JHEP03(2019)186 Springer , March 12, 2019 March 27, 2019 : : January 16, 2019 : , dominated by the 505 , Accepted Published 10 Received [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP03(2019)186 [email protected] training region, allowing for reliable extrapolation. We , 1 , 1 value. h 1 , 1 h . 3 1811.06490 The Authors. c Superstring Vacua, Compactification and String Models

We provide the first estimate of the number of fine, regular, star triangulations , [email protected] Department of Physics, NortheasternBoston, University, MA 02115, U.S.A. E-mail: [email protected] Open Access Article funded by SCOAP ArXiv ePrint: triangulations far beyond the estimate that number ofpolytope triangulations with the in highest the KS dataset isKeywords: 10 Abstract: of the four-dimensional reflexive polytopes,provides as an upper classified bound by on Kreuzer theeties. and number Skarke of The (KS). Calabi-Yau estimate threefold This is hypersurfaces performed in(EQL) toric with architecture. vari- deep learning, We specifically demonstrate the that novel equation EQL learner networks accurately predict numbers of counts with equation learners Ross Altman, Jonathan Carifio, James Halverson and Brent D. Nelson Estimating Calabi-Yau hypersurface and triangulation JHEP03(2019)186 9 22 19 – 1 – 21 6 5 21 20 14 7 12 5 11 15 22 10 = 491 polytope 1 1 24 , 1 4 h 5.1 Classification of5.2 the 2d facets Training data 5.3 Machine learning and extrapolation 4.5 Model selection 4.6 Estimate for4.7 the total upper The bound 4.1 Training data 4.2 Initial attempts 4.3 Neural networks 4.4 The EQL architecture 3.1 Distinguishing unique3.2 facets Performing the3.3 classification Classification results in metric moduli space; thesethe may be case utilized of to Calabi-Yau generate hypersurfacesstructure large in of ensembles toric of a geometries. varieties, triangulation, Calabi-Yautransitions In and data in operations the is Calabi-Yau on hypersurface. encoded the in Thegenerate transitions triangulation the a between often these large objects induce network naturally that topological encodes ensemble structure. Triangulations and Calabi-Yau manifoldsin are combinatorics objects andoperations of algebraic on intrinsic triangulations geometry, such as mathematical respectively. flipsare may interest be canonically In used to related the determine to whenSimilarly, former two elementary one triangulations topology case, another, changing operations elementary or onconifold Calabi-Yau to manifolds transitions such populate may as relate flop large two or such ensembles manifolds of to one triangulations. another by finite distance motion 1 Motivation 6 Conclusion 5 Comparison to 3d polytopes 4 Machine learning numbers of triangulations 3 Facet classification Contents 1 Motivation 2 Approach JHEP03(2019)186 2- ≥ ], and 20 – 16 ] in this directions 23 – ] on a network of string 21 8 , 8 ] also give rise to large ensem- ] transitions induce space-time 5 3 , 4 ] selects models with large numbers 10 ], computational complexity [ 15 – 11 ] and conifold [ , – 2 – 2 9 ]. There, flop [ 1 ], could then lead to concrete statistical predictions for physics in 7 ]. It is therefore natural to expect that, in addition to the formal progress 17 ] for additional promising results. ] in a well-studied bubble cosmology [ 9 29 – ] by Kreuzer and Skarke. With 473,800,776 reflexive polytopes in four dimensions, 24 ]. The associated collection of four-dimensional effective potentials and metastable , 30 6 8 Let us comment further on the desingularization process, since we will be studying the Instead, in this paper we take a modest but necessary first step in this direction. We One concrete goal is to understand the full ensemble of Calabi-Yau threefolds, network- A natural direction is to understand cosmological dynamics on the landscape and In string theory, Calabi-Yau manifolds are some of the simplest and best-studied back- singularities. In thedimensional language singularities, of which toric in turn geometry,all corresponds this 2-codimensional to corresponds cones a to in fineintegral the blowing (i.e. points corresponding maximal) up on subdivision 4d all the of reflexive convexthe polytope. hull toric for By variety, this and considering thereby subdivision, only the we class preserve of the the canonical Calabi-Yau divisor hypersurface of itself. Furthermore, place via a seriessignificant of growth blowups, in of the which numberto there of estimate are compactification this growth. geometries. typically many The possibilities, goal resulting in in thismanifolds work resulting is from this process.be In non-singular, order for its the resolved Calabi-Yau ambient threefold toric hypersurface to variety must have no worse than 1-dimensional associated to a triangulation of awork 4d [ reflexive polytope; the latter werethe classified in number a seminal of possibilitiesfurther since already many borders of these onin toric order intractible. to varieties contain obtain However, singularities a smooth that the Calabi-Yau must number hypersurface. first grows be The desingularization resolved process takes mological dynamics and vacuumthis selection is in far compatifications out of of reach string currently theory. for reasons However, will of estimate enormity the and complexity. numberare of one of Calabi-Yau the threefold most-studied hypersurfaces ensembles in in string toric theory. varieties, Many such which manifolds are naturally that is clearly required,be data necessary science to techniques understand such theand as landscape; [ see supervised for machine initial learning works will [ like structures induced by transitions between them, and associated implications for cos- role in the dynamics.geometries For example, [ dynamical vacuum selectionof [ gauge groups andlandscape axions, are difficult as due well to its asundecidability enormity [ strong [ coupling. However, concrete studies of the vacua form the so-calledtions string of landscape, string which theory is for central particle to physics understanding and the cosmology. associated implica- mechanisms forproperties vacuum of selection. vacua [ four Such dimensions. dynamics, Global together structures with in statistical the landscape, such as large networks, may play a grounds on which to compactifymetry the in extra four dimensions dimensions of [ topology space while change preserving in supersym- aensembles physically of consistent string compactifications; manner generalized andbles fluxes may [ [ be utilized to generate large JHEP03(2019)186 4 we (1.1) FRST 3 n ] for the related 32 in the analogous problem 1 , 1 h , 6 . 292 , we give an overview of our approach  2 2 . 505 , it is interesting to study the interpretabil- , 10 for quadratic functions associated to trained – 3 – 10 than a standard feed-forward neural network. In FRST 5 n 1 ' , 1 h and FRST 4 n ], which we demonstrate is significantly better at extrapolat- 31 , we found that an EQL that utilizes quadratic functions makes FRST n is the number of FRSTs arising in the Kreuzer-Skarke ensemble. We obtain FRST predictions. That is, sufficiently complex phenomena in complex systems may not n This paper is organized as follows: in section In the case of Given these accurate predictions of The number of such fine, regular, star triangulations (FRST) grows exponentially as FRST admit descriptions in terms of simple equations in human-understandablefor variables. estimating the numberdiscuss of the FRSTs method in by theresults which 4d of we Kreuzer-Skarke this classified database. classification, the and In 3d our success section facets in of obtaining the FRTs of 4d these reflexive facets. polytopes, In the section EQLs. By studying an associatedables heat and map cross-correlations of are coefficients,of clearly we demonstrate of a that more large some importance number vari- pretability than of difficult. others, warm but spots This the suggest couldmay existence that be be many an suboptimal, variables artifact but matter, ofn it which having could makes used also inter- a be quadratic the function, case which that there is no simple interpretation of an interpretable function thatby allows utilizing for simpler understanding functions andlikelihood than of standard generalization. intepretability. architectures, Accordingly, EQLs in principle increase the accurate predictions; see sections artificial intelligence, interpretability is aand major it current is goal of oneidea machine of of learning the research, conjecture reasons generation,into for by developing rigorous which the results. interpretable EQL numericalral architecture; In decisions sciences see the may such [ EQL be as context, physics, turned where the a idea physical is phenomenon to is mimic often what described happens in in terms of natu- lation significantly beyond the regime inprovides which an training estimated data upper is available.in bound Predicting toric on varieties. the number of Calabi-Yau threefold hypersurfaces ity of the predictions made by the neural network. Sometimes refered to as intelligible this estimate with deep learning,an specifically equation a learner novel (EQL) neural [ networking architecture triangulation known predictions as to large particular, by demonstrating accurate extrapolationfor to 3d high reflexive polytopes, we lend credibility to the 4d prediction, which requires extrapo- by the necessitythe to number group of subsetsupper distinct of bound FRSTs triangulation gives into a topologically-equivalent concrete classes, upper bound. Ourwhere main result is this tutes a triangulation, and each givesvariety. rise Finally, to by a considering distinct only choice ofby regular desingularization projecting triangulations of down such the toric that from they a can higher be dimension, obtained we preservethe projectivity complexity of of the the variety. singular toric variety increases. While this