MOFFlow: Flow Matching for Structure Prediction of Metal-Organic Frameworks
Authors: Nayoung Kