelling with AI. An idea, for instance, could be fluid neural networks: ANNs that are not organized within a fixed layered for instance,instance, could be fluidfluid neuralneural networks:networks: ANNs ANNs that are not organized within a fixedfixed layered structure, but, similar to a fluid, change their local structure and connectivity with time. Additionally, structure, but, similar to a fluid, fluid, change their loca locall structure structure and and connectivity connectivity with with time. time. Additionally, Additionally, the particle–neuron duality could also find application in swarm-intelligence, where agents, akin to the particle–neuron duality could also findfind applicationapplication in swarm-intelligence,swarm-intelligence, where agents, akin toto particle–neuron hybrids, interact with each other following rules determined by (i) the (multi)physics particle–neuron hybrids, interact with each other following rules determined by (i) the (multi)physics of the environment and (ii) the information they exchange with each other. of the environment and (ii) the information they exchange with each other. Besides the applications mentioned above, there is an aspect of deep multiphysics that has deep Besides the applications mentioned above, there is an aspect of deep multiphysics that has deep theoretical implications. As discussed above, DMP combines particle methods such as SPH, LSM or theoretical implications. As discussed above, DMP combines particle methods such as SPH, LSM or DEM. The particle–neuron duality shows that, if we look at physical interactions as a form of DEM. The particle–neuron duality shows that, if we look at physical interactions as a form of Appl. Sci. 2019, 9, 5369 13 of 14 particle–neuron hybrids, interact with each other following rules determined by (i) the (multi)physics of the environment and (ii) the information they exchange with each other. Besides the applications mentioned above, there is an aspect of deep multiphysics that has deep theoretical implications. As discussed above, DMP combines particle methods such as SPH, LSM or DEM. The particle–neuron duality shows that, if we look at physical interactions as a form of information exchanged between particles, the mathematics behind these particle-methods and ANNs can be unified under the same computational paradigm. Within this paradigm, particle-based FPM (such SPH, LSM or DEM) and ANN-based ML algorithms are just two faces of the same coin: the AI algorithm does not need to be linked to the physical model anymore because the AI algorithm and the physical model are the same algorithm. One of the main conclusions of this work, therefore, is that, thanks to the particle–neuron duality, the coupling between ANNs and particle methods is more promising than the coupling of ANNs with mesh-based models. This study only scratches the surface of the possible implications of deep multiphysics, but, considering the recent upsurge of research activity dedicated to the coupling of FPM with ML, I believe it can have a far-reaching impact in the field.

Supplementary Materials: The following are available online at http://www.mdpi.com/2076-3417/9/24/5369/s1, video 1: rigid_cell.avi, video 2: soft_cell.avi. Funding: This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) grant number: EP/S019227/1. Conﬂicts of Interest: The author declares no conﬂict of interest.

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