# generative_hierarchical_materials_search__7ba2f855.pdf

Generative Hierarchical Materials Search

Sherry Yang , Simon Batzner, Ruiqi Gao, Muratahan Aykol, Alexander Gaunt, Brendan Mc Morrow, Danilo Rezende, Dale Schuurmans, Igor Mordatch, Ekin Dogus Cubuk Google Deep Mind

Generative models trained at scale can now produce text, video, and more recently, scientiﬁc data such as crystal structures. In applications of generative approaches to materials science, and in particular to crystal structures, the guidance from the domain expert in the form of high-level instructions can be essential for an automated system to output candidate crystals that are viable for downstream research. In this work, we formulate end-to-end language-to-structure generation as a multi-objective optimization problem, and propose Generative Hierarchical Materials Search (Gen MS) for controllable generation of crystal structures. Gen MS consists of (1) a language model that takes high-level natural language as input and generates intermediate textual information about a crystal (e.g., chemical formulae), and (2) a diffusion model that takes intermediate information as input and generates low-level continuous value crystal structures. Gen MS additionally uses a graph neural network to predict properties (e.g., formation energy) from the generated crystal structures. During inference, Gen MS leverages all three components to conduct a forward tree search over the space of possible structures. Experiments show that Gen MS outperforms other alternatives of directly using language models to generate structures both in satisfying user request and in generating low-energy structures. We conﬁrm that Gen MS is able to generate common crystal structures such as double perovskites, or spinels, solely from natural language input, and hence can form the foundation for more complex structure generation in near future.

1 Introduction

Modern technologies increasingly rely on the development of materials, such as semiconductors [1], solar cells [2], and lithium batteries [3]. Large-scale generative models, trained on expansive internet data, exhibit intriguing generalization capabilities. For example, these models can synthesize a highly realistic image of an astronaut riding a horse by merging two distant concepts [4]. This raises a compelling question: can the generalization capabilities of large generative models, pretrained on existing materials science knowledge, be harnessed to combine knowledge from existing materials systems to propose candidate crystals?

Previous research has demonstrated that generative models can output crystal structures that are not in the the training data [5, 6, 7]. However, these works typically require either a vast number of unconditional samples to generate an unknown material [5, 8] or a chemical formula provided during inference [6, 9]. It is difﬁcult for end users to come up with new chemical formulae, as it is hard to know which compositions will result in what material properties. Therefore, it is highly desirable to develop an interface that allows users to describe the desired characteristics of crystal structures such as properties, compositions, space groups, and geometric characteristics in natural language. For example, a user might specify a stable chalcogenide with atom ratio 1:1:2 that is not on ICSD. Ideally, a model should automatically interpret these high-level language instructions to search for,

Correspondence to sherryy@google.com and cubuk@google.com. 38th Conference on Neural Information Processing Systems (Neur IPS 2024).

Generate a stable, chalcogenide with atom ratio 1:1:2 that is not in ICSD

Instruction following

Chalcogenides containing one or more chalcogen elements, e.g., S, Se, Te.

Instruction following

Figure 1: Overview of Gen MS. Gen MS takes a high-level language instruction as input, retrieves relevant information from the internet, and samples from a high-level LLM (πhi) to generate candidate formulae that satisfy user requirement. Gen MS then samples from a low-level diffusion model (πlo) to generate structures conditioned on candidate formulae. Sampled structures then go through a property prediction module for selection.

generate, and validate a wide range of potential structures, ultimately producing one that best meets the user s speciﬁcations.

However, developing an end-to-end language-to-structure generative model presents several challenges, for which we make a few key observations. First, there are no existing labeled datasets that map language descriptions directly to crystal structures. Nevertheless, we observe that there is a wealth of language-to-formula data available online, including Wikipedia articles, research papers, and textbooks. This data can be complemented by formula-to-structure information from specialized materials databases such as the Materials Project [10], ICSD [11], OQMD [12], etc. Second, the task of converting language into structures is inherently multimodal, requiring the transformation of discrete linguistic inputs to continuous structural outputs. Nevertheless, it has been shown that semantic-level autoregressive models combined with low-level (pixel-level) diffusion models are effective for cross-modal generation, such as in text-to-video applications [13, 14]. Lastly, user descriptions of desired crystal structures can often be vague users may not articulate all relevant details about the crystal they wish to generate. We observe that one can leverage generative models to infer missing information, and rely on additional search and selection mechanisms to identify structures that best satisfy a user s requirement.

Based on these observations, we propose Generative Hierarchical Materials Search (Gen MS) for end-to-end language-to-structure generation. Gen MS consists of (1) a large language model (LLM) pretrained on high-level materials science knowledge from the internet, (2) a diffusion model trained on low-level crystal structure databases, and (3) a graph neural network (GNN) for property prediction. To improve the efﬁciency of (2), Gen MS proposes a compact representation of crystal structures for diffusion models. During inference, Gen MS prompts the LLM to generate candidate chemical formulae according to user speciﬁcation, samples structures from the diffusion model, and uses the GNN to predict the properties of the sampled structures. To sample structures that best satisfy user requirements during inference, we formulate language-to-structure as a multi-objective optimization problem, where user speciﬁcations are transformed into objectives that can be optimized at both the formula and structure level.

We ﬁrst evaluate Gen MS s ability to generate crystal structures from language instructions, and ﬁnd that Gen MS can successfully generate structures that satisfy user requests more than 80% of the time for three major families of structures, while proposing structures with low formation energies, as veriﬁed by DFT calculations. In contrast, using pretrained LLMs to directly generate crystal structures from user instructions in a zero-shot manner often results in close to a 0% success rate. Qualitative evaluations show that Gen MS is able to generate complex structures, such as layered structures, double perovskites, and spinels, solely from natural language. We next study the effect of each individual component of Gen MS. Here we ﬁnd that language instructions have a signiﬁcant impact on the structures generated, that the novel compact representation of crystals proposed by Gen MS improves the DFT convergence rate of diffusion generated crystal structures by 50% over previous work, and that using a pretrained GNN to select samples leads to lower energy structures more than 80% of the time. Given such experimental evidence, we believe the development of language-to-structure models are promising for enabling users to ﬁnd viable crystal structure candidates, complementing existing databases in utility

2 Generative Hierarchical Materials Search

We begin by formulating the problem of generating crystal structures from high-level language as a multi-objective optimization task. Given this formulation, we then propose a hierarchical, multimodal tree search algorithm that leverages language models, diffusion models, and graph neural networks as submodules. Lastly, we discuss the speciﬁc design choices for each of the submodules.

2.1 Language to structure as a multi-objective optimization

Given some high-level language description g G of desired structures, we want to learn a conditional crystal structure generator π( |g) : G 7 (X)2 that can be used to sample crystal structures x X conditioned on language. One option is to parametrize π with a pretrained LLM. However, pretrained LLMs alone are not able to predict sufﬁciently accurate crystal structures, due to the lack of low-level structural information about crystals (e.g., 3D atom coordinates) in the pretraining data.

If we had access to a paired language-to-structure dataset, D = {gi, xi}N i=1, π could be trained using a maximum likelihood objective. However, materials data naturally exist at different levels of abstraction and are segregated into different sources: high-level symbolic knowledge is documented in sources like Wikipedia articles, research papers, and textbooks, whereas detailed low-level crystal information, including continuous-valued atom positions, is stored in specialized crystal databases like the Materials Project [10] and ICSD [11]. Even though a direct language-to-structure dataset D remains unavailable, the pretraining data for LLMs, including Wikipedia articles, research papers, and textbooks, can be viewed as a high-level symbolic dataset Dhi = {gi, zi}m i=1, where z Z denotes symbolic textual information such as chemical formulae. Meanwhile, many crystal databases already feature paired data, Dlo = {zi, xi}n i=1, linking chemical formulae to detailed crystal structures.

Given this observation, we propose to factorize the crystal generator as π = πhi πlo, where πhi : G 7 (Z) and πlo : Z 7 (X), so that πhi and πlo can be trained using different datasets Dhi and Dlo. Furthermore, we consider two heuristic functions, Rhi(g, z) : G Z 7 R and Rlo(z, x) : Z X 7 R, where the high-level heuristic function Rhi can be used to select formulae that satisfy the language input at a high level, and the low-level heuristic function Rlo can be used to select structures that are both valid and exhibit desirable properties such as low formation energy. To this end, we propose to search for crystal structure given language input by ﬁnding a chemical formula / space group z with a corresponding crystal structure x that jointly optimize

z , x = arg max z,x πhi,πlo Ez πhi,x πlo(z)[λhi Rhi(g, z) + λlo Rlo(z, x)], (1)

where λhi and λlo are hyperparameters to control how much weight to put on high and low-level heuristics. Note that Rhi and Rlo can also be combinations of multiple objectives. For instance, Rhi can be a weighted sum of instruction following and simplicity, where Rlo can be a weighted sum of properties such as band gap, conductivity, and formation energy.

2.2 Searching through language and structure

Given the objective in Equation 1, it is clear that a pretrained LLM (even with ﬁnetuning) is insufﬁcient to optimize for the best structure x . Instead, we propose to ﬁrst sample a set of intermediate chemical formulae from a pretrained LLM πhi(g) conditioned on language input g. We then use the high-level heuristic function Rhi to prune and rank the intermediate formulae. In practice, Rhi is a combination of (i) a regular expression checker (to ensure sampled formulae are valid chemical formulae), (ii) a uniqueness checker against formulae from existing crystal datasets such as Materials Project and ICSD, and (iii) a formula compliance checker to ensure the sampled formulae are compatible with user request (e.g., atom ratio 113 for perovskites, 227 for pyrochlore, and 124 for spinel). For formulae that pass these checks, we prompt a pretrained LLM as Rhi to rank the formulae by how likely they are to comply with the user request g. We then select the top W ranked formulae to generate L crystal structures each using πlo parametrized by a diffusion model, and use a graph neural network Rlo to rank the W L structures by their predicted formation energy. Note that additional checkers can be integrated in Rlo, such as structural and compositional validity deﬁned in [5]. We illustrate the overall search procedure in Algorithm 1.

2We use ( ) to denote a probability simplex function.

Algorithm 1 Generative Hierarchical Materials Search

1: Input: Language input g 2: Functions: High-level language policy πhi(z|g), high-level heuristic function Rhi(g, z), low-level diffusion policy πlo(x|z), low-level heuristic function Rlo(z, x). 3: Hyperparameters: High-level language branching factor H, low-level structure branching factor L, max width for formulae W. 4: plans [[g] i {1 . . . H}] # Initialize H different plans starting with language input. 5: for h = 1 . . . H do 6: g plans[h][ 1] # Get the high-level language speciﬁcation from the tree. 7: {zi}H i=1 πhi(g) # Generate H different intermediate formulae. 8: z = argmax({g, zi}H i=1, Rhi) 9: plans[h].append(z ) # Add formula with the best heuristic value to plan. 10: end for 11: plans sort(plans, Rhi) # Sort formulae based on heuristic. 12: for w = 1 . . . W do 13: z plans[w][ 1] # Get the best intermediate formula from the tree. 14: {xi}L i=1 πlo(z) # Generate L low-level structures. 15: x = argmax({z, xi}H i=1, Rlo) 16: plans[w].append(x ) # Add structure with the best heuristic value to plan. 17: end for 18: return plans[0][0] # Return the best structure.

Alternative search strategies. The search algorithm described above, Algorithm 1, follows the best-ﬁrst search strategy, i.e., intermediate formulae and ﬁnal structures are sorted and searched over based on the preference of a heuristic function. Alternative search strategies such as breadth-ﬁrst or depth-ﬁrst can also be employed. The most suitable search strategy depends on the downstream application and computational resources available. For instance, if large-scale density function theory (DFT) calculations are available downstream, we can employ breadth-ﬁrst search to devise more diverse composition.

Prevent heuristic exploitation. One concern of using a heuristic GNN to select structures with the lowest formation energy is that the GNN might exploit irregularities in the predicted structures, especially when a predicted structure lies outside of the training manifold of the energy GNN. To mitigate this issue, we use the GNN pretrained by [15] on DFT energies and forces of unrelaxed structures (hence the GNN has seen more irregular structures prior to relaxation.) Furthermore, we discard sampled structures from πlo if they result in energy predictions from Rlo that lie outside of a threshold range.

2.3 Choices of parametrization for the submodules

Since controllable crystal structure generation from language input is multimodal by nature, there are various design choices for the parametrization of the submodules in Equation 1, namely the generators πhi, πlo and the heuristic functions Rhi, Rlo. In this section, we discuss the parametrization choices we have found to be the most effective.

