Amorphous ice phases are key constituents of water’s complex structural landscape. This study investigates the polyamorphic nature of water, focusing on the complexities within low-density amorphous ice (LDA), high-density amorphous ice, and the recently discovered medium-density amorphous ice (MDA). We use rotationally invariant, high-dimensional order parameters to capture a wide spectrum of local symmetries for the characterization of local oxygen environments. We train a neural network to classify these local environments and investigate the distinctiveness of MDA within the structural landscape of amorphous ice. Our results highlight the difficulty in accurately differentiating MDA from LDA due to structural similarities. Beyond water, our methodology can be applied to investigate the structural properties and phases of disordered materials.

Water is ubiquitous in everyday life, and yet, due to its numerous anomalous behaviors, a complete understanding of its properties remains elusive. The complexity of water is reflected in the intricacy of its phase diagram, the most complex of any pure substance.1 Since the discovery of polymorphism in crystalline ice by Bridgman in 1912,2 over 15 new crystalline phases have been reported,1 with three discovered in the last five years.3–5 

At deeply super-cooled conditions, water also possesses polyamorphism (amorphous polymorphism), a signature (but not a proof) of the existence of a liquid–liquid critical point.6 Low-density amorphous (LDA) and high-density amorphous (HDA) ices7,8 are two classes of amorphous ices that encompass a broader set of sub-families characterized by differing structures, densities, and/or preparation routes. LDA is believed to be the most abundant form of ice in the universe, formed from the cooling of water droplets onto the cold surfaces of dust in the interstellar medium. LDA can also be prepared via rapid quench of liquid water at ambient conditions, and its family comprises LDA-I and LDA-II, obtained via heating HDA and (very-high-density amorphous) vHDA, respectively,9 as well as a more-ordered low-density phase obtained upon heating ice VIII.10 HDA is formed via a first-order phase transition from compression of LDA or hexagonal ice Ih, resulting in a ∼20% density increase.11 Two other phases fall under the HDA bracket, namely, annealed and very high-density amorphous ice, eHDA12 and vHDA.13 From a computational point of view, the differences between the sub-families of LDA and HDA are too small to be easily distinguished from their parents and, therefore, we will only refer to LDA and HDA.

Amorphous ices are very complex. For instance, the short-range order in LDA and HDA is remarkably different. LDA shows high tetrahedrality resulting from well-separated first and second hydration shells. HDA, in contrast, is more disordered, with water molecules populating the space between the first and second shells of neighbors and organized in motifs reminiscent of ice IV.14–16 LDA and HDA are also remarkably different in terms of the topology of the hydrogen bond network17 but have comparable degrees of suppression of long-range density fluctuations.17,18

The polyamorphism of water is of great interest for several reasons. For one, LDA and HDA are supposedly the glassy states of the low-density liquid (LDL) and the high-density liquid (HDL), which may be separated by a liquid–liquid phase transition at deeply under-cooled conditions. The existence of a liquid–liquid critical point has been proven by computer simulations for several models of water,19,20 but the definitive experimental proof is still lacking (although experiments strongly support the hypothesis21,22). A division line of the first-order between LDA and HDA, as found both experimentally23 and from computer simulations,17,24,25 might extend at higher temperatures and end at the liquid–liquid critical point.26 Besides canonical hysteresis cycles, LDA and HDA embed signatures of metastable criticality in their long-range order.17,18,27

Recently, a joint experimental and computational investigation reported the existence of another amorphous phase with a density in between that of LDA and HDA.28 This new amorphous ice, named medium-density amorphous ice (MDA), can be obtained by ball milling hexagonal ice at low temperatures,28 a process that can be simulated via random shearing of layers of hexagonal ice.28 The authors suggest various possible explanations, including MDA being the true glass of liquid water. Were this to be the case, it would question the validity of the liquid–liquid critical point hypothesis.

