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1 Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to be learned, and b atom R d s the bas vector to be learned. Note that W atom and b atom are not characterzed by the layer l. Other learnng parameters n the followng descrbed neural networks have the same property. The neural network g n Eq. 1 s gven by g x l, V l, αl = α l f x l + v l, 3 where f s the same neural network descrbed n Eq. 2 and v l R d s the vector representaton of the potental V l α l R s the weght on f x l R between the -th and -th atoms. The scalar varable + v l, whch we assumed to be the nteracton strength or bond strength between the -th and -th atoms. In other words, we consder the potental wthn the atom vector.e., a feature of the atom and the nteracton strength by weghtng the atom vector.e., a computaton for the atom. Such modelng provdes a nterpretable machne learnng; that s, we can descrbe a curve of learned potental based on V l see Fgure 3 and we can analyze a strength of learned nteracton based on α l see Fgure 4. Snce these are characterzed wth l, we can observe the dfferently learned potentals and nteractons n all layers n a deep network. The vector representaton of the potental n Eq. 3 s gven by v l = ReLU w potental V l + b potental, 4 where w potental R d s the weght vector and b potental R d s the bas vector. In ths paper, 17
2 we use the Morse potental V l,.e., V l = D l 1 exp a l d r l 2, 5 where d s the Eucldean dstance between the -th and -th atoms,.e., d = r r 1. In our model, D l, al neural networks as follows:, and rl are also learnng parameters and are computed by the D l = σ wd l x a l = σ w a r l = σ w r x l x l x l + bd, 6 + ba, 7 x l x l + br, 8 where σ s the sgmod functon: σx = 1/1 + e x and s the concatenaton of two vectors. Note that, n order to descrbe the potental curves and compare ts steepness see Fgure 3, we consder the constrants,.e., postve values for D and r wth the sgmod functon and a = 1/a + ϵ, where we set ϵ = 0.2. Therefore, our model consders the potental between the -th and -th atoms to be the vector v l, whch s obtaned from x l, x l, and d. Because the atom vectors are characterzed by the layer l and contan nformaton concernng the global molecular structure, the model allows us to learn atomc many-body potentals mplctly or ndrectly wthn the neural network. Note that, for the scalar-valued potental we consder the constrants that the parameters n the Morse potental equaton are postve; we then apply the ReLU functon to the vector-valued representaton.e., feature vector of the potental, whch allows us to correctly propagate the feature nformaton n the forward computaton of a deep network. In the followng, we refer to f x l l and f x + v l l as the hdden vectors h and h l, 1 We can also consder other potentals such as Lennard Jones and learn ts parameters,.e., ϵ the depth of the potental well and σ the fnte dstance. However, we used the Morse type potental n ths paper; we beleve that the type of potental s not sgnfcantly mportant. 18
3 respectvely. We then re-wrte Eq. 1 as follows: x l+1 = h l + M\ α l hl. 9 The motvaton to ntroduce α l s that, f we smply sum or mean over the hdden vectors h l wthout α l, the model consders all the hdden vectors to be equally mportant. However, ther relatve mportance, whch can be assumed to be ther nteracton strengths, are ndeed dfferent and are determned by the -th and -th atom states and ther dstances. In ths paper, we consder the nteracton strength to be α l, whch s represented as a non-lnear dot product n a proected space wth a neural network, used as a weght descrbed n Eq. 9, and learned by backpropagaton. More precsely, usng the hdden vectors h l and h l as nputs, two new vectors y l and y l are obtaned as follows: y l y l = ReLU W nt h l + b nt, 10 = ReLU W nt h l + b nt, 11 where W nt R d d s the weght matrx and b nt R d s the bas vector. Note that, as seen n Eq. 10, we use the same neural network for h l and h l because we wsh to consder the nteracton strengths n a common space proected va a neural network. Then, takng the non-lnear dot product between y l s l and y l, we obtan = σ y l y l. 12 Then normalzng s l wth a softmax functon, we obtan = exp s l k exp. 13 s l α l k 19
4 Fnally, we obtan the weghted sum of the hdden vectors,.e., M\ αl n Eq. 9. Therefore, α l s also a functon of x l and x l hl, as descrbed as s the potental V l ; ths leads to the learnng of atomc many-body nteractons mplctly or ndrectly wthn the neural network. Note that the above computaton s nspred by neural attenton mechansm, whch s wdely used n deep learnng based machne translaton systems. In lnear regresson, the tranng obectve s to mnmze the mean squared errors MSEs between the model output t M = w z M + b and the quantum chemcal property t M n the tranng dataset,.e., the loss functon s LΘ = 1 2 N =1 t M t M 2, where Θ s the set of all learnng parameters n our model, N s the number of data samples molecules, and M s the -th molecule n the tranng dataset. Note that each property t M n the tranng dataset s normalzed to have a mean of 0 and a varance of 1. Then, we use the mean absolute errors MAEs to evaluate the predcton performance. We mplemented the above model usng PyTorch 25 verson and the tranng detals are as follows: the optmzaton s acheved va Adam 26, whch s a stocastc gradent descent SGD-based algorthm; the dmensonalty of the atom vector s d = 100; and the number of layers s L = 6. In addton, we used the dev decay scheme; we kept track of the best performance on the development or valdaton set and decayed the learnng rate by a constant factor f the model dd not obtan a new best performance. In our settngs, the constant factor s 0.5. Note that, whle neural networks are usually traned usng mnbatches, ts sze for our mplementaton s 1 because we acheved the best performance n terms of the convergence of accuracy. The accuracy MAE saturated after approxmately 20 epochs see the learnng curves n Fgure 7, whereas, for example, SchNet 11 requres from 750 to 2,400 epochs wth 32 mnbatch szes to saturate. There are some related studes modelng the atomc potentals wthn machne learnng; 4,17 19 n partcular, Smth et al., developed neural network potentals NNPs. The NNP computes each atomc potental n a molecule, and then the total molecular energy s obtaned by the sum of atomc energes produced by the NNP. On the other hand, our 20
5 model computes the potentals between two atoms n a molecule, whch are always consdered n all layers of the deep network. Ths allows us to learn the atomc potentals consderng the global molecular structure. We beleve that ths s a huge dfference between ther model and ours. In addton, Smth et al., also nvestgated the extrapolaton n terms of the molecular sze and energy predcton; ther evaluaton settng s smlar to ours. They also traned the proposed model called ANI-1 wth small 8 heavy atoms molecules and test the traned ANI-1 model wth larger 10 heavy atoms molecules. The predcton performance for the molecular energy s poorer when the model s tested wth larger molecules,.e., RMSE = heavy atoms and RMSE = heavy atoms. 21
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