Retrieval augmentation and long-context deduplication. One important recent advance in LLMs is increased context length [16]. The factorization π = πhi πlo provides a natural way to integrate additional context in πhi via long-conext generation. Speciﬁcally, we further factorize πhi into πhi = πretrival hi πRAG hi , where πretrival hi ( |g) is a deterministic retrieval function that uses the Wikipedia API to retrieve textual information related to language input g, while πRAG hi is a retrieval augmented generative (RAG) model that proposes chemical formulae and space groups conditioned on the information retrieved from the internet. Another use case for long-context LLMs is to further encourage the generation of new compositions by providing the formulae for all known crystals in the context, then asking πhi to produce a formula that is not in the context. As we will see in Section 3.2, this drastically improves the efﬁciency of the search, as a large subset of the search space with known crystals can be eliminated.

Compact crystal representation. In order to support efﬁcient tree search at inference time, we need to ensure that sampling from both πhi and πlo are efﬁcient. Previous work on diffusion models

Noisy atoms

MLP Selfattention n, m*k

MLP n, m x, y, z, a

(n, 4) n, m*k n, m*k n, m

Denoised atoms

Figure 2: Diffusion architecture with compact crystal representation. The diffusion model in Gen MS represents crystal structures by the x, y, z location of each atom plus the atom number a represented as a continuous value. Each atom undergoes blocks consisting of multi-layer perceptrons followed by order-invariant self-attention. The MLP and self-attention blocks are repeated k times where each repetition increases the dimension of the hidden units. The concatenation of skip connections are employed as in other U-Net architectures.

for crystal structure generation has leveraged sparse data structures, such as voxel images [17, 18, 19], graphs [5], and periodic table shaped tensors [6]. These existing representations of crystals incur computational overhead due to sparsity (voxel images, padded tensors) or quadratic complexity as the number of atoms in the system increases (graphs). Instead, we propose a new compact representation of crystal structures, where each crystal x X RA 4 is represented by a A 4 tensor, with A being the number of atoms in the crystal, and the inner 4 dimensions representing the x, y, z location of an atom along with its atom number. Here we directly represent the atom number as a continuous value normalized to the range of the input in the diffusion model to further improve inference speed, as opposed to representing the atom number using a one-hot vector. In addition, we use another 2 3 vector to represent the lattice structure (i.e., angles and lengths of the unit cell). Figure 2 illustrates the architecture for the diffusion model with compact crystal representations, where each atom undergoes multi-layer perceptron (MLP) followed by order-invariant self-attention (without positional encoding) across atoms. Different from typical U-Net architecture for image generation, there is no downsampling or upsampling passes that change the input resolution. Nevertheless, we follow the concatenation of skip connections commonly used in U-Net architectures [20]. Additional details and hyperparameters for the diffusion model can be found in Appendix A.3.

3 Experimental Evaluation

We now evaluate the ability of Gen MS to generate crystal structures from high-level language descriptions. First, we evaluate the success of end-to-end generation in Section 3.1. We then investigate the individual components of Gen MS in Section 3.2. See details of experimental setups in Appendix A.

3.1 End-to-end evaluation

Baselines and metrics. We aim to evaluate Gen MS s ability to generate unique, valid, and potentially stable crystal structures from well-known crystal families that satisfy high-level language speciﬁcations. We consider few-shot prompting of LLMs to generate crystal information ﬁles (CIF) as a baseline. Speciﬁcally, we give the Gemini long context model [16] a number of CIF ﬁles from a particular crystal family, as speciﬁed by language input as prompt, with the number of CIF ﬁles ranging from 1, 5, 25 to as many as can ﬁt in the context. We ask the LLM to generate 100 samples given each language instruction. See additional details of baselines in Appendix A.2. We do not compare to ﬁnetuning LLMs to generate CIF ﬁles in this section, as there are no high-level language to low-level crystal structure datasets available for ﬁnetuning such an instruction following LLM. Nevertheless, we will compare the diffusion model in Gen MS to formula-conditioned structure generation using ﬁnetuned LLM in Section 3.2. We consider language input that directs the model to generate unique and stable crystals from a particular crystal family (perovskite, pyrochlore, and spinel). We consider the following metrics for evaluation: (i) CIF validity, which measures whether the generated CIF ﬁle can be properly parsed by pymatgen parser [21]. (ii) Structural and composition validity, which verify atom distances and charge balances using SMACT [22], following [5]. (iii) Formation energy per atom (Ef) in the unit of e V/atom, which measures the stability of predicted structures using a pretrained GNN. We further conduct DFT calculations to compute Ef (see details in Appendix A.4) for structures predicted by Gen MS. (iv) Uniqueness, which measures the percentage of generated formulae that do not exist in Materials Project [10] or ICSD [11]. Finally, (v) the match rate, which measures the percentage of generated structures that can be matched (according to the pymatgen

Family Metric Prompting CIF Gen MS 1 shot 5 shot 25 shot Max

Perovskites

CIF validity 0.94 1.00 0.98 0.88 1.00 Structural validity 0.04 0.28 0.66 0.22 1.00 Composition validity 0.07 0.17 0.45 0.00 0.85 Ef (GNN/DFT) N/A -0.19 0.28 0.53 -0.47/-1.32 Uniqueness 0.07 0.29 0.68 0.16 0.90 Match rate 0.00 0.09 0.36 0.19 0.93

CIF validity 0.40 0.60 0.64 0.88 1.00 Structural validity 0.36 0.36 0.28 0.23 0.95 Composition validity 0.00 0.22 0.18 0.00 0.89 Ef (GNN/DFT) 1.28 1.19 0.63 -1.22 -1.37/-2.56 Uniqueness 0.36 0.38 0.45 0.08 0.49 Match rate 0.00 0.00 0.04 0.00 0.86

CIF validity 0.73 0.96 0.97 0.96 1.00 Structural validity 0.47 0.61 0.71 1.00 1.00 Composition validity 0.29 0.92 0.92 1.00 1.00 Ef (GNN/DFT) 1.09 -0.85 -0.97 -1.37 -1.38/-1.77 Uniqueness 0.48 0.13 0.51 0.08 0.44 Match rate 0.00 0.12 0.18 0.08 0.89

Table 1: End-to-end evaluation of generating crystal structure from natural language. Gen MS signiﬁcantly outperforms LLM prompting baselines in producing unique and low-energy (predicted by GNN) structures that satisfy user request. We further conduct DFT calculation to compute Ef (formation energy in e V/atom) averaged across structures generated by Gen MS. Values before / in the row (GNN/DFT) represent GNN predicted Ef, and after / represent DFT computed Ef. We report Ef from GNN prior to relaxation, and Ef from DFT post relaxation. DFT calculations for baselines are eliminated as many structures from the baselines do not follow user instruction. N/A represents Ef predicted by GNN falling outside of the reasonable range.

structure matcher) to one of the structures of the corresponding family in Materials Project. More details of these metrics can be found in Appendix A.1.

Results on specifying crystal family. The evaluation of Gen MS and baselines are shown in Table 1. Since Gen MS does not rely on an LLM to directly generate CIF ﬁles, the compact crystal representation (described in Section 2.3) always results in structures that can be parsed by pymatgen (100% CIF validity). In addition, structures generated by Gen MS have a much higher validity and match rate compared to those generated by the baselines. Gen MS struggles slightly with uniqueness, as less than half of the generated formulae for pyrochlore and spinel are unique with respect to MP and ICSD. Structures produced by Gen MS have lower average Ef. Increasing the number of CIF ﬁles in the context generally improves the performance of the baselines (1, 5, and 25-shot), but including too many ﬁles in the context can hurt performance (Prompting CIF Max).

Qualitative evaluation. In addition to the three families of structures evaluated above, we qualitatively evaluated Gen MS s ability to generate structures that satisfy ad hoc user requests, such as a layered material , an elpasolite , and so on. Gen MS can consistently produce structures that satisfy user request as shown in Figure 3, and have plausible initial geometries. Interestingly, we observe that Gen MS can understand semantic-level request, suggesting more ﬂuoride like chemistries when asked for elpasolite , which is reasonable as elpasolite is associated with the mineral K2Na Al F6.

Effect of search. Next, we aimed to understand the effect of search in Gen MS, especially in producing low-energy structures. For each of the family of crystals in Table 1, we analyzed the effect of the language and structure branching factors (H and L in Algorithm 1). Only crystals that match input speciﬁcation were considered for energy computation. We found that increasing the branching factor of both language and structure enables Gen MS to generate structures with lower formation energies (at a higher inference cost).

Cl perovskite Double perovskite Pryochlore

Figure 3: Qualitative evaluation. We test Gen MS on a set of ad hoc language inputs to generate plausible examples from well-known crystal families. Gen MS is able to search for the corresponding structures that satisfy user requests and have plausible initial geometries. Visualization provided by VESTA [23].

Perovskite Pyrochlore Spinel

Language Structure Ef Language Structure Ef Language Structure Ef branch (H) branch (L) (DFT) branch (H) branch (L) (DFT) branch (H) branch (L) (DFT)

1 1 -0.55 1 1 -2.40 1 1 -1.67 1 100 -0.79 1 100 -2.51 1 100 -1.74 25 1 -2.76 25 1 -3.02 25 1 -1.82 25 100 -2.91 25 100 -3.24 25 100 -1.95

Table 2: Ef (computed by DFT) vs. branching factor. Gen MS can generate structures with lower formation energy (computed by DFT) at the cost of slower inference when language and structure branching factors are increased.

3.2 Evaluating individual components of Gen MS

Next, we evaluate the individual component of Gen MS, including the effect of using language to narrow down the search space, the choice of the compact representation of crystal structures, and ﬁnally the best-of-N sampling strategy for choosing the crystal structures with low formation energy.

Effect of language. We want to understand whether Gen MS can provide effective control over formulae proposed by the LLM at the semantic-level through natural language. In Table 3, we ﬁrst show that requesting a particular element to be in the formula always results in formulas with that particular element being proposed by the pretrained LLM πhi. We then show that when a user requests for metal, the model is 4 times more likely to generate formulae for metal. The model also respects a user s request for the generated formulae to be unique (with respect to either a user provided list of known formulae in the context of the LLM, or the name of some crystal database).

Next, we study the effect of retrieval augmented generation (RAG). We use Gen MS with and without RAG to propose 25 formulae for each of the three major crystal families from Section 3.1 and generates 4 structures per formula using the diffusion model. We report the rate of valid formulae proposed by the LLM and the structures that can be matched with existing structures from the corresponding family in Table 4. RAG improves both the rate of valid formulae and matched structures.

Element Metal Unique Unique constraint only (custom list) (Materials Project)

Not asking N/A 0.25 0.24 0.16 Asking 1.00 1.00 0.88 0.96

Table 3: Effect of language. Asking for a speciﬁc element from the periodic table results in formulae that always contain that element. Asking for metal and formulae unique with respect to some existing formula sets result in formulae that are more likely to satisfy user requests.

Valid Match formula rate

Without RAG 0.97 0.72 With RAG 1.00 0.89

Table 4: Effect of RAG. Using retrieval augmented generation improves the percentage of valid formulae and matched structures. See details for the structure matcher used in Appendix A.1.

Uni Mat [6] Gen MS

DFT converge 0.62 0.93 Mean Ef -0.40 0.06 -0.49 0.03

Table 5: DFT evaluation of Gen MS vs Uni Mat. Structures proposed by Gen MS result in much high DFT convergence rate and lower average Ef than structures proposed by Uni Mat. Error bars reﬂect standard error.

Small LLM [9] Large LLM Gen MS

Success 0.86 0.87 0.93 Match (unseen) 0.26 0.37 0.48

Table 6: Comparison to ﬁnetuned LLMs. Gen MS s diffusion model with compact representations achieves high success rate of generating valid crystals, as well as a high matching rate to holdout structures compared to Crystal LM [9].

Energy diff

Composition Composition

Ef best-of-10 < Ef single Ef best-of-10 > Ef single

Ef best-of-10 < Ef single Ef best-of-10 > Ef single

Formation energy difference (GNN) Formation energy difference (DFT)

Figure 4: Formation energy between Best-of-N and a single sample. Both according to energy predicted by GNN and calculated by DFT, best-of-N with N = 10 leads to improvements in energy compared to single samples for 80% of 1,000 compositions considered.

Compact crystal representation. We now evaluate the diffusion model πlo trained using the compact representation of crystals structures described in Section 2.3. We compare diffusion model with compact crystal representation against two prior work for generating crystal structures conditioned on composition. Uni Mat [6] proposed a periodic table representation of crystals which requires a large amount of paddings to handle atoms that do not exist in the structure. Crystal LM [9] proposes to ﬁnetune an LLM to directly generate CIF ﬁles from input compositions. In Table 5, we report the DFT convergence rate and DFT calculated Ef on a set of holdout structures following [6]. We observe that the compact crystal representation results in both higher convergence rate and lower Ef than the sparse representation in [6]. To compare Gen MS s diffusion model against ﬁnetuning LLMs to generate CIF ﬁles directly, we follow the experimental setting of Crysta LLM where we train a composition conditioned diffusion model on a combination of Materials Project [10], OQMD [24], and NOMAD [25], and test the success rate of generating matching structures for unseen compositions following [9]. In Table 6, we see that Gen MS has signiﬁcantly higher rate in producing a valid crystal and a crystal that can be matched to the test set in [9].

Best-of-N structure sampling. To better understand the effect of high structure branching factor in Algorithm 1 across different compositions, we measure the difference in the formation energy, using a holdout test set of 1,000 compositions, between using the energy prediction GNN to select the best of 10 samples compared to only predicting a single structure. The energy difference with and without best-of-N sampling is shown in Figure 4. Using best-of-N with N = 10 results in improved energy for over 80% of structures (as also veriﬁed by DFT calculations). We found the energy prediction GNN to be a good indicator of the true energy of the crystal structures, i.e., the GNN predicted energy difference (left) and the DFT calculated energy difference (right) are very similar in Figure 4.

4 Related work

Hierarchical and latent image and video generation. Image and video generative models have exhibited an impressive ability to synthesize photorealistic images or videos when given text description as input. Many of the state-of-the-art models adopt a hierarchical modeling approach that inspired the design of with Gen MS. For example, latent diffusion models [26, 27] contains (1) a language model that converts text to high-level text embeddings, (2) a diffusion model takes the text embeddings as input and output latents in a compressed latent space, and (3) a feed forward decoder network [26] or a diffusion decoder [28, 14] that given the generated latents generates full-resolution signals in the pixel space. Cascaded diffusion models [29, 30, 31] instead proposed to generate signals at the lowest resolution with a standard diffusion model, followed by a few super-resolution

models that successively upsample signals and add high-resolution details. Similar to Gen MS, by breaking down complicated image or video generation into a hierarchy of less challenging problems, these models can generate high quality samples more efﬁciently and effectively.

Generative models for crystal structures. A number of works [9, 8, 32] have proposed to train or ﬁne-tune language models to generate output ﬁles containing crystal information or low-level atom positions. However, it remains expensive and challenging to train and generate detailed structural information with LLMs. On the other hand, diffusion models, as a powerful class of generative model in vision, have been applied to generate crystal structures [5, 7, 6]. However these methods either reply on training with a large set of unconditional samples and brute-force sampling for new materials not in the training set, or necessitate predetermined compositions as conditioning information during inference. Handling of candidate structure generation requires a model capable of independent reasoning about chemical compositions based on high-level user speciﬁcations and structure optimization, as done in Gen MS.

Hierarchical search and planning. The problem of learning to generate low-level continuous output from high-level language instructions, while employing intermediate search and planning steps, has been studied in other domains such as continuous control [33], self-driving [34], and robotics [35]. While some works have focused on purely using LLMs to search and plan through complex output spaces [36, 37], other research has shown that solely relying on LLMs to search and plan can fail short due to the lack of low-level information (e.g., locations, precise motions) captured in the model [38]. Recently, video generation models have been applied to provide additional details about the physical world so that low-level control actions can be extracted more accurately [39, 40, 41, 42]. Gen MS follows a similar approach but focuses on generating crstyal structures, using diffusion models on top of LLMs to provide additional details about crystal structure, enabling high-level plans (i.e., symbolic chemical formulae) to be veriﬁed at a low-level (i.e., crystal structures with precise atom locations).

Large language models for science. Recently, there has been a surge of interest in applying large langauge models in domains of science, such as physics [43], biology [44], chemistry [45, 46], and materials science [47]. In these settings, LLMs generally serve as a conversational [44] or educational [48] tool, where LLMs output natural language to be consumed by human users (e.g., an answer to a scientic question asking about the property of some existing crystal structure). On the other hand, we are interested in the ability of a pretrained LLM to propose intermediate textual information such as chemical formulae for interesting crystal structures. Closest to our work are [49, 50] which leverage an LLM to generate SMILES or other chemical strings for molecular design. Nevertheless, we are interested in generating not just the formulae, but the actual crystal structures with continuous-valued atom locations, as many materials property can only be calculated and veriﬁed once the full structure available.

5 Conclusion and future work

We have introduced Gen MS, an initial attempt at enabling end-to-end generation of candidate crystal structures that look physically viable and satisfy instructions expressed in natural language. Gen MS can generate examples from families such as pyrochlores and spinels purely from natural language prompts. We hope the design principles of Gen MS will initiate broad interest in exploiting language as a natural interface for ﬂexible design and generation of crystal structures that meet user-speciﬁed criteria, and enable the domain experts to work more efﬁciently. Gen MS has a few limitations that call for future work:

Generating complex structures. While Gen MS is able to generate simple structures such as those shown in Figure 3, we found that Gen MS is less effective in generating complex structures such as Mxenes and Kagome lattices. Controllable generation of highly complex crystal structures is an interesting area of future work.

Impact on experimental exploration. While we have shown that Gen MS is effective in generating crystal structures that are not in public databases and that satisfy user requirements, its effectiveness in suggesting speciﬁc materials with target properties (e.g., battery electrodes or electrolytes, semiconductors, superconductors etc.) requires further experimental veriﬁcation.

Synthesizability. While the goal of Gen MS is to provide an end-to-end generative framework from natural language instructions to realistic crystal structures, synthesizability of the generated crystals is not currently part of the pipeline. We foresee development in multimodal models and integration of other computational tools from materials science to allow predicted structures to be assessed for synthesizability.

Extension to other chemical systems. We have shown that Gen MS can effectively generate crystal structures from natural language. We note that Gen MS can also potentially be extended to generating molecules and protein structures from natural language (e.g. generate a protein with an alpha-helix ). We leave these explorations for future work.

[1] Lev I Berger. Semiconductor materials. CRC press, 2020. 1

[2] Martin A Green, Anita Ho-Baillie, and Henry J Snaith. The emergence of perovskite solar cells. Nature photonics, 8(7):506 514, 2014. 1

[3] KJPC Mizushima, PC Jones, PJ Wiseman, and John B Goodenough. Lixcoo2 (0< x<-1): A new cathode material for batteries of high energy density. Materials Research Bulletin, 15(6):783 789, 1980. 1

[4] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In International conference on machine learning, pages 8821 8831. Pmlr, 2021. 1

[5] Tian Xie, Xiang Fu, Octavian-Eugen Ganea, Regina Barzilay, and Tommi Jaakkola. Crystal diffusion variational autoencoder for periodic material generation. ar Xiv preprint ar Xiv:2110.06197, 2021. 1, 3, 5, 9, 14

[6] Mengjiao Yang, Kwang Hwan Cho, Amil Merchant, Pieter Abbeel, Dale Schuurmans, Igor Mordatch, and Ekin Dogus Cubuk. Scalable diffusion for materials generation. ar Xiv preprint ar Xiv:2311.09235, 2023. 1, 5, 8, 9

[7] Claudio Zeni, Robert Pinsler, Daniel Zügner, Andrew Fowler, Matthew Horton, Xiang Fu, Sasha Shysheya, Jonathan Crabbé, Lixin Sun, Jake Smith, et al. Mattergen: a generative model for inorganic materials design. ar Xiv preprint ar Xiv:2312.03687, 2023. 1, 9

[8] Daniel Flam-Shepherd and Alán Aspuru-Guzik. Language models can generate molecules, materials, and protein binding sites directly in three dimensions as xyz, cif, and pdb ﬁles. ar Xiv preprint ar Xiv:2305.05708, 2023. 1, 9

[9] Luis M Antunes, Keith T Butler, and Ricardo Grau-Crespo. Crystal structure generation with autoregressive large language modeling. ar Xiv preprint ar Xiv:2307.04340, 2023. 1, 8, 9, 14

[10] Anubhav Jain, Shyue Ping Ong, Geoffroy Hautier, Wei Chen, William Davidson Richards, Stephen Dacek, Shreyas Cholia, Dan Gunter, David Skinner, Gerbrand Ceder, et al. Commentary: The materials project: A materials genome approach to accelerating materials innovation. APL materials, 1(1), 2013. 2, 3, 5, 8, 14

[11] Mariette Hellenbrandt. The inorganic crystal structure database (icsd) present and future. Crystallography Reviews, 10(1):17 22, 2004. 2, 3, 5, 14

[12] Scott Kirklin, James E Saal, Bryce Meredig, Alex Thompson, Jeff W Doak, Muratahan Aykol, Stephan Rühl, and Chris Wolverton. The open quantum materials database (oqmd): assessing the accuracy of dft formation energies. npj Computational Materials, 1(1):1 15, 2015. 2

[13] William Peebles and Saining Xie. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4195 4205, 2023. 2

[14] Tim Brooks, Bill Peebles, Connor Holmes, Will De Pue, Yufei Guo, Li Jing, David Schnurr, Joe Taylor, Troy Luhman, Eric Luhman, Clarence Ng, Ricky Wang, and Aditya Ramesh. Video generation models as world simulators. 2024. 2, 8

[15] Amil Merchant, Simon Batzner, Samuel S Schoenholz, Muratahan Aykol, Gowoon Cheon, and Ekin Dogus Cubuk. Scaling deep learning for materials discovery. Nature, 624(7990):80 85, 2023. 4

[16] Machel Reid, Nikolay Savinov, Denis Teplyashin, Dmitry Lepikhin, Timothy Lillicrap, Jeanbaptiste Alayrac, Radu Soricut, Angeliki Lazaridou, Orhan Firat, Julian Schrittwieser, et al. Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context. ar Xiv preprint ar Xiv:2403.05530, 2024. 4, 5

[17] Jordan Hoffmann, Louis Maestrati, Yoshihide Sawada, Jian Tang, Jean Michel Sellier, and Yoshua Bengio. Data-driven approach to encoding and decoding 3-d crystal structures. ar Xiv preprint ar Xiv:1909.00949, 2019. 5

[18] Juhwan Noh, Jaehoon Kim, Helge S Stein, Benjamin Sanchez-Lengeling, John M Gregoire, Alan Aspuru-Guzik, and Yousung Jung. Inverse design of solid-state materials via a continuous representation. Matter, 1(5):1370 1384, 2019. 5

[19] Callum J Court, Batuhan Yildirim, Apoorv Jain, and Jacqueline M Cole. 3-d inorganic crystal structure generation and property prediction via representation learning. Journal of Chemical Information and Modeling, 60(10):4518 4535, 2020. 5, 14

[20] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical image computing and computer-assisted intervention MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, pages 234 241. Springer, 2015. 5

[21] Shyue Ping Ong, William Davidson Richards, Anubhav Jain, Geoffroy Hautier, Michael Kocher, Shreyas Cholia, Dan Gunter, Vincent L Chevrier, Kristin A Persson, and Gerbrand Ceder. Python materials genomics (pymatgen): A robust, open-source python library for materials analysis. Computational Materials Science, 68:314 319, 2013. 5, 15

[22] Daniel W Davies, Keith T Butler, Adam J Jackson, Jonathan M Skelton, Kazuki Morita, and Aron Walsh. Smact: Semiconducting materials by analogy and chemical theory. Journal of Open Source Software, 4(38):1361, 2019. 5, 14

[23] Koichi Momma and Fujio Izumi. Vesta 3 for three-dimensional visualization of crystal, volumetric and morphology data. Journal of applied crystallography, 44(6):1272 1276, 2011. 7

[24] James E Saal, Scott Kirklin, Muratahan Aykol, Bryce Meredig, and Christopher Wolverton. Materials design and discovery with high-throughput density functional theory: the open quantum materials database (oqmd). Jom, 65:1501 1509, 2013. 8

[25] Claudia Draxl and Matthias Schefﬂer. The nomad laboratory: from data sharing to artiﬁcial intelligence. Journal of Physics: Materials, 2(3):036001, 2019. 8

[26] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. Highresolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10684 10695, 2022. 8

[27] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. Advances in neural information processing systems, 34:11287 11302, 2021. 8

[28] Aditya Ramesh et al. Hierarchical text-conditional image generation with clip latents, 2022. 8

[29] Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. Cascaded diffusion models for high ﬁdelity image generation. Journal of Machine Learning Research, 23(47):1 33, 2022. 8

[30] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily L Denton, Kamyar Ghasemipour, Raphael Gontijo Lopes, Burcu Karagol Ayan, Tim Salimans, et al. Photorealistic text-to-image diffusion models with deep language understanding. Advances in neural information processing systems, 35:36479 36494, 2022. 8

[31] Jonathan Ho, William Chan, Chitwan Saharia, Jay Whang, Ruiqi Gao, Alexey Gritsenko, Diederik P Kingma, Ben Poole, Mohammad Norouzi, David J Fleet, et al. Imagen video: High deﬁnition video generation with diffusion models. ar Xiv preprint ar Xiv:2210.02303, 2022. 8

[32] Nate Gruver, Anuroop Sriram, Andrea Madotto, Andrew Gordon Wilson, C Lawrence Zitnick, and Zachary Ulissi. Fine-tuned language models generate stable inorganic materials as text. ar Xiv preprint ar Xiv:2402.04379, 2024. 9