In the present study, we investigate the structural fingerprints of LDA, HDA, and MDA using Steinhardt bond orientational order parameters29 (BOOs). We exploit the fact that BOOs provide a representation of the rotation group SO(3), relating the irreducible representation of SO(3) and the symmetries of crystalline structure, hence allowing for a systematic investigation of crystalline symmetries. We follow Ref. 30 and encode all possible crystalline symmetries in a 30-dimensional vector containing the following order parameters (OPs): ql(i) and q ̄ l ( i ) with l ∈ [3, 12], wl(i) and w ̄ l ( i ) with l even and l ∈ [4, 12] (see Sec. I A of the supplementary material for further details). For example, the icosahedral symmetry is sensitive to l = 6, 10, while bcc, hcp, fcc, and simple cubic symmetries are all sensitive to l = 4, 6, 8, 10, but with different intensities. Enhanced separations between similar symmetries can be obtained by looking at the intermediate range order and including the second shell of neighbors, as for q ̄ l and w ̄ l . From these, we can gain an understanding of the structural properties of amorphous ices and elucidate the differences between LDA, HDA, and MDA. These BOOs are completely general and applicable to all disordered materials.

We consider the classification task to identify the class of structure (HDA, LDA, or MDA) from which a given atomic environment has been sampled. This approach builds upon prior work by Martelli et al.,30 wherein a neural network (NN) was trained in a similar manner on HDA, LDA, and liquid structures. Herein, we train on pairs (xi, yi), where {xi} are the 30-dimensional BOOs describing the local environment of atom i, and {yi} are one-hot encoded vectors relating the environment to exactly one of the three classes.

The database in this work consists of HDA, LDA, and MDA structures. Configurations for LDA and HDA were taken from Ref. 30; they include thermodynamic conditions spanning from 1 to 20 000 bar and from 80 to 140 K. MDA configurations were taken from Ref. 28. The training, validation, and test sets were assembled in a two-step manner. First, each structure was assigned to one of the training, validation, or test sets, resulting in 1285, 20, and 1245 structures, respectively. Second, within each set, individual atomic environments were randomly sampled in equal proportion from each structure type (HDA, LDA, and MDA), totaling 24 000 training environments and 7500 environments each for validation and testing. This sampling strategy aimed to mitigate potential data imbalances and ensure equal representation of each structure type in each dataset. Full details regarding the generation and composition of the database can be found in Sec. I B of the supplementary material.

To learn the mapping from BOO to structure type, we adopt the simplest neural network (NN) architecture: a feed-forward multi-layer perceptron (MLP). We used rectified linear unit (ReLU) activation functions31 for the hidden layers and a softmax activation function for the final layer. The latter transforms raw output values into predicted probabilities over the input classes (Fig. 1). To optimize the weights of the network, we minimized the cross-entropy loss using the Adam optimizer32 as implemented in PyTorch.33 We optimized the number of hidden layers, number of neurons per layer, learning rate, and weight decay using Bayesian optimization, as implemented in Optuna.34 The sampled ranges and optimization outcomes are found in Table I.

FIG. 1.

Schematic representation of the neural network employed in the present study. The input layer (x) consists of 30 nodes, each representing one of the structural bond orientational order parameters. The output layer has three nodes for the HDA, LDA, and MDA phases, respectively. A softmax activation function is used on the output layer to convert the raw outputs, f(x), to predicted probabilities, P (x).

FIG. 1.

Schematic representation of the neural network employed in the present study. The input layer (x) consists of 30 nodes, each representing one of the structural bond orientational order parameters. The output layer has three nodes for the HDA, LDA, and MDA phases, respectively. A softmax activation function is used on the output layer to convert the raw outputs, f(x), to predicted probabilities, P (x).

Close modal
TABLE I.

Optimized NN hyperparameters.

Hyperparameter Range Optimized value
Number of hidden layers  [1, 5] 
Number of neurons per layer  [8, 256]  82 
Weight decay  [10−8, 0.1]  1.36 × 10−4 
Learning rate  [10−5, 0.1]  6.36 × 10−3 
Hyperparameter Range Optimized value
Number of hidden layers  [1, 5] 
Number of neurons per layer  [8, 256]  82 
Weight decay  [10−8, 0.1]  1.36 × 10−4 
Learning rate  [10−5, 0.1]  6.36 × 10−3 

Interestingly, we observe that such a model is very insensitive to hyperparameter selection. Among 110 optimization iterations, 97% of models achieved test set accuracy within 3% of the best model’s performance. Our final model comprises three hidden layers, each containing 82 neurons.

We use a confusion matrix to quantify the accuracy of our classification model [Fig. 2(a)]. The model shows very good performance in identifying HDA-like atoms but is notably worse at differentiating the two other target states, LDA and MDA. Martelli et al.30 demonstrated that a similar NN trained exclusively on HDA, LDA, and liquid water achieved a misclassification ratio of under 5% between these three phases. These results are discussed in Sec. II A of the supplementary material. In contrast, the misclassification ratio between MDA and LDA increases to 18 % 23 % upon the inclusion of MDA in the training dataset. This result suggests that LDA and MDA are quite hard to distinguish, at least from a structural point of view. This is further demonstrated by Fig. 2(b), which shows the confidence with which the model is classifying each target state. We observe a stark contrast between the model’s confidence in classifying HDA environments compared to LDA and MDA environments. The much broader distribution of confidences demonstrates that the model is unable to accurately differentiate between LDA and MDA environments based on local order parameters alone.

FIG. 2.

Analysis of amorphous ices. (a) Confusion matrix for local environments classification in a test set containing HDA, LDA, and MDA. (b) Prediction confidence for the three target states. (c) KDE plots of BOOs q ̄ 4 , q ̄ 6 , and q ̄ 8 for the test set.

FIG. 2.

Analysis of amorphous ices. (a) Confusion matrix for local environments classification in a test set containing HDA, LDA, and MDA. (b) Prediction confidence for the three target states. (c) KDE plots of BOOs q ̄ 4 , q ̄ 6 , and q ̄ 8 for the test set.

Close modal

The model’s performance can be rationalized by inspecting the distribution of three BOOs for the three target states [Fig. 2(c)]. Excluding MDA, the distributions of HDA and LDA are distinctly separable, facilitating the NN’s ability to categorize local environments based on their BOO values. However, with MDA included, this distinction becomes blurred. There is a substantial overlap between the LDA and MDA, distributions indicating high structural similarities. Consequently, a clear delineation among these phases in the structural space becomes ambiguous. This hampers the network’s capability and confidence in classifying local environments, resulting in a greater misclassification rate.

We benchmark our model’s performance against various established classification algorithms, as implemented in scikit-learn.35  Table II shows the results of this analysis for the three most common classification models: k-Nearest Neighbors (k-NN), logistic regression, and random forest. All three baseline models exhibit similar performance to our NN in classifying HDA, achieving recall rates of ∼98%. As expected, when dealing with the more challenging task of differentiating LDA and MDA environments, all models struggle to achieve recall rates surpassing 80%, with k-NN notably underperforming in this regard. The balanced accuracy scores (BAS), which summarize overall model accuracy, show that our NN outperforms the benchmark models for this classification task. For additional details on benchmarking models, we refer the reader to Sec. I C of the supplementary material.

TABLE II.

Recall and balanced accuracy score (BAS) for various classification methods. Boldface emphasizes the highest values.

Recall (%)
HDA LDA MDA BAS (%)
MLP  98.5  77.8  80.6  85.6 
k-NN  98.0  72.4  70.6  80.3 
Logistic regression  98.0  76.7  80.4  85.0 
Random forest  97.7  77.4  79.7  84.9 
Recall (%)
HDA LDA MDA BAS (%)
MLP  98.5  77.8  80.6  85.6 
k-NN  98.0  72.4  70.6  80.3 
Logistic regression  98.0  76.7  80.4  85.0 
Random forest  97.7  77.4  79.7  84.9 

To further probe the key characteristics of the different classes of amorphous ices, we performed sensitivity analysis on the input vectors (Fig. 3). To do so, we used permutation feature importance (PFI) as implemented in scikit-learn.35 PFI gauges feature importance by measuring the reduction in a model’s accuracy when a single feature value is randomly shuffled.36 By disrupting the relationship between the feature and the target, this process reveals the extent to which the model relies on that specific feature. Given that each feature in the NN corresponds to a specific crystalline symmetry, this analysis allows us to determine the key symmetries that characterize each individual phase.

We built three distinct NN models using the optimized hyperparameters detailed earlier. The models were individually trained to distinguish between the presence or absence of each structure type (i.e., binary classification). Subsequently, we computed PFIs for each model. This approach allows us to discern the unique set of features and, consequently, the local symmetries that the model deems most important when classifying a local environment for each structure type.

Figure 3 highlights the universal importance of q ̄ 4 and q ̄ 12 in characterizing the three amorphous ice phases. Beyond these, however, differences emerge. In particular, in the classification of HDA, the value of q7 carries considerably more weight compared to its importance in the LDA and MDA classifications. Additionally, LDA and MDA show reduced dependency on features outside of q ̄ 4 and q ̄ 12 , with perhaps a slight preference for q ̄ 8 over other features. The strong similarity in feature importance between the LDA and MDA underscores again the high structural resemblance present in the local environments of both phases. This similarity serves as an explanation for the high misclassification ratios observed by the NN. Interestingly, the cubic order parameters show negligible influence in discerning the local environments across these phases. We also performed PFI calculations using Random Forest models (Sec. II B of the supplementary material).

FIG. 3.

Permutation feature importance for the three NN models of amorphous ice phases.

FIG. 3.

Permutation feature importance for the three NN models of amorphous ice phases.

Close modal

To delve deeper into the nature and characteristics of MDA and its relation to the other amorphous forms of water, we applied our NN to structures sampled from LDA compression trajectories taken from Ref. 30. These trajectories were obtained via isothermal compression of LDA at three temperatures (T = 100 K, T = 120 K, and T = 140 K), from 10−4 to 2.0 GPa, at a compression rate of 0.01 GPa/ns. Applying the order parameters to these compression structures, we use our trained NN to compute the fraction of each target state present within the structures. An illustrative set of structure snapshots along the trajectory is shown in Fig. 4(a), where each atom is color-coded according to the class of amorphous ice as which the environment was classified. The compositions of the different populations are shown in Fig. 4(b) for T = 100 K. The graph shows expected behavior, with LDA-like atoms dominating at low pressures. As pressure increases, this population declines to zero, coinciding with a sharp increase in the proportion of HDA-like atoms. This behavior is consistent with the known phase change from LDA to HDA.14,18

FIG. 4.

Classification analysis for the isothermal compression of LDA ice at T = 100 K. (a) Structure snapshots at various pressures along the compression trajectory from Ref. 30. Atoms are color-coded according to which class of amorphous ice the NN model classified them as. Hydrogen atoms are omitted for clarity. Structures were visualized using OVITO.37 (b) Fraction of local environments as classified by the NN as a function of pressure. The gray band shows the region of the phase transition. The dashed line represents the pressure at which the MDA structure most closely resembles a structure along the compression trajectory. (c) Evolution of the BOOs as a function of pressure.

FIG. 4.

Classification analysis for the isothermal compression of LDA ice at T = 100 K. (a) Structure snapshots at various pressures along the compression trajectory from Ref. 30. Atoms are color-coded according to which class of amorphous ice the NN model classified them as. Hydrogen atoms are omitted for clarity. Structures were visualized using OVITO.37 (b) Fraction of local environments as classified by the NN as a function of pressure. The gray band shows the region of the phase transition. The dashed line represents the pressure at which the MDA structure most closely resembles a structure along the compression trajectory. (c) Evolution of the BOOs as a function of pressure.

Close modal

Interestingly, the proportion of MDA-like environments exhibits very different behavior. At lower pressures, there is a gradual and steady rise in the fraction of MDA-like environments present within the structures. During the phase change, a notable spike in the MDA-like population is observed [gray shaded area in Fig. 4(b)], followed by a rapid decrease to zero once the transition to HDA is complete. This behavior suggests that MDA might occur as a transient state during the transition from LDA to HDA.

In Fig. 4(c), we show the evolution of all BOOs considered herein as a function of pressure for T = 100 K. We standardize the BOOs across all atomic environments in the trajectory. This standardization highlights the behavior of all BOOs with pressure, revealing two distinct clusters: one at low pressures (purple) and another at high pressures (yellow), with a clear gap coincident with the phase transition. The plot distinctly showcases unique characteristics between LDA-like and HDA-like environments. These align with our knowledge of the short-range order in these two phases. The set of BOOs for each phase forms a structural signature, which the NN can easily recognize. BOOs characterizing MDA-like environments, however, notably overlap with low-pressure BOOs resembling LDA.

This plot also aligns with the outcomes of the sensitivity analysis (Fig. 3). Specifically, q ̄ 4 and q ̄ 12 maximally separate the LDA and HDA phases, enabling the NN to most easily use these features to distinguish between the two phases, thereby explaining the high PFI scores. In contrast, the w and w ̄ parameters exhibit much greater noise and fewer defining characteristics, resulting in negligible predictive power, as shown by their extremely low PFI scores.

By comparing the trajectory BOOs with the MDA descriptor, we can identify the structure and, consequently, the thermodynamic conditions along the LDA compression trajectory that best match the MDA structure. Our approach involves determining the minimum root-mean-square distance (RMSD) between the mean BOO from all MDA environments and the mean BOO of the environments at each pressure along the compression trajectory. We observe that the RMSE reaches its minimum during the phase transition [dashed line in Fig. 4(b)], indicating that the closest resemblance to MDA local environments is found during the LDA to HDA shift. The same analysis for T = 120 K and T = 140 K is shown in Sec. II C of the supplementary material.

In conclusion, we combined high-dimensional order parameters and a NN model to probe the characteristics of amorphous ices. We showed that our optimized NN outperforms classical baseline classification models. Our analysis afforded insights into the nature of LDA, HDA, and the newly discovered medium-density amorphous ice (MDA). While previous work successfully categorized HDA, LDA, and liquid environments with minimal misclassification errors,30 we showed that including MDA structures in the training dataset significantly reduces the model’s performance. The difficulty of the problem lies in similarities in local structures between LDA and MDA that challenge clear delineation among the phases on structural grounds alone. Using sensitivity analysis on the high-dimensional OPs and our NN, we were able to extract key structural markers for the amorphous phases. Beyond water, we expect such approaches to be more general: for example, complex structural transitions under pressure are known for amorphous silicon,38,39 and order parameters initially used for water40 have been applied to study tetrahedrality in phase-change memory materials.41 Hence, our approach of combining BOOs and NN models could lead to a systematic framework for characterizing amorphous materials.

The online supplementary material is divided into two sections. In the Methods section, we report details of the definitions of Steinhardt BOOs, of datasets, and benchmarking against six classification methods. In the Results section, we report extended results on sensitivity analysis and the compression trajectories of LDA at T = 120 K and T = 140 K.

Z.F.B. acknowledges J.L.A. Gardner for useful discussions and advice. F.M. is grateful to Pablo Debenedetti for insightful discussions. Z.F.B. was supported through an Engineering and Physical Sciences Research Council DTP award and IBM Research. F.M. acknowledges support from the Hartree National Centre for Digital Innovation, a collaboration between STFC and IBM.

The authors have no conflicts to disclose.

Zoé Faure Beaulieu: Conceptualization (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Writing - original draft (equal). Volker L. Deringer: Conceptualization (equal); Methodology (supporting); Supervision (equal); Writing - original draft (equal). Fausto Martelli: Conceptualization (lead); Methodology (lead); Supervision (equal); Writing - original draft (equal).

Data and code supporting the present study are openly available in GitHub at https://github.com/ZoeFaureBeaulieu/NN-amorphous-ices, Ref. 42.