number is partially offset each configuration of simplexes in the maximally subdivided (i.e. fine, star) polytope consti- JHEP03(2019)186 (2.1) whose n Z ]. It would 38 ]. Even with } ∈ v 37 associated with { 10 [ 1 , > ∼ 1 h 1 , 1 h 776 reflexive polytopes, as , } ∆ 800 ] can, in principle, produce all , ∈ 35 [ v ∀ 1 TOPCOM ≥ − ]. The results were in good agreement with v · 32 – 4 – w ] and | 34 n [ Z , we perform a similar analysis for the 3d reflexive ∈ ]. The ambient 4d space described by these polytopes ], all triangulations of all 4d reflexive polytopes with 5 w 30 PALP 36 { = ◦ ∆ ] that a hypersurface in a toric variety can be chosen to be Calabi- 33 6 were obtained. The nearly 652,000 unique triangulations required approximately ], the number of FRSTs for many of the 3d reflexive polytopes was computed, and an ≤ There is reason to believe that application of machine learning techniques may allow for Crucially, there are typically very many inequivalent FRSTs for a given polytope. A reflexive polytope ∆ is defined as the convex hull of a set of points 39 1 , 1 In [ estimation of the total number was obtained,thereafter, using another standard estimate triangulation techniques. was obtained Soon a using decision-tree supervised regression machine model learning.number was Specifically, of trained FRSTs on for known the results, remaining cases and [ used to estimate the polytope dataset must remainpower unobtainable, or barring dramatic a advances vastlymotivation in for superior computational the triangulation current work. algorithm. It isan this estimate of impasse the total which numberof forms of principle triangulations already the in exists the in KS dataset the to KS be set achieved. of A 3d proof reflexive polytopes, of which there are 4319. 120,000 core-hours of processinggated time somewhat to by obtain. exploitingthis the The assistance, reflexivity computational however, property burden obtaining of can morepolytope these be than (in polytopes a miti- a reasonable [ single, computational canonical time)appear, becomes triangulation therefore, difficult for that for a an given enumeration of the unique triangulations in the KS 4d reflexive possible triangulations of aingly problematic given computationally polytope; as in thefor overall practice, size this such of computational the a load polytopethe request is grows. polytope. becomes A the increas- good value Forh proxy of example, the in topological [ quantity hypersurface in a suitably de-singularizedis ambient space. equivalent The to process of obtainingpolytope. de-singularization a fine, regular, star triangulation (FRST)While of software packages the such 4d as reflexive dual polytope is itself a lattice polytope.classified In by four Kreuzer dimensions, there and are Skarkeis 473 [ generally a singular toric variety. A proper Calabi-Yau manifold will therefore be a Batyrev has shown [ Yau if the objectcondition underlying of the reflexivity. construction of the variety, a lattice polytope, obeys the successful EQL model.polytopes and In show section that aits model training of range. a similar architecture is able to extrapolate far outside 2 Approach we discuss our unsuccessful initial machine learning attempts, our input features, and our JHEP03(2019)186 . 6 12 when ≥ 1 , 1 h ], Kreuzer and Skarke 30 – 5 – . 3 ]. The objective of the current work was to obtain such an 39 . The data associated with these explicitly-triangulated facets then 1 . , 1 4 h For their classification of the 3d and 4d reflexive polyhedra [ Any such machine learning estimate will obviously have limitations, and ours come With this classification in hand, we then explicitly computed the number of FRTs for As was done for the 3d case, the method employed here is to estimate the number of reflexive polytopes, since individual facetswe may appear must determine in under multiple what polytopes. conditionsand In two then facets particular, should find be a considered method to to be determine equivalent, whether thedefined a conditions normal are form, satisfied. with the property that two reflexive polyhedra have the same normal In this section weand review then methods apply those that methods easily to determine the when classification3.1 two of facets facets are in Distinguishing 4d equivalent, unique reflexiveWe facets polytopes. must distinguish between three-dimensional (3d) facets of the four-dimensional (4d) training, we were able tonumber of see cases that as our thelimitations, model number and extrapolated of the with triangulations potential stable increased. overcounting results associated We over with comment a our further large approach, on in3 these section Facet classification with fewer triangulations, thisthe problem range involved seen extrapolation inour to estimate the is output training sound. values set.ultimately The beyond employed was equation However, developed learning as especially (EQL)set. for we neural By extrapolation network will withholding outside architecture the of discuss that subset a below, we of given training our we known believe results that that first appeared at subject of section from the two main issuesknown that results we faced. for 1.03% First,nearly of as noted 99%. cases, above, meaning Second, we that were as it only the able was facets to necessary that obtain to were estimate able the to value be for triangulated were necessarily ones at low values of became the training datawas for employed to supervised construct machine aFRTs learning. model of which each Ultimately, predicts facet. a thethe neural natural From 4d logarithm network this reflexive of we polytopes. the arrive number at The of machine an learning estimate techniques, of and the FRST total estimate, number are of the FRSTs of billion facets. Incomputational order repetition, the to first step better was afor get classification doing of a these so 3d is handle facets. the on The subject procedure this of set, section as as many well of as the facets avoid1.03% as unnecessary of possible. the Due to total computational — constraints, this primarily consisted consisting of only of facets that first appear in dual polytopes estimate for the 4d reflexive polytopes via similar techniques. FRSTs of a given(FRTs) of polytope each as of the itsdataset product facets. (which of However, equals counting the the the number number of number of fine, vertices of in regular facets the triangulations 4d by reflexive duality) shows that there are over 7.5 the original estimate of [ JHEP03(2019)186 ) 1 , ). 1 Z } h 0 (3.5) (3.2) n, ∪ { . ) F ) (3.4) ) (3.3) }} 1 }} . }} , 1 ) 0 0 , − , ) (3.1) }} , 0 0 1 , 1 , = conv( }} 1 0 0 − − , { F , , , 1 , 0 1 1 C } , − 57%. − 0 1 {− . , , both of which appear , , , is an associated Calabi- , 2 1 2 1 1 ]. The origin is interior to } , { F − 0 , 0 X , , 40 {− , } 2 1 , 0 0 , { } , { and , 0 0 , 0 , } , , } 1 1 0 1 0 0 , F , , , 0 0 1 0 {− , , ), where , { , ) transformation. 0 1 , 1 } , Z X , } 1 ) transformation. While powerful, this 1 ( 0 , 0 1 {− { Z n, , , { 0 , , 1 , , 1 } } n, h , } 0 1 0 , 0 0 . , , , , 1 0 1 have the same normal form if and only if their 1 0 }} , , – 6 – , 2 1 0 0 {− . This also means that we need not worry about 0 F , , , , {− , 0 1 1 , F C } 1 , } 0 { \ 0 , 1 , {− {− , , 0 , , F } 0 , 0 and = 2 polytope (given by POLYID 21 in the ToricCY } } 0 { C , 0 , 0 1 , 1 , 0 below, this particular facet, known as the standard 3- , , 0 1 } F , , 1 , 0 0 0 1 1 C , , 0 , h , 0 0 1 {− are related by a GL( 3.3 , , 0 , , 1 1 2 0 } }{− {{ , 0 0 F 0 , , { 0 0 {{− {{− , , , ) = } 0 0 and 2 , , 0 F 1 , 0 0 C 0 F , {{ {{ , we constructed the associated subcone defined by 1 = conv( = conv( F , 0 1 2 ]). Each of the facets are the convex hulls of four vertices, as below: { F F , 36 } ) = NF( 0 1 = conv( = conv( , F 2 1 0 C F F , 0 As discussed in section As an example, consider the following two facets To circumvent this issue, and thereby be able to use the normal form, we employ the C C , 45% of all facets, while no other facet accounts for more than 8 1 . NF( Yau hypersurface. As itCalabi-Yau, we is identified the the dual facetsvalue, polytope of we the that identified dual is the polytope facets triangulated inlist that when each of appeared case. constructing unique in this facets Hence, each as forwith dual well each polytope, which as multiplicity). keeping recording both which a facets master appeared in each dual polytope (and 3.2 Performing the classification The primary practical challenge of thedataset facet classification itself. was the volume of Inthrough the 4d order the polytope to polytopes have in a order consistent of direction increasing for our computation, we worked We see that the two subcones haveRemoving the the same origin, normal form, we see and thus that{{ the both facets facets are are equivalent. equivalent to the 3d polytope with vertices simplex, is the20 most common facet among the 4d reflexive polytopes. It accounts for Computing the normal form for each subcone, we find that Adding the origin to each facet, we obtain the associated subcones as dual facetsdatabase to [ the same The subcone isAdditionally, a as full-dimensional the polytope, originorigin and is is so the the only itslattice sole lattice normal translations point interior form when in lattice comparing cantransformation, point subcones. be two of subcones As computed. a theassociated reflexive origin facets is polytope, fixed the under any GL( normal form has the restriction that it can onlymethod be of Grinis used and on Kasprzyk, full-dimensionalevery as reflexive polytopes. described polytope, in and section thus we 3.2for know of that each [ none facet of the facets contain the origin. Thus, form if and only if they are related by a GL( JHEP03(2019)186 , 1 , 1 1 The h = 27, 1 , 1 , allowing h (Right) Sage value. 1 , 1,1 value. The dual and 1 h 1 h , 1 h 6% of the total number of facets, . value. 1 ,

. The shape of the two distributions 1 lattice polytope implementation of our 0 100 200 300 400 500 h

1

8 6 4 2 0

10 log Polytopes ) ( ++ , and we recorded the smallest such value. value is shown in the left panel of figure 1 , = 33. 1 1 1 , h , 1 1 – 7 – h value; i.e., the same facet may appear in many h 1 , 1 h source code for many of its underlying calculations. This PALP 1,1 h 557 unique 3d facets, which is an order of magnitude less than the , 487. We found that a relatively small number of facets accounted for , The logarithm of the number of new facets at each 990 , 985 , (Left) 471 , .

0 100 200 300 400 500 As we performed the classification procedure, we recorded the number of facets which Computation was done on Northeastern University’s Discovery cluster using the

7 6 5 4 3 2 1 0

10 Facets log ) ( The distribution of thesewhich new should facets be per comparedvalue, to which the is distribution given ofare the in very number similar, the of albeit right decreased polytopesdistributions panel by of have around of different a an figure peaks: given orderwhile of the the magnitude. number number We of of note new reflexive that facets polytopes the peaks two peaks at at the majority of theaccounting facets for that just appear over across 20% the of polytopes, the total. with theappeared most for common the facet firstdifferent polytopes time with at different each values of 3.3 Classification results Using the above method forare the a facets total of of everynumber 45 of 4d polytopes reflexive themselves. polytope, This wewhich is found is approximately that 7 0 there normal form computations were doneown creation using which a used C the offered significantly faster performancethe than classification to the be LatticePolytope finished class on in the timespan of a few weeks. SLURM workload manager. Asfor identifying the each dual polytope, facets we isrunning an were time independent able computation by to severalwas use orders performed distributed after of computing identification magnitude. to had decrease been Filtering our completed and real-world for removal each of duplicate facets Figure 1 logarithm of the number of reflexive polytopes at each JHEP03(2019)186 1 , 1 h ). 6 value. We value, as a 1 10 1 , , 1 1 × h h 1,1 h . 3 value, as a fraction of the number of

1 , 1 0 100 200 300 400 500 , the ratio quickly saturates to a value

h

1

1.0 0.8 0.6 0.4 0.2 0.0

Polytopes Facets 250 represents the relatively small number that a small number of facets dominate the > ∼ 3 . In the left panel, we show the number of new , in which we show a histogram of the number 1 , – 8 – 2 1 3 h shows the number of new facets at a given values. Coupled with the information contained in 2 1 , number of polytopes to that particular 1 h value, as a fraction of the total number of polytopes at that 1 , The total number of facets found through each bins. 1 h 1 , 1 h cumulative 1,1 , we were able to achieve all possible FRTs for a very small set of (Right) h 1 . 1 , 1 The number of new facets at each h ). The 100 most common facets account for 74% of the total number. As 3 (Left) ) to the number of 4d reflexive polytopes in the KS database (470 . 6 , it is clear that the overwhelming majority of facets are encountered well before 250, though (as we will see), the number of triangulations are dominated by the

10 1 ' 0 100 200 300 400 500 value. The scatter that emerges for In understanding the reliability of our estimation methods, it is important to ask how One can see from the left panel of figure Finally, the right panel of figure Along with the cumulative distribution, we also show these distributions as fractions of

×

0 1 2 3 4

1 1

, , Facets Polytopes 1 1 database, with the frequencypanel counts of decreasing figure linearly onmentioned a in logarithmic section scalethe (the total right number of 3dThis facets, is but indicated these by cases the represent blue over shading 88% in of both facets panels by of appearance. figure often the unique facets thatKS we have database. enumerated actually This appear is inof the the appearances subject 4d of of polytopes a figure in given the facet. value, normalized by the see that after the peakof in approximately the 0.1. right panel(47 of This figure is nothing more than the ratio of total unique facets found h of polytopes at thesefigure rather large h outliers in the very large the relevant numbers of polytopes infacets that figure appear at a given Figure 2 polytopes at that fraction of the total number of polytopes up to that point. JHEP03(2019)186 , 45% (3.6) . (left panel) ), the output of the . , which takes a specified }} 1 , Number of appearances 0 model , 0 , 0 { , } 0 supervised learning ,

1e+00 1e+03 1e+06 1e+09

1 e0 e0 e0 e0 1e+08 1e+06 1e+04 1e+02 1e+00 ,

0 Count , 0 { , } – 9 – 0 , 671 times in the database, accounting for 20 0 , , 1 , 0 150 , { , } 0 528 . The orange bars represent the total number of facets, while the , , 0 output pairs, and attempts to minimize a predetermined loss , 0 → , 1 {{ (right panel) . However, the S3S, being itself a simplex with no interior points, has , which are adjusted by the machine learning algorithm. The algorithm is 4 Number of appearances weights . Histograms of the frequencies of facet appearances, with bins spaced linearly

= 1 and maintains a consistent presence through the database, as evidenced by the 0.0e+00 5.0e+08 1.0e+09 1.5e+09 2.0e+09 In this section we will estimate the number of FRSTs of 4d reflexive polytopes by using The most common facet is the 3d polyhedron known as the standard 3-simplex (S3S), e0 e0 e0 e0 1e+08 1e+06 1e+04 1e+02 1e+00

1

, Count 1 supervised learning to predictinput the number is of a FRTs set ofFRTs of the for features facets. that which facet. For describe thisthe We a application, structure describe the facet, of in and the this the model section desired that the output we features obtained is which via the we supervised number used learning. of as input and machine learning algorithm is aset function, of commonly inputs called and a produceseters, a unique or output value. Thistrained function on contains numerical a param- setfunction of by input adjusting the weights of the function. only one FRT, and thus has essentially no effect on the total number4 of polytope FRSTs. Machine learning numbers ofIn triangulations supervised machine learning (often called simply The S3S appears aof total all of facets, 1 andh appears in 87.8%graphs of in the figure polytopes. It first appears as a dual facet at which has normal form vertices Figure 3 and logarithmically blue bars show the amount for which the number of FRTs is explicitly computed, and thus known. JHEP03(2019)186 120 The > ∼ values. 1 , 1 1 , h 1 (Right) h for 4 value. 1 1,1 , 1 h b) shows more clearly h 4 value at which they appear. 1 , 100% 1 1.652% 0.003% 1.028% 99.983% 82.097% 69.625% h % Triangulated

0 20 40 60 80 100 120

0 03% of the total. These consist almost . 80 60 40 20

100

= 14 was the last value at which we were Percentage 1 , 1 h 3,907 1,360 92,162 142,257 108,494 124,700 472,896 – 10 – Triangulated 120. We note that the erratic portion of the figure ≤ 92,178 Facets 142,257 132,153 180,034 236,476 1 , 45,207,459 45,990,557 1 1,1 h h 1 11 , 15 1 12 13 14 15 − h > Total 1 The percentage of dual polytopes that contain S3S at each 880 of the facets, which represents 1 , (Left) . . Dual facet FRT numbers obtained, binned by the first

shows our progress. As one can see, 0 100 200 300 400 500 1

0 Due to computational restrictions, we were only able to find the actual number of

80 60 40 20

100 Percentage Table 1 the prevalence of S3S. Figure 4 same graph, truncated at accounts for fewer than 1 million polytopes. The truncated graph (figure entirely of facetsTable which first appearable in to the obtain dual anmay appreciable polytope constitute fraction only at of a the relatively smallfacet real low fraction appearances. FRT of values. the While unique facets, the they triangulated account set for over 88% of all The first step inmodel. our In machine this learning case,possible, process this in meant was it order to was to generate necessary obtain to data the triangulate on best as possible which many training to ofFRTs set. the for train 3d 472 a facets as 4.1 Training data JHEP03(2019)186 1 , 1 h (4.1) . executable 2 value, indicating ]: Linear Discrim- 1 TOPCOM , 1 32 h values that it had trained 1 , 1 h ,

i P i are the predicted and actual values values was poor. As the majority of − A i Python package, using 35 estimators. i 1 , A A 1 facet.

h th n =1 i and i X i value and the number of triangulations. This × P 1 , 1 at which they appeared as a dual facet, which – 11 – value of the facet. Our reason for this was that h n 1 1 , 100 , 1 1 ) for the scikit-learn h h FRT N increased, fewer cases were able to complete as the facets MAPE = 1 , 1 values, the inability to extrapolate well was unacceptable. ]. As was done in the 3d case, for each polytope we constructed h 1 , 32 ), consisting of the number of points, interior points, boundary 1 v in order to obtain the FRT number. As with the classification h , n : as b 1 , n i up to some maximum value (typically 11). This allowed us to see how , n 1 p , 1 n h 10. On the training set, the model achieved a MAPE of 5.723. It performed ≤ is the number of data points, and 1 , n 1 code. The n-skeleton of a lattice polytope consists of all points which are not interior h As one can see, the MAPE value consistently increases with the As an example, we show here the results from one of these attempts. The model is an We trained models using the same four algorithms as discussed in [ When training with this data, and throughout our process of finding a good model, To obtain the FRT values, we first computed the 2-skeleton of each facet using our ≤ ++ If we could beacceptable confident model. of However, a when similar wevalues used whose accuracy the FRT for numbers model the are to rest known, predict we of on obtained the facets the from facets,that results higher this the shown model could in is table be less accurate an farther away from its training region. Further, the rightmost the facets lie at higher ExtraTreesRegressor model from the We trained the model on5 a data set consistingsimilarly of on 60% the of test set the consisting known of values the from other the 40% range of these values, with a MAPE of 5.821. inant Analysis (LDA), k-Nearest(CART), Neighbors and (KNNR), Naive Classification Bayes andmance (NB). Regression on Trees the Of training these, set,on. the as well CART However, as algorithm its on gave ability the to impressive test extrapolate perfor- data for to the higher where for the output, which here is ln( Our first attempt atpolytopes obtaining in section a 3 of modelthe [ was 4-tuple to ( followpoints, the and same vertices, pattern respectively. Ourmean used metric absolute for percent for the error determining 3d (MAPE), the which model’s is success defined was as the facets with well the model would extrapolatewould to prove FRT numbers to higher be than an it issue had throughout seen,4.2 something our that attempts. Initial attempts we organized the facetswe by will the refer to first there from is here a on rough correlation asis between the this reflected in table became more complex. For this reason, the training set for each model only contained C to any (n+1)-dimensional face.points2nfinetriangs We then passedprocess, the 2-skeleton we to used the several Northeastern’s hundred facets Discovery at cluster a to time. run simultaneous calculations for JHEP03(2019)186 · values neuron 1 ), and , is called 1 activation h f offset (or ) value greater than bias , such that there are no = 22 were triangulated, FRT 1 , N 1 h itself is called the layers , while the set whose output f 9.903 9.189 10.179 10.067 . All other layers are referred to is called the = 11, and for a majority of cases up Predicted mean b 1 value is 491. , is the input. The value of 1 , 1 i , h input layer 1 x h neural network, there are no cycles, and so output layer weights is 35. Conversely, in the 4d case, the number of – 12 – 1 9.582 , 12.638 11.755 10.882 1 ), using the ExtraTreesRegressor model, for h Actual mean FRT feed-forward N are called i ) whose argument w is one of the characteristics that define the neuron. b f + 9.065 6.566 17.403 11.566 i MAPE x · is a (finite) directed graph, each node of which is a neuron. For each 1 i , 1 w 14 13 12 11 h k i . P ( f . The parameters and the choice of . Prediction results for ln( = 14, in a data set where the maximum neural network 1 , In the general case, a neural network may contain cycles, allowing for complicated A We recall briefly here the definitions of a neuron and a neural network. A This example is emblematic of the problems we faced with traditional machine learning. output 1 hidden layers h connections between layers. Ininformation a moves only in oneoutput direction. to Cycles be ingiven influenced a facet neural by is network the independent allow previousfeed-forward for of case. the the output. current number for As other the facets, we number can of restrict triangulations ourselves to of the a at the head. Conceptually, aconnections neural between network nodes is organized in into theinvolve same the layer. original The set inputinvolves of data the nodes actual is whose output arguments known explicitly dataas as is the called the the represents the appropriate tensor contraction.function The function arrow in the graph, the output of the neuron at the tail is used as the input for the neuron Given the poor performance ofwe traditional supervised shifted machine our learning focus asseemed described to the above, most artificial suited neural to networks. our purposes. In particular,is a a feed-forward function network in a set where theFRTs was maximum only value obtained of forto all facets up through 4.3 Neural networks Models either fit thetraining regions. training data The primary poorly,worked), difference and or between the had the 4d case, 3d trouble is casecase, the extrapolating the (where inability number outside to this of generate simple of FRTs a for approach representative their all training facets set. that appear In the up 3d through outside of its training region. two columns illustrate the problemhigher with number this of model: triangulations. it12.467, The is which under-predicting model is on never facets similar predicts with to a a 12.595, ln( the highest value in its training set. Table 2 JHEP03(2019)186 (4.3) (4.2) = 8, and has . 1 , 1 0) , h , 0 } , 0 . 0 , , 1 , }} 16 1 1 , , , 2- and 3-simplices, respec- 1 0 0 , { , 3 N , 1 , } , 6 0 0 , , { , 1 13 , } , 0 0 , , 32 0 2 , { − , , 29 } , 2 0 , , 3 12 0 , { , , with normal form is given by 2 } 48 , – 13 – , 1 1 F { − 72 , , , } 1 0 13 , − , , 0 , 3 16 , 0 , , 3 1 { 10 , { , , } 8 1 } , , 0 0 , 10 , 0 up to 5 , 0 , 9 , 0 N , 0 , 8 value at which the facet appears in a dual polytope 0 { , 1 , {{ 10 1 , h (1 The total numbers of 1-, 2-,The and numbers 3-simplices of in 1- the andredundancy triangulation 2-simplices between higher-dimensional in simplices the triangulation, withoutThe numbers accounting of for 1- andtively, 2-simplices for shared between – – – , we find that it contains 72 1-simplices, 48 2-simplices, and 12 3-simplices. Of the Several quantities obtained from a single fine, regular triangulation of the facet: The number of edges The number of flipsdiffer of by a a seed flipinserting triangulation another. if of one the can 2-skeleton. be obtained Two triangulations from the other by removing one edge and The number of points in the 1-The and first 2-skeletons The number of faces Adding this additional data to our original inputs improved results, but not to the As as example we consider the facet We initially tried applying a feed-forward network to our 4-tuples of data, but despite • • • • • • point of satisfaction. As anwe example show of here the the struggles results faced from by one a such traditional model. neural This network, neural network has two hidden layers, TOPCOM 1-simplices, there were 29 unique13 ones, 1-simplices while there shared were between 324, two unique and 2-simplices, 2-simplices. 1 6 shared There shared between were none 5. between shared three, There between were 3 more 16 shared 2-simplices than between shared two. between Hence two 3-simplices, the with input vector describing this facet was This facet has 11 latticeOf points, these of 10 which 10 boundary lie9 points, on and 8 the 10 boundary are points, (leaving vertices.10 1 respectively. faces in The and the It 1- 16 interior). first and edges. appears 2-skeletons Its in of flip a this graph facet dual has contain 13 polytope nodes. with Obtaining one FRT for this facet from trying a variety ofmodels architectures consistently faced underpredicting similar outsidein issues of their the as prediction training innumbers regime. the attempts, of previous additional To points, aid subsection, interior input our points,were with data computed boundary networks for points, was each and generated. facet vertices, to the be In following used quantities addition as inputs to to the the neural network: JHEP03(2019)186 (4.4) values 1 , 1 h . This belief values. The =14 1 , 1,1 1 1 h , 1 Percent error Percent h . h 7 −40 −20 0 20 40 , with a MAPE of 6.304. 1

values, our results were as

, 0 0 0 400 300 200 100 0 ) to be positive.