[33] Jacky Liang, Wenlong Huang, Fei Xia, Peng Xu, Karol Hausman, Brian Ichter, Pete Florence, and Andy Zeng. Code as policies: Language model programs for embodied control. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pages 9493 9500. IEEE, 2023. 9

[34] Xingcheng Zhou, Mingyu Liu, Bare Luka Zagar, Ekim Yurtsever, and Alois C Knoll. Vision language models in autonomous driving and intelligent transportation systems. ar Xiv preprint ar Xiv:2310.14414, 2023. 9

[35] Can Cui, Yunsheng Ma, Xu Cao, Wenqian Ye, Yang Zhou, Kaizhao Liang, Jintai Chen, Juanwu Lu, Zichong Yang, Kuei-Da Liao, et al. A survey on multimodal large language models for autonomous driving. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 958 979, 2024. 9

[36] Yaqi Xie, Chen Yu, Tongyao Zhu, Jinbin Bai, Ze Gong, and Harold Soh. Translating natural language to planning goals with large-language models. ar Xiv preprint ar Xiv:2302.05128, 2023. 9

[37] Karthik Valmeekam, Matthew Marquez, Sarath Sreedharan, and Subbarao Kambhampati. On the planning abilities of large language models-a critical investigation. Advances in Neural Information Processing Systems, 36:75993 76005, 2023. 9

[38] Karthik Valmeekam, Alberto Olmo, Sarath Sreedharan, and Subbarao Kambhampati. Large language models still can t plan (a benchmark for llms on planning and reasoning about change). ar Xiv preprint ar Xiv:2206.10498, 2022. 9

[39] Mengjiao Yang, Yilun Du, Kamyar Ghasemipour, Jonathan Tompson, Dale Schuurmans, and Pieter Abbeel. Learning interactive real-world simulators. ar Xiv preprint ar Xiv:2310.06114, 2023. 9

[40] Yilun Du, Sherry Yang, Bo Dai, Hanjun Dai, Oﬁr Nachum, Josh Tenenbaum, Dale Schuurmans, and Pieter Abbeel. Learning universal policies via text-guided video generation. Advances in Neural Information Processing Systems, 36, 2024. 9

[41] Yilun Du, Mengjiao Yang, Pete Florence, Fei Xia, Ayzaan Wahid, Brian Ichter, Pierre Sermanet, Tianhe Yu, Pieter Abbeel, Joshua B Tenenbaum, et al. Video language planning. ar Xiv preprint ar Xiv:2310.10625, 2023. 9

[42] Anurag Ajay, Seungwook Han, Yilun Du, Shuang Li, Abhi Gupta, Tommi Jaakkola, Josh Tenenbaum, Leslie Kaelbling, Akash Srivastava, and Pulkit Agrawal. Compositional foundation models for hierarchical planning. Advances in Neural Information Processing Systems, 36, 2024. 9

[43] Jason Holmes, Zhengliang Liu, Lian Zhang, Yuzhen Ding, Terence T Sio, Lisa A Mc Gee, Jonathan B Ashman, Xiang Li, Tianming Liu, Jiajian Shen, et al. Evaluating large language models on a highly-specialized topic, radiation oncology physics. Frontiers in Oncology, 13, 2023. 9

[44] Rachel K Luu and Markus J Buehler. Bioinspiredllm: Conversational large language model for the mechanics of biological and bio-inspired materials. Advanced Science, 11(10):2306724, 2024. 9

[45] Andres M. Bran, Sam Cox, Oliver Schilter, Carlo Baldassari, Andrew D White, and Philippe Schwaller. Augmenting large language models with chemistry tools. Nature Machine Intelligence, pages 1 11, 2024. 9

[46] Di Zhang, Wei Liu, Qian Tan, Jingdan Chen, Hang Yan, Yuliang Yan, Jiatong Li, Weiran Huang, Xiangyu Yue, Dongzhan Zhou, et al. Chemllm: A chemical large language model. ar Xiv preprint ar Xiv:2402.06852, 2024. 9

[47] Ge Lei, Ronan Docherty, and Samuel J Cooper. Materials science in the era of large language models: a perspective. ar Xiv preprint ar Xiv:2403.06949, 2024. 9

[48] Liangtai Sun, Yang Han, Zihan Zhao, Da Ma, Zhennan Shen, Baocai Chen, Lu Chen, and Kai Yu. Scieval: A multi-level large language model evaluation benchmark for scientiﬁc research. In Proceedings of the AAAI Conference on Artiﬁcial Intelligence, volume 38, pages 19053 19061, 2024. 9

[49] Hisaki Ikebata, Kenta Hongo, Tetsu Isomura, Ryo Maezono, and Ryo Yoshida. Bayesian molecular design with a chemical language model. Journal of computer-aided molecular design, 31:379 391, 2017. 9

[50] Michael Moret, Irene Pachon Angona, Leandro Cotos, Shen Yan, Kenneth Atz, Cyrill Brunner, Martin Baumgartner, Francesca Grisoni, and Gisbert Schneider. Leveraging molecular structure and bioactivity with chemical language models for de novo drug design. Nature Communications, 14(1):114, 2023. 9

[51] Özgün Çiçek, Ahmed Abdulkadir, Soeren S Lienkamp, Thomas Brox, and Olaf Ronneberger. 3d u-net: learning dense volumetric segmentation from sparse annotation. In Medical Image Computing and Computer-Assisted Intervention MICCAI 2016: 19th International Conference, Athens, Greece, October 17-21, 2016, Proceedings, Part II 19, pages 424 432. Springer, 2016. 14

[52] Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J. Fleet. Video diffusion models, 2022. 14

[53] Georg Kresse and Jürgen Furthmüller. Efﬁcient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical review B, 54(16):11169, 1996. 15

[54] Georg Kresse and Jürgen Furthmüller. Efﬁciency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational materials science, 6(1):15 50, 1996. 15

[55] John P Perdew, Matthias Ernzerhof, and Kieron Burke. Rationale for mixing exact exchange with density functional approximations. The Journal of chemical physics, 105(22):9982 9985, 1996. 15

[56] Peter E Blöchl. Projector augmented-wave method. Physical review B, 50(24):17953, 1994. 15

[57] Georg Kresse and Daniel Joubert. From ultrasoft pseudopotentials to the projector augmentedwave method. Physical review b, 59(3):1758, 1999. 15

[58] Kiran Mathew, Joseph H Montoya, Alireza Faghaninia, Shyam Dwarakanath, Muratahan Aykol, Hanmei Tang, Iek-heng Chu, Tess Smidt, Brandon Bocklund, Matthew Horton, et al. Atomate: A high-level interface to generate, execute, and analyze computational materials science workﬂows. Computational Materials Science, 139:140 152, 2017. 15

A Experiment details

In this section, we provide additional experimental details, including metrics used for evaluation, baselines, architecture and training of the diffusion model with the compact crystal representation, and details of the setup for the DFT calculations.

A.1 Details of evaluation metrics

Structure and composition validity. The structure and composition validity metrics follow [5]. The structure validity determins that a structure is valid as long as the shortest distance between any pair of atoms is larger than 0.5 Å [19]. The composition is valid if the overall charge is neutral as computed by SMACT [22].

Uniqueness. We determine a generated formula is unique if the reduced form of the formula does not exist in either Materials Project [10] or ICSD [11]. For instance, if ICSD contains formula in the form of AB2, we consider A2B4 generated by the model as a duplicate (thus not unique) structure.

Match rate. To compute the match rate, we use the Structure Matcher module from pymatgen s analysis package. We set the hyperparameters of the matcher following [9], speciﬁcally with stol = 0.5, ltol = 0.3, angle_tol = 10. For each family of crystals in perovskite, pyrochlore, and spinel, we ﬁrst curate the reference set by downloading CIF ﬁles from Materials Project [10] that is likely to belong to each family based on formula and space group. We then use fit_anonymous method of the matcher to compare each generated structure to the structures in the reference set. A generated structure is considered matched if fit_anonymous returns true for at least one reference structure of the corresponding family. Note that this approach might result in false positive matches. For example, when we selected the reference set for pyrochlore, we downloaded CIF ﬁles Material Project that have composition A2B2O7. However, not all A2B2O7 are pyrochlore, so generated structures may still not be a pyrochlore despite being matched to one of the reference structures.

A.2 Details of baselines

We use the following prompts in Table 7 to generate the CIF ﬁles for the end-to-end prompting baseline or to generate the chemical formulae for Gen MS.

Method Prompt

Prompt CIF (baseline) I want you to generate another crystal information (CIF) ﬁles for a stable and potentially realistic material that belongs to {category}. [(Optional) Here are some information about {category} from Wikipedia.] Below are some examples of CIF ﬁles from this category: {example1, example2, ...} Please generate one more ﬁle for a crystal that is not in existing materials databases like Materials Project and ICSD. Please make sure the CIF ﬁle is valid. Just generate the ﬁle and do not say anything else.

Prompt formula (Gen MS) [(Optional) Here are some information about {category} from Wikipedia.] Please give me a list of chemical formulae for a hypothetical material for {category}. I want the formula to be stable, and potentially realistic and do not exist in dataset like Materials Project or ICSD. Please just give the formula and do not say anything else."

Table 7: LLM prompts for baseline and Gen MS.

A.3 Compute, architecture, and training

We repurpose the 3D U-Net architecture [51, 52] into modeling atoms within a crystal structure by their x, y, z locations concatenated with atom number (number of protons) a. As a result, we can

represent each crystal structure using an Ax4 matrix where A is the total number of atoms in the structure, and the dimension with size 4 represents the x, y, z location and atom number of each atom. We repurpose the spatial downsampling and upsampling passes from typical U-Net for images or videos, and keep the resolution (number of points) the same, but still employ residual network with concatenating skip connections (see Figure 2 from the main text). Below we show the architecture and hyperparameters used in the diffusion model for crystals with compact representation.

Hyperparameter Value

Learning rate 5e-5 Optimizer Adam (β1 = 0.9, β2 = 0.99) Base hidden dimension 256 Hidden dimension multipliers 1, 2, 4 Number of mlp and self-attention blocks 9 Batch size 512 EMA 0.9999 Weight decay 0.0 Prediction target ϵ Attention head dimension 64 Dropout 0.1 Training hardware 64 TPU-v4 chips Diffusion noise schedule cosine Noise schedule log SNR range [-20, 20] Training steps 200000 Sampling timesteps 256 Sampling log-variance interpolation γ = 0.1

Table 8: Hyperparameters for training the diffusion model in Gen MS.

A.4 Details of DFT calculations

In all our density functional theory (DFT) calculations, we employ the Vienna ab initio simulation package (VASP) [53, 54] with the Perdew-Burke-Ernzerhof (PBE) [55] functional and projectoraugmented wave (PAW) potentials [56, 57]. Our computational settings align with those used in the Materials Project, as implemented in pymatgen [21] and atomate [58]. These settings include the application of the Hubbard U parameter to selected transition metals in DFT+U calculations, a planewave basis cutoff of 520 e V, speciﬁc magnetization settings, and the use of PBE pseudopotentials. However, we opt for updated versions of potentials for Li, Na, Mg, Ge, and Ga, maintaining the same valence electron count. For structural optimization, our protocol involves a two-stage relaxation of all geometric parameters, followed by a ﬁnal static computation. We utilize the custodian package [21] to manage any issues with VASP and to make necessary adjustments to the simulations. Additionally, we generate gamma-centered k-points for hexagonal cells, deviating from the conventional Monkhorst-Pack scheme. We initialize our simulations with ferromagnetic spin, observing that attempts to explore alternative spin conﬁgurations were computationally too demanding. In our ab initio molecular dynamics (AIMD) simulations, we disable spin polarization and employ the NVT ensemble with a 2 fs timestep. For systems containing hydrogen, we reduce the timestep to 0.5 fs to ensure accuracy.

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The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) The assumptions made should be given (e.g., Normally distributed errors). It should be clear whether the error bar is the standard deviation or the standard error of the mean. It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not veriﬁed. For asymmetric distributions, the authors should be careful not to show in tables or ﬁgures symmetric error bars that would yield results that are out of range (e.g. negative error rates). If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding ﬁgures or tables in the text. 8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufﬁcient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justiﬁcation: We report this in our experiment section. Guidelines:

The answer NA means that the paper does not include experiments. The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn t make it into the paper). 9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines? Answer: [Yes] Justiﬁcation: We have checked. Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [NA] Justiﬁcation: This work is exploratory and is at a too early of a stage to have broad impacts. Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake proﬁles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact speciﬁc groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efﬁciency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justiﬁcation: No risk as far as we can see.

Guidelines:

The answer NA means that the paper poses no such risks. Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety ﬁlters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justiﬁcation: We have properly cited the models (Gemini) we used in the paper.

Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset s creators. 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justiﬁcation: No new assets. Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip ﬁle. 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justiﬁcation: Not involved. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is ﬁne, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [NA] Justiﬁcation: no crowdsourcing nor research Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary signiﬁcantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.