1.
C. G.
Salzmann
, “
Advances in the experimental exploration of water’s phase diagram
,”
J. Chem. Phys.
150
,
060901
(
2019
).
2.
P. W.
Bridgman
, “
Water, in the liquid and five solid forms, under pressure
,”
Proc. Am. Acad. Arts Sci.
47
,
441
558
(
1912
).
3.
M.
Millot
,
F.
Coppari
,
J. R.
Rygg
,
A.
Correa Barrios
,
S.
Hamel
,
D. C.
Swift
, and
J. H.
Eggert
, “
Nanosecond x-ray diffraction of shock-compressed superionic water ice
,”
Nature
569
,
251
255
(
2019
).
4.
R.
Yamane
,
K.
Komatsu
,
J.
Gouchi
,
Y.
Uwatoko
,
S.
Machida
,
T.
Hattori
,
H.
Ito
, and
H.
Kagi
, “
Experimental evidence for the existence of a second partially-ordered phase of ice VI
,”
Nat. Commun.
12
,
1129
(
2021
).
5.
V. B.
Prakapenka
,
N.
Holtgrewe
,
S. S.
Lobanov
, and
A. F.
Goncharov
, “
Structure and properties of two superionic ice phases
,”
Nat. Phys.
17
,
1233
1238
(
2021
).
6.
H.
Tanaka
, “
Liquid–liquid transition and polyamorphism
,”
J. Chem. Phys.
153
,
130901
(
2020
).
7.
K.
Amann-Winkel
,
R.
Böhmer
,
F.
Fujara
,
C.
Gainaru
,
B.
Geil
, and
T.
Loerting
, “
Colloquium: Water’s controversial glass transitions
,”
Rev. Mod. Phys.
88
,
011002
(
2016
).
8.
T.
Loerting
,
K.
Winkel
,
M.
Seidl
,
M.
Bauer
,
C.
Mitterdorfer
,
P. H.
Handle
,
C. G.
Salzmann
,
E.
Mayer
,
J. L.
Finney
, and
D. T.
Bowron
, “
How many amorphous ices are there?
,”
Phys. Chem. Chem. Phys.
13
,
8783
8794
(
2011
).
9.
K.
Winkel
,
D. T.
Bowron
,
T.
Loerting
,
E.
Mayer
, and
J. L.
Finney
, “
Relaxation effects in low density amorphous ice: Two distinct structural states observed by neutron diffraction
,”
J. Chem. Phys.
130
,
204502
(
2009
).
10.
J. J.
Shephard
,
S.
Klotz
,
M.
Vickers
, and
C. G.
Salzmann
, “
A new structural relaxation pathway of low-density amorphous ice
,”
J. Chem. Phys.
144
,
204502
(
2016
).
11.
O.
Mishima
,
L. D.
Calvert
, and
E.
Whalley
, “‘
Melting ice’ I at 77 K and 10 kbar: A new method of making amorphous solids
,”
Nature
310
,
393
395
(
1984
).
12.
R. J.
Nelmes
,
J. S.
Loveday
,
T.
Strässle
,
C. L.
Bull
,
M.
Guthrie
,
G.
Hamel
, and
S.
Klotz
, “
Annealed high-density amorphous ice under pressure
,”
Nat. Phys.
2
,
414
418
(
2006
).
13.
T.
Loerting
,
C.
Salzmann
,
I.
Kohl
,
E.
Mayer
, and
A.
Hallbrucker
, “
A second distinct structural ‘state’ of high-density amorphous ice at 77 K and 1 bar
,”
Phys. Chem. Chem. Phys.
3
,
5355
5357
(
2001
).
14.
F.
Martelli
,
N.
Giovambattista
,
S.
Torquato
, and
R.
Car
, “
Searching for crystal-ice domains in amorphous ices
,”
Phys. Rev. Mater.
2
,
075601
(
2018
).
15.
J. J.
Shephard
,
S.
Ling
,
G. C.
Sosso
,
A.
Michaelides
,
B.
Slater
, and
C. G.
Salzmann
, “
Is high-density amorphous ice simply a ‘derailed’ state along the ice I to ice IV pathway?
,”
J. Phys. Chem. Lett.
8
,
1645
1650
(
2017
).
16.
H.
Kobayashi
,
K.
Komatsu
,
H.
Ito
,
S.
Machida
,
T.
Hattori
, and
H.
Kagi
, “
Slightly hydrogen-ordered state of ice IV evidenced by in situ neutron diffraction
,”
J. Phys. Chem. Lett.
14
,
10664
10669
(
2023
).
17.
M.
Formanek
,
S.
Torquato
,
R.
Car
, and
F.
Martelli
, “
Molecular rotations, multiscale order, hyperuniformity, and signatures of metastability during the compression/decompression cycles of amorphous ices
,”
J. Phys. Chem. B
127
,
3946
3957
(
2023
).
18.
F.
Martelli
,
S.
Torquato
,
N.
Giovambattista
, and
R.
Car
, “
Large-scale structure and hyperuniformity of amorphous ices
,”
Phys. Rev. Lett.
119
,
136002
(
2017
).
19.
J. C.
Palmer
,
F.
Martelli
,
Y.
Liu
,
R.
Car
,
A. Z.
Panagiotopoulos
, and
P. G.
Debenedetti
, “
Metastable liquid–liquid transition in a molecular model of water
,”
Nature
510
,
385
388
(
2014
).
20.
P. G.
Debenedetti
,
F.
Sciortino
, and
G. H.
Zerze
, “
Second critical point in two realistic models of water
,”
Science
369
,
289
292
(
2020
).
21.
J. A.
Sellberg
,
C.
Huang
,
T. A.
McQueen
,
N. D.
Loh
,
H.
Laksmono
,
D.
Schlesinger
,
R. G.
Sierra
,
D.
Nordlund
,
C. Y.
Hampton
,
D.
Starodub
,
D. P.
DePonte
,
M.
Beye
,
C.
Chen
,
A. V.
Martin
,
A.
Barty
,
K. T.
Wikfeldt
,
T. M.
Weiss
,
C.
Caronna
,
J.
Feldkamp
,
L. B.
Skinner
,
M. M.
Seibert
,
M.
Messerschmidt
,
G. J.
Williams
,
S.
Boutet
,
L. G. M.
Pettersson
,
M. J.
Bogan
, and
A.
Nilsson
, “
Ultrafast x-ray probing of water structure below the homogeneous ice nucleation temperature
,”
Nature
510
,
381
384
(
2014
).
22.
K. H.
Kim
,
K.
Amann-Winkel
,
N.
Giovambattista
,
A.
Späh
,
F.
Perakis
,
H.
Pathak
,
M. L.
Parada
,
C.
Yang
,
D.
Mariedahl
,
T.
Eklund
,
T. J.
Lane
,
S.
You
,
S.
Jeong
,
M.
Weston
,
J. H.
Lee
,
I.
Eom
,
M.
Kim
,
J.
Park
,
S. H.
Chun
,
P. H.
Poole
, and
A.
Nilsson
, “
Experimental observation of the liquid–liquid transition in bulk supercooled water under pressure
,”
Science
370
,
978
982
(
2020
).
23.
O.
Mishima
,
L. D.
Calvert
, and
E.
Whalley
, “
An apparently first-order transition between two amorphous phases of ice induced by pressure
,”
Nature
314
,
76
78
(
1985
).
24.
N.
Giovambattista
,
H.
Eugene Stanley
, and
F.
Sciortino
, “
Phase diagram of amorphous solid water: Low-density, high-density, and very-high-density amorphous ices
,”
Phys. Rev. E
72
,
031510
(
2005
).
25.
J.
Engstler
and
N.
Giovambattista
, “
Heating- and pressure-induced transformations in amorphous and hexagonal ice: A computer simulation study using the TIP4P/2005 model
,”
J. Chem. Phys.
147
,
074505
(
2017
).
26.
O.
Mishima
and
H. E.
Stanley
, “
The relationship between liquid, supercooled and glassy water
,”
Nature
396
,
329
335
(
1998
).
27.
T. E.
Gartner
,
S.
Torquato
,
R.
Car
, and
P. G.
Debenedetti
, “
Manifestations of metastable criticality in the long-range structure of model water glasses
,”
Nat. Commun.
12
,
3398
(
2021
).
28.
A.
Rosu-Finsen
,
M. B.
Davies
,
A.
Amon
,
H.
Wu
,
A.
Sella
,
A.
Michaelides
, and
C. G.
Salzmann
, “
Medium-density amorphous ice
,”
Science
379
,
474
478
(
2023
).
29.
P. J.
Steinhardt
,
D. R.
Nelson
, and
M.
Ronchetti
, “
Bond-orientational order in liquids and glasses
,”
Phys. Rev. B
28
,
784
805
(
1983
).
30.
F.
Martelli
,
F.
Leoni
,
F.
Sciortino
, and
J.
Russo
, “
Connection between liquid and non-crystalline solid phases in water
,”
J. Chem. Phys.
153
,
104503
(
2020
).
31.
A. F.
Agarap
, “
Deep learning using rectified linear units (ReLU)
,” arXiv:1803.08375 [cs.NE] (
2019
).
32.
D. P.
Kingma
and
J.
Ba
, “
Adam: A method for stochastic optimization
,” arXiv:1412.6980 [cs.LG] (
2017
).
33.
A.
Paszke
,
S.
Gross
,
S.
Chintala
,
G.
Chanan
,
E.
Yang
,
Z.
DeVito
,
Z.
Lin
,
A.
Desmaison
,
L.
Antiga
, and
A.
Lerer
, “
Automatic differentiation in PyTorch
,” in
Proceedings of the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA
(
NeurIPS
,
2017
), see https://note.wcoder.com/files/ml/automatic_differentiation_in_pytorch.pdf.
34.
T.
Akiba
,
S.
Sano
,
T.
Yanase
,
T.
Ohta
, and
M.
Koyama
, “
Optuna: A next-generation hyperparameter optimization framework
,” in
Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 19
(
ACM
,
New York
,
2019
), pp.
2623
2631
.
35.
F.
Pedregosa
,
G.
Varoquaux
,
A.
Gramfort
,
V.
Michel
,
B.
Thirion
,
O.
Grisel
,
M.
Blondel
,
P.
Prettenhofer
,
R.
Weiss
,
V.
Dubourg
,
J.
Vanderplas
,
A.
Passos
,
D.
Cournapeau
,
M.
Brucher
,
M.
Perrot
, and
E.
Duchesnay
, “
Scikit-learn: Machine learning in Python
,”
J. Mach. Learn. Res.
12
,
2825
2830
(
2011
).
36.
L.
Breiman
, “
Random forests
,”
Mach. Learn.
45
,
5
32
(
2001
).
37.
A.
Stukowski
, “
Visualization and analysis of atomistic simulation data with OVITO—The Open Visualization Tool
,”
Modell. Simul. Mater. Sci. Eng.
18
,
015012
(
2009
).
38.
S. K.
Deb
,
M.
Wilding
,
M.
Somayazulu
, and
P. F.
McMillan
, “
Pressure-induced amorphization and an amorphous–amorphous transition in densified porous silicon
,”
Nature
414
,
528
530
(
2001
).
39.
V. L.
Deringer
,
N.
Bernstein
,
G.
Csányi
,
C.
Ben Mahmoud
,
M.
Ceriotti
,
M.
Wilson
,
D. A.
Drabold
, and
S. R.
Elliott
, “
Origins of structural and electronic transitions in disordered silicon
,”
Nature
589
,
59
64
(
2021
).
40.
J. R.
Errington
and
P. G.
Debenedetti
, “
Relationship between structural order and the anomalies of liquid water
,”
Nature
409
,
318
321
(
2001
).
41.
S.
Caravati
,
M.
Bernasconi
,
T. D.
Kühne
,
M.
Krack
, and
M.
Parrinello
, “
Coexistence of tetrahedral- and octahedral-like sites in amorphous phase change materials
,”
Appl. Phys. Lett.
91
,
171906
(
2007
).
42.
Z. F.
Beaulieu
,
V. L.
Deringer
, and
F.
Martelli
(
2023
) “
Research data supporting `High-dimensional order parameters and neural network classifiers applied to amorphous ices
,'” GitHub. https://github.com/ZoeFaureBeaulieu/NN-amorphous-ices

Supplementary Material