1

1 Frequency , h 1 FRT h N 10.324 10.753 11.094 = 14. However, the MAPE values . 1 ) , Predicted mean 1 values which make up the bulk of our h , x 1 , 1 h =13 1,1 h Percent error Percent – 14 – ) = max(0 10.882 11.755 12.638 x ), using the traditional neural network, for ( Mean value −40 −20 0 20 40 act FRT f

N

0020 0040 5000 4000 3000 2000 1000 0 increases, the percent error distribution skews to the left, Frequency 1 , 1 h 5.904 6.550 is equal to the positive part of its argument: 10.915 MAPE 11 to avoid biasing the model towards higher act . As , and the mean predicted values are falling behind the true mean f 1 ≤ , 1 5 ], and will find that it has significantly better extrapolation ability. , 1 1 1 12 13 14 h , h 31 1 h . This model performs much better than the ExtraTreesRegressor outside =12 ≤ 1,1 3 h Percent error Percent . Histograms of the percent error of the feed-forward neural network’s predictions in the . Prediction results for ln( −40 −20 0 20 40

The model was trained on a data set consisting of an equal number of randomly chosen

0010015000 10000 5000 0 Frequency data set. 4.4 The EQLIn architecture this section wefirst will introduced instead in utilize [ an equation learner (EQL) architecture, which was are increasing with values, leading one to believeis that confirmed the by model examining is therange, underpredicting shown histograms at in of higher figure the percentindicating error that values the in model theis is extrapolation not not suitable keeping up for with extrapolation the to increasing the values. higher Hence this model points from 6 model performed well on theHowever, when test we set evaluated of the values model’sshown from progress in the on table same higher of the training region, with a MAPE of 10.915 at its activation function A ReLU unit was chosen for the final layer as we want ln( Figure 5 extrapolation region. Compare to extrapolations in the EQL networkeach in with figure 30 nodes. Thea first tanh layer activation has function. a sigmoid The activation final function, layer while is the a second rectified has linear unit (ReLU), meaning that Table 3 outside of its training region. JHEP03(2019)186 are (4.5) = 30 z 2 30 -dimensional -dimensional o of output val- m n o n n . = 3 sandwiched between m o n n to the final R z of the elements, an activation ∈ b u elements are pairwise multiplied, = 4 and v to an intermediate m x n 1 2 b x f f + , and outputs some number i W x n – 15 – = z and some bias vector i n -dimensional input × i m n n R ∈ , while the remaining two elements are multiplied together. i . f W 6 via an affine transformation. That is, z via a non-linear transformation. For this transformation, the elements of output values. These nodes are called binary units. This pairwise multiplication . A representation of a simple EQL layer with y 2 v The second stage takes this intermediate representation Each layer in an EQL network consists of two stages: a standard linear stage followed 4.5 Model selection Our final neural networkinput model and output is layers. simple, Thelinear with input stage layer only consists of one of the 22 hiddenother EQL nodes 30 EQL as layer being layer previously contains the between described. 45 binary the The nodes. units. The Of output these, from 15 this layer are thus unary consists units, of with 15 + the step is thenodes key of aspect the network. of Morepairwise the generally, multiplication nonlinear layer, could interactions between as also thelayer be it nodes is utilized. other shown allows than in A for figure diagrammatic nonlinear representation interactions of an between EQL the output divided into two parts which arefunction acted is upon applied. differently. These To function nodes can are be called unary applied units. togiving each In unary principle, unit. a different The activation other representation for some weight matrix by a non-linear stage.in a The standard neural non-linearof network stage the layer, essentially linear but replaces stage’s differslayer output the accepts significantly some tensor. activation in number function that of Likeues. input it any The values feed-forward changes linear neural the stage network shape maps layer, the an EQL Figure 6 two fully-connected layers. The firston two by elements of activation functions the intermediate representation are each acted JHEP03(2019)186 = 11 1 , . 99, and 1 max . 4 h = 0 = 6, and examined 2 1 , β 1 , 1 min 1 max = 14 h h 9, 1 . values, from 6 to 11, , 4.934 7.469 4.416 5.834 7.469 6.598 1 ≤ 1 values. , h 1 1 1 , , = 0 h 1 1 h h 1 β ≤ 001 was used on all weights . = 13 1 , 1 , 1 min 4.463 7.626 4.490 5.172 7.626 6.699 ), we can see that this is not the 1 = 0 h 11. h 7 λ ≤ values; there are, for example, over 37 1 , 1 1 max , ranges. The model with 1 = 12 h 1 Extrapolation MAPE , h 1 1 , 4.444 7.512 4.393 5.048 7.512 6.647 h 1 h = 6. We then tested each model on the full set – 16 – 1 , 1 h and 14, respectively. We also note that, regardless of 5 have few triangulations and constitute relatively few , ≤ 13 5.551 6.967 5.643 7.184 6.001 7.297 , 1 , 1 6 in each case. In order to have an adequate range on which increases. h = 11 as at Test MAPE 1 ≥ = 12 1 , , 1 1 1 1 , h , h 1 1 min 1 , h h 1 max 11 11 11 10 10 10 h 1 , value. This was done rather than taking a percentage of the data to avoid 8 7 6 8 7 6 1 min h 1 , . At each layer, an L1 regularizer with 1 8 h − 10 ) and the distributions of percent errors (figure . Results of training our model on various 5 × One might worry that, despite the promising MAPE values, this model is persistently As the table shows, the model trained on the largest range of The training data sets consisted of an equal number of randomly chosen data points The Adam optimizer was used, with hyperparameters To select a model, we trained across multiple ranges = 1 underpredicting like the previous(table model. However, examiningcase. the The mean mean predicted predicted values centered values around stay close zero to as the true means, and the percent errors stay performed the best onthe both model the that we test used set toextrapolation generate and range predictions when contains for a the extrapolating large rest to numbertriangulated of of higher the facets points: data values. at set. there This are We note 92162,the is that 108494, training the and range, 124700 the models have stable MAPE values in the extrapolation region. biasing the model towards fittingtimes to as the many facets higher at of values across itsmean training absolute range percentage to error see (MAPE),how how as well it this the performed. gives model a is Our size-independent performing. metric measurement The of of results choice of was our the training are shown in table the results. As thedata facets points, at we chose to test our model’s extrapolation, we also chose from each  in order to selectused for for the the most EQL important hidden features. layer to In help addition, select a the dropout most rate optimal of neuron 0.1 configuration. was nodes. The unaryeach activation of which the was unarythe units. found natural to The logarithm be output offunction stage the the to consisted most number ensure of of successful that one the FRTs. was node, output to This whose was square final nonnegative. value represents output had a ReLU activation Table 4 performs well on the test set and the best on extrapolation to higher JHEP03(2019)186 ) for each | c | ( =14 1,1 10 h Percent error Percent , makes any stronger 1 , −40 −20 0 20 40 1 . This formula does not values that are not in the h

1 0010010020000 15000 10000 5000 0 ,

1 4.3 Frequency h . 5 10.722 11.591 12.492 significantly beyond the training set. 1 , 1 h . We can see that there are two brightest Predicted mean =13 9 1,1 h Percent error Percent the formula learned by our EQL model. The values significantly beyond that in the training 8 1 – 17 – , 1 h = 491. For this reason we will perform an analogous 1 , −40 −20 0 20 40 1 , and will demonstrate that in that case the EQL accu- 10.733 11.755 12.638 h

5 0010010020000 15000 10000 5000 0

= 25. This suggests that the EQL may also make accurate Mean value Frequency 1 , 1 = 11, we will accurately predict numbers of FRTs for all avail- 1 h , 1 1 , 12 13 14 1 h h , which has 1 , 1 h =12 1,1 polytopes in section h Percent error Percent in the normalized expression to determine their relative importance on the are in the same order as in the example in section d c i x . Histograms of the percent error of our chosen model’s predictions in the extrapolation . The true mean values and the mean predicted by our model in the extrapolation region. −40 −20 0 20 40

For completeness, we present in figure On the other hand, the extrapolation here is only to three

0010015000 10000 5000 0 Frequency without accounting for redundancy.there However, are while many these others terms atterms a have are similar the more order highest important ofa values, than magnitude. lack others, of Thus, the while knowledge complicatedstatements we of difficult. form can how say of these that the variables some will function, change along at with larger we normalized each variable sothe that mean it value has of each ancoefficient variable expectation in value the of training 1. data.value. This We then was The calculated done results log using ofsquares, this which are correspond shown to inseed the figure triangulation interactions of with the both number the of number unique of 1-simplices in 3-simplices, the and the number of 1-simplices predictions in the 4d polytope case at values of variables seem to lend itselfthe to fact any that its intuitive variables interpretation. will generally Its be interpretability of is different scales. also To hindered attempt by to adjust for this, tope with maximal analysis of 3 rately predicts the number ofset; FRTs for e.g., training up to able facets, which go up to Figure 7 region. Compare to extrapolations in the EQL network in figure training set, which provides some cause for concern since we will be interested in the poly- Table 5 JHEP03(2019)186 − + − + + − − + − + − + − + + + + + − − − − − + + − − + + + + + + − + − − 3 9 9 8 7 5 6 0 18 10 x 21 17 x 19 20 15 14 20 21 x 17 2 18 16 11 x 20 21 17 10 13 18 x 18 16 2 13 19 14 x x 11 2 12 12 15 x x 1 x x x x x 2 x x 3 x x 3 x x x x 0 x x x x x x 2 x x x x x x x x x 0 x 0 0 x 0 x 7 7 7 7 6 x 3 3 6 6 x x 3 5 5 14 14 13 14 12 13 x 10 10 10 x x x x x x x x x x x x x x x x x x x x x x x x 44341 383880 0707 01711 00333 . . 00375 548990 . . . . . 20046 03016 19461 35742 08904 0459 . 0 . . . . 0311 0 02683 1 . 00024 24571 16209 01789 04801 10175 00198 01352 16816 01429 00204 01481 33024 . . 01197 0 . . 0 0 . 00012 00073 ...... 0 00039 . . 00556 00365 . +1 +0 +0 . 01379 00151 11095 . . . 0 . . . . 0 0 . − 0 0 0 0 + 0 3 8 − − 0 0 − 0 − 0 0 0 − + 0 − 17 x x + 0 − + 0 − − 9 + 0 8 + 0 + 0 − + 0 + 0 + 0 − 7 x 0 − − 18 − − 2 4 + 0 + 0 + 0 + 0 − 6 13 2 9 − − 2 21 x − 12 x x x x x x 20 x x 1 x 2 2 20 2 19 x 2 15 2 16 21 17 20 21 17 10 13 19 1 18 16 x 19 14 15 14 x 11 x 17 x x 5943 16 11 12 x x x 18 12 x x x x x x x . x x x x x x x 11 x x x x x 4 x x 13 6 6 6 6 5 x 1 2 2 5 5 x 70274 2 9 9 4 13 13 x 9 12 . x x x x x x 11 12 x x 0257 x x x 72012 − . x x 02246 x x x x 0 x 11071 . x x . . 5381 0025 7 . 00072 . 0 05497 00125 02539 0 − . x . . . 02357 0 0125 07068 . +0 0 . 01206 0 0 + 0 . − − . 0 16 01621 2 2 09782 01556 04606 05244 03714 02332 0 08977 00186 + 0 02334 03319 . − 00715 ...... − . . 00924 11092 9 00223 00541 00468 . . 4 x + 0 x 04658 00253 − − . 03737 02735 ...... 0 21 7 0 0 − . . x 9 x 0 0 8 − 0 + 0 x x 0 + 0 16 8 3 x 6 x 19 20 − 1 + 0 + 0 + 0 + 0 − + 0 x 829660 − + 0 x 0 x + 0 + 0 + 0 + 0 1 x + 0 − . − + 0 x x 20 18 + 0 − − + 0 x 3368 12 x 13 0 . 2 x x 15 20 x x 20 21 17 10 13 18 19 21 17 x 0 x 16 18 15 19 14 19 253460 x 16 11 x 15 14 − 11 18 . x x x x x x x x x 12 x x 16 x x x x 10 x x − x x 6 11 x x 1 5 5 5 5 4 1 1 3 x 12 x 4 4 08246 8 8 00599 x 2 1 8 12 12 x x 0095 . x x x x x . 3 11 x x x 05763 x x x . 10 11 x x 29528 − x . x x x x 00003 . x x x . x 1 19063 0 00165 . 0 15 + 0 x . 00997 01115 + 0 0208 + 0 0 . . . x + 0 − 9 0037 0 01227 4 8 00442 − . 06183 08754 01284 00025 04584 . 00257 02935 11551 7 x 2 8 05976 . 02568 00244 03144 . . . . . − 0266 . . . x 05335 540610 . x 01278 00317 00919 0 . + 0 . . 03125 . 7 02014 x . . 0 x . . . + 0 − 0 0 0 0 00161 2 . . 0 + 0 . 2 5 0 0 x 0 0 0 21 0 6916 0 17529 x x 0 − x . . 16 x + 0 x − + 0 − − − − + 0 20 19 + 0 + 0 +1 + 0 − + 0 + 0 + 0 18 0 − − x − + 0 − x x 5 − − 18 x 19 − +0 20 13 12 x 20 21 17 10 13 14 00301 18 16 21 17 14 x 16 11 0789 18 17 14 . x 15 19 2 x 18 2482 23766 15 x x . x x x x x 11 15 x x x x x x . x x . 3 02359 x x x 12 x 14 x x 9 x 0 4 4 4 4 3 20861 x . x 0 0 x 7 7 0 3 . x 11 2 x 10 0 x 7 x x x x x x 11 11 10 0 9 x x + 0 x x 10 x 2 x x x x x − x x x x x + 0 − 8 x x + 0 00001 4 + 0 9 . x 591030 00033 5 – 18 – 2 7 . 01116 03959 . x 0 x 7 2 6 02117 . . x 2 x . 0 00549 1 6 48053 x 07238 00042 00004 00375 00554 00045 x 4 . 07638 21655 . 03147 08456 . 08978 . . . . . 08197 x 00701 x . . 04768 02462 − 07719 . . − . 00006 05868 05699 34428 03075 x . . . . 0 . 0 0 00025 01155 − . . . . . 0 + 0 + 0 4 . 0 . 0 0 0 + 0 0 0 0 +0 0 09 x 0 + 0 21 . + 0 + 0 − − + 0 − +0 0609 + 0 − 16 + 0 + 0 − 15195 + 0 19 20 . + 0 + 0 − x − − − − 00153 . 2 1 18 13 − − x . x x 0 00008 − 13467 70712 0 19 x x x . . . 20 21 17 10 13 17 20 17 2 15 2 16 11 12 21 17 + 0 14 13 14 15 x 13 0 18 0 16 17 19 x x x x x 15 − x x x x x x x 2 10 − x x 14 2 x x x + 0 x x 23 x 3 3 3 3 2 5 12 x x x x 6 6 8 x 6 2 x 9 − 110140 8 x 7 9 − 6 10 9 8 x x x x x 1 x x 10 10 . x x x x 2304 x x . − x x x 9 x 3 x 1 x x x . 4 x x 5 04856 6 x 6 x x 0 . x x +1 x 02185 5 21 02151 01246 x 0 . 0495 03973 . . − 01978 04463 3 . x x . 00099 x . . 4.3 00152 0 0 1 . +0 00222 21444 18455 02883 01278 x 0 . 03652 08848 02864 0 . . . . . 00677 06852 00938 . . . 04967 0 05542 x 00014 01531 07933 0031 0 . . 00023 00019 . . . 0 . . . . 0 . . + 0 − − 0 12 35135 0 + 0 − 0 − + 0 0 21098 . 0 0 12497 x x 06078 . − . − 0 . + 0 + 0 − + 0 + 0 + 0 − 25023 + 0 − + 0 21 01146 17 15 − + 0 − + 0 . + 0 18 16 + 0 − 20 19 . 170990 − + 0 x x x . − x x 10 0 x x 006350 19 + 0 20 21 17 10 13 + 0 16 11 + 0 14 . 15 1 20 19 12 21 17 x 14 13 x 18 5 x x x 16 x x 16 14 x x + 0 6 1 12 12 7 x x 16 15 8 15 12 x x x 9 x x − 29447 72398 1 x x x 2 2 2 2 1 x 5 5 x x x . . − x 8 7 x x 9 x 9 8 x x x 7 x x 9 9 x 1 8 − x 5 2 x x x x x 2 3 x x 4 5 x x 5 x 0 0 x x x x x x x 2 x x x x 4 20 x x x x +0 +0 x x x 2 0 11 x 00592 01574 01548 02977 02373 00623 01261 . 0372 . . 20665 12461 . . 01513 02516 00744 02553 00116 . . x 06464 16713 . . 00392 01862 . 01132 00468 . . . . . 0 0 0 02409 10988 . . 01724 07052 . . 06696 0 . 39255 . 40825 0 . . 04321 . 41043 . . 8572 10282 . 07426 02349 0 . . 0 425 . 0 . . . . 0 0 . 0 01136 0 − 0 − − + 0 0 + 0 156280 − + 0 . 1 0 0 0 − . + 0 + 0 + 0 + 0 + 0 + 0 + 0 − + 0 − + 0 + 0 − + 0 + 0 0 − − + 0 − − − − − 18922 − − 21 19 20 17 18 15 16 . − + 0 − . The formula learned by our model for the number of FRTs of a facet. The meanings of +4 21 21 19 19 20 20 17 17 15 16 14 12 13 8 10 10 11 3 5 6 7 x x x x x x x 18 13 14 9 19 1 x x x x x x x x x x x x x x x x x x x x x +0 15 1 8 15 0 7 14 1 8 11 1 8 15 2 14 6 13 4 11 7 11 6 0 4 1 8 4 2 2 1 3 4 x x x x ) = x x are described in section 4 7 0 3 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 10 x x x x i x x FRT N 167680 01386 01325 00012 819380 551 02498 04685 01064 01247 01527 01378 00283 02091 02233 0052 07151 00757 03323 00167 02235 00638 00624 0065 0373 00051 00373 00692 2274 00254 00105 01736 07556 00583 00696 20651 00197 ...... 1 Figure 8 the 0 0 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ln( 0 0 0 0 JHEP03(2019)186 (4.7) (4.6) , the triangulation of 2 F 4 2 0 −2 −4 . and 
1 ) F i , in the formula, after rescaling each F ( ) 20 c i , and these induced triangulations 2 F 18 ( F FRT N T 16 , we know that FRT 1 i ln( F N F 14 i i 12 X Y  ≤ 10 – 19 – 8 (∆) ) for each coefficient 6 | c | ( 4 FRST (∆) = exp 10 N 2 0 FRST c N are the number of FRSTs and FRTs of the 2-skeleton of a given c 0 2 4 6 8 18 20 10 12 14 16 FRT N and . A heatmap showing log FRST N In practice, since our neural network predicted the natural logarithm of the number of Naively, for a given reflexive polytope, its FRSTs would be given by the set of all the number of FRSTs is left to a futureFRSTs, work. we calculated the estimate for each polytope as where reflexive polytope and facet, respectively.the Via product this of inequality, the theand facet upper sum FRTs. bound the is One results. given can by Determining then the calculate degree the to estimate which for the each product 4d of polytope FRTs overestimates though the individual facet FRTsfor are each regular. facet, we Thus, are usingSo, only the for able estimated a to number given estimate of reflexive an FRTs polytope upper ∆ bound with for facets each 4d reflexive polytope. possible combinations of its facets’by FRTs. two considerations. But in The first reality,each the is induces number that a of given triangulation two FRSTs on 3dmay is the facets not reduced intersection overlap. Theoverlap, the second aggregated consideration triangulation is of that the even reflexive if polytope the may fail induced to triangulations be do regular even Having obtained a satisfactory model,facets. we generated We the then necessary fedFRTs input for the data each data for facet. into all of the the neural network to obtain the estimated number of Figure 9 variable to have expectation value 1.and The the top linear row corresponds terms. to the constant term (top left square) 4.6 Estimate for the total upper bound JHEP03(2019)186 ) and (4.9) }} . The . 42 4670 ◦ 491 − 505 , , 10 10 are given by 48 × i − 10 F , × = 1 56 5 , the triangulated . 0 have the most and − . , 3 3.3 753 63 F , . ) = 1 ) 10 ) {− e ) }} , )) whenever it was known, 4771 and i }} } 433, as the percent error for )}} 84 (4.8) FRTs. As it appears twice, it F . 4 10 , 42 42 ( F , , }} 84 12 × , 21 , 36 36 84 FRT 10 , , , 14 25 . , 15 N 6 × 14, our model had MAPE values of , 28 , 8 28 , , , 7 , 4 )(1 41 , 7 { . = 23, and appears twice in ∆ 13 , 21 3 { 21 , , } { { { 1 , 4670 , , } 0 , 1 = 5 , and nowhere else in the database. } } 0 } h 10 , 0 0 0 . This polytope has 680 integral points and value) dominated the count. Our estimation 32 , , 21 , = 12 . × , 1 6 14 1 1 ◦ 491 , ◦ 491 1 29 , , , , FRSTs. This is more than any other polytope by 1 , – 20 – e 0 4 8 15 0 1 h )(1 , , , , , h 0 3 7 7 0 505 , { { { { { to the triangulation number for this polytope, which , , , , , 1038 10 } } } } } has a larger contribution, as our model predicts that it 25 0 0 0 0 0 10 10 , , , , , . For 2 10 0 0 0 0 0 × × will have F , , , , , 4 1 1 1 1 1 1 × FRTs. However, the triangulation number for the polytope , , , , , )(4 465 F . 0 0 0 0 0 , which our model predicts to have 93 25 { { { { { . 4 , , , , , 1038 F } } } } } 10 FRTs, respectively. Combining these, we get the estimate of the 0 0 0 0 0 = 2 10 , , , , , × 0 0 0 0 0 32 × each appear once in ∆ , , , , , and = 491 (the greatest . 4771 93 0 0 0 0 0 . 416. Using the average of these values, 4 4 1 3 . 29 , , , , , , 10 F 1 1 1 1 1 1 F × = 4 = 491 polytope 2 h × e {{ {{ {{ {{ {{ 5 1 . , first appears as a dual facet at ) = (2 and 1 25 1 . 3 h 2391 ◦ 491 F e ,F = 1 490, and 4 2 (∆ . = conv( = conv( = conv( = conv( = conv( 9 F 4 . 3 4 1 2 , = 491 polytope, which we will call ∆ To estimate a range of uncertainty for this value, we can use the extrapolated MAPE Our model predicts that Performing this calculation for each polytope, it was found that the single polytope F F F F ◦ 491 985 FRST , 1 , ∆ 393 1 . N 10 Finally, we note thatsecond-most according FRTs among to all our facets. model, the facets values from our model4 in table e number of FRSTs for this polytope: will contribute is a subdominant contribution. will have is dominated by The facet facets 4.7 The The polytope whose FRST counth dominates the database is thefive polytope facets, dual of to which the only single the four convex are hulls unique. The polytope and its four facets over 1300 orders of magnitude, andnumber so for this the value entire effectively set. servesthan as any Considering the other, that total it triangulation this is polytope unsurprisingthe that has most our 680 triangulations. method integral has points, identified this 40 polytope more as possessing and the neural networkfacets prediction account for otherwise. over 88% As of all noted facets. in section whose dual has method predicts that it has 1 In these calculations, we used the actual value for ln( JHEP03(2019)186 1 , is 6. 1 . h B 292  values — 2 . 1 , — for each 1 h 2 505 , ]. 8 . ) + 4, where B 797 ( , lattice polytope im- 1 , ) = 10 10 1 = 15. As all evidence nevertheless accurately ++ h 1 10 in the smallest polytope , , FRT 1 1 | , 6 . 1 h N ∆ ( h | , . 212 , 7 0 , 10 . . 0 . 10 , value at which they appeared. As 476 487 319 3d reflexive polytope contain 30 . 106 , 1 , 1   1 = [10  0 9 h  . . 6 5 . . 32 we have log . 292  2 . ◦ 491 505 , ) = 29 ) = 2391 ) = 10753 ) = 10985 – 21 – 10 FRT FRT FRT FRT 4. N N N N | − values, lending some credence to our above assumption ∆ | , we have 1 , : ln( : ln( : ln( : ln( 1 3 4 1 2 ◦ 491 h . ) = F F F F B ( 10 1 , 1 h , the calculations were performed using our C ) predictions on data with relatively low 2 we organized the facets by the first FRT , we began by classifying the 2d facets of the 3d reflexive polytopes to avoid 2 N ) estimate in ∆ 2 FRT N we will mean 1 , 1 Performing our classification, we found that the 4 As in section In section Though obtaining the true number of FRTs for these largest facets is currently infeasi- , as obtained by Batyrev’s formula, is equal to 20 for every 3d reflexive polytope, we h 1 , 1 plementation. Due tocompleted in the just small over 20 size seconds. of the344 3d unique reflexive 2d dataset, facets.value the The are numbers shown classification in of was figure facets and polytopes that first appear at each h instead organized the facets by∆ the in number which of they integral appeared.the points smooth We note weak that Fano this toricby quantity threefold is associated equal to to ∆. For the remainder of this section, As in section redundancy in our calculations.facet, The same we classification obtained method a as 3dremoved in polytope the section origin by attaching afterwards. the origin, calculated the normal form, and to predict ln( extrapolate to much higher the 4d case. 5.1 Classification of the 2d facets ble, we hope to gainfor confidence the in our 2d model facets byFRTs of showing for the that the we 3d majority can of reflexive achieveability the polytopes. the can facets same be can As goal more be these thoroughly triangulated. facets tested. Therefore are Specifically, a smaller, we model’s will the extrapolation demonstrate number that of EQLs trained The biggest point of uncertaintyinability involving to our test analysis of oursuggests the model that 4d on these reflexive facets facets polytopes — thatwill is and first be the in particular responsible appear those for beyond thatbe the first preferable dominant appear to contribution at to high have some the confidence total in number our of model’s FRSTs, accuracy it in would this region. Propagating these errors, weThis find gives that a for range ∆ for our estimate of 10 5 Comparison to 3d polytopes each ln( JHEP03(2019)186 = 30. 1 , 1 h 1,1 h The number of reflexive (Right) value.

1 , 0 10 20 30 40

1 0

h to obtain the FRT count for each facet.

400 300 200 100 Polytopes – 22 – TOPCOM = 25 being completed. 1 , 1 h 1,1 h value. 1 , 1 The number of new facets at each h (Left) .

The average side length The number of vertices The length of the longest side The length of the shortest side The number of integral points The number of boundary points The number of interior points 0 10 20 30 40 Additionally, for 6 of the remaining 12 facets, as well as all of the other 322, we were

• • • • • • • 5 0

25 20 15 10 Facets Thus, we used the numberadditional of facets. FTs as This an gave approximation us for values the for number5.3 all of facets FRTs that for appear the Machine six up learning through Having and triangulated extrapolation as manylayer. facets Our as input possible, data was we simpler trained than models for with the an 3d facets, EQL and hidden consisted of facets that appear up to able to obtain thenumber number of FRTs of was found, fine there triangulations isFTs. very (FTs). close More agreement specifically, between For the the the numbers number of 322 FRTs of and facets FTs for was which at the most 1.026 times the number of FRTs. 5.2 Training data Having classified the 344order unique to facets, test we our setWe model. were out able to We to again triangulate successfully used obtain as the many number as of possible FRTs for in 322 of the 344 facets, with all Figure 10 polytopes at each JHEP03(2019)186 . 6 6% of the facets. . ). As the goal of this exercise . From this graph we see that FRT 11 0.146 0.295 0.380 0.644 2.228 0.373 1.075 6.504 0.798 1.158 2.050 1.314 0.619 0.610 0.755 0.538 0.722 0.330 0.334 0.213 0.087 0.075 0.055 0.061 0.032 MSE 11. There are 243 facets that first N of the 3d reflexive polytopes. ≤ 1 , 1 1 , 1 h h 2.074 5.258 6.900 3.985 2.649 7.894 9.167 9.259 7.850 9.071 9.103 8.756 9.128 8.754 5.808 5.851 8.805 7.865 ≤ 12.112 14.333 10.962 10.491 10.200 10.678 16.583 MAPE – 23 – 1 1 3 5 6 3 7 7 5 14 23 13 21 13 22 14 22 21 18 15 19 19 12 15 13 Facets 1 , 1 9 8 7 6 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 h , we obtain the graph shown in figure . Performance of our model across 12, meaning that our extrapolation region contains 70 1 = 25 is the highest value for which all of the FRT values are known; for , 1 1 ≥ , h 1 1 , 30 we approximate the number of FRT by the number25. of FTs With for some our facets. trained model, we predicted the number of FRTs for all 344 h 1 h Table 6 ≤ ≤ 1 1 , , 1 1 h h Our trained model contains a single EQL layer, which used a combination of linear and As we can see, the model performs well through the extrapolation region of As before, we had the neural network train on ln( ≤ ≤ 17 facets, and then estimatedof the the number predictions of fornumbers FRSTs each for for facet. polytopes any where given Weeach all also polytope value facets computed of as have been these the triangulated. estimates product using Taking the the mean real value FRT at appear at quadratic unary activation functions.Recall The that performance of the26 model is shown in table was to mimic the 4d case,training where data we to only have facets data that for the first smallest appear facets, we at restricted 4 our JHEP03(2019)186 − − − − − 2 2 2 and 4 6 5 x x (6.1) x x x 1 2 2 x x x (blue) 00206 59766 . . in the order 0 00929 11917 04087 i . . +0 . − x 2 0 0 1 x x 1 − − + 0 x 4 6 5 x x x 11. In particular, we 0 1 . Again, we see many 1 46716 x x . x ≤ 00201 13 . 1 , + 0 1 0 +0 h 03399 04438 04939 2 x . . . x 0 0 ≤ 0 x − + 0 − . 6 28707 2 3 6 . 5 . = 30, our model is in nearly perfect x x 0 x 0 1 0 , 292 530190 11134 x . 1 − x .  h 2 6 2 . x 05023 , using predicted facet FRT values +0 + 5 . 1,1 1 1 2 , 505 6 , h 1 07105 x 04551 . x . h 10 0 + 0 14731 . 10 3 – 24 – − + 0 x ) at 02165 2 2 5 ' . + 0 2 4 x x 354040 0 6 . x x − FRST 5 1 . N + 1 FRST x x ( 0694 n 0 5 00993 00774 . . . 10 x x 0 007 the formula learned by our model, with the (red) . 0 value available to us, + 0 + 0 − 12

1 − 4 3 5 , 0 5 10 15 20 25 30 35 03435 49381 1 6 x x x . .

3 h x 1 4 5 0 0

0

4

x 20 15 10 x x

FRST 10 − x N log ) ( − 2 0 4 x x 11072 02813 00719 . . . 14734 . 0 0 01418 . 35609 + 0 − + 0 . − 0 3 4 5 6 . The formula learned by our model for the number of FRTs of a facet. The meanings . The mean value of log x x x x − 3 3 0 2 are described earlier in this section. ) = 0 3 x x x x i x x FRT We give below in figure N 00532 07082 4975 03566 01405 . . . . . 6 Conclusion In this paper westar have triangulations provided of the 4d first reflexive concrete polytopes, estimate of the number of fine, regular, of the list at theto beginning have of expectation this value section. 1,warm As with spots, in the making the results further 4d being interpretation case, show of we in also the figure rescaled formula the difficult. variables the estimates from thefrom model known predictions values are well in outsidenote that good of at the agreement the with training highest agreement region the with of estimate the 4 made true result. 0 0 0 Figure 12 of the Figure 11 known facet FRT and FT values ln( 0 0 JHEP03(2019)186 and = 35, 1 , 1 FRST h n = 491 that it 1 , 1 h −2 −3 2 1 0 −1 in the formula, after rescaling each 6 c = 11. This provides some evidence = 11 and made accurate predictions 1 , 1 1 , 1 h polytopes and found that the EQL was h d 4 at which we were able to compute FRTs 1 , 1 h – 25 – ] in the first place. 2 31 ) for each coefficient = 30, where the maximum possible value is | c | 1 , ( 1 10 h 0 c , it is so much smaller than the maximum 1 , 1 c 6 0 2 4 h = 14, which was the highest . A heatmap showing log 1 , 1 h In the process of making the prediction, we have demonstrated the overall utility of After training successful models, we extracted the equations that they learned. The We attempted a variety of supervised learning techniques for predicting equation difficult. This may be due to the functional formdeep utilized neural in networks in our this EQLs. higher context, topological in complexity. particular This their provides motivation abilityand for to further Calabi-Yau extrapolate studies manifolds of to using triangulations regions techniques of from data science. motivation for the EQL architecture [ variables were rescaled to haveequations. an expectation We value found ofbut that 1 the in some majority an of variables attempt variables and to made terms interpret significant these were contributions, making more interpretation important of than the others, performed an analogous analysisable in to the extrapolate case accurately of to 3 despite the fact that it wasthat only the trained corresponding up through extrapolationto in extrapolate the far beyond 4d the case training may region. be Indeed, trustworthy, such despite extrapolations needing were part of the where the EQL was trained onup 4d to polytopes up to of facets for theextrapolated sake to of higher validatingis the hard trained to trust EQL. such a While high encouraging extrapolation that without the further EQL evidence. For that reason, we This provides an upper boundhypersurfaces on the in number toric of varieties. topologically distinct Calabi-Yau threefold found that abest. neural network The with estimation the was equation computed learning by (EQL) taking architecture products performed of numbers of FRTs of facets, Figure 13 variable to have expectation value 1.and The the top linear row corresponds terms. to the constant term (top left square) JHEP03(2019)186 , ] (1995) Vacuum (1999) (2018) 08 B 451 121 ]. ]. hep-th/0307049 JHEP SPIRE SPIRE [ , IN IN (1994) 414 ][ ][ Nucl. Phys. (2003) 046 , 05 ]. Phys. Rev. Lett. Probabilities in the inflationary (2004) 060 B 416 ]. , Vacuum Configurations for 01 JHEP SPIRE , SPIRE IN IN ]. ][ hep-th/0004134 hep-th/0105097 ][ JHEP [ [ , Calabi-Yau moduli space, mirror manifolds Nucl. Phys. , SPIRE Hierarchies from fluxes in string IN [ – 26 – (2000) 006 M theory, orientifolds and G-flux ]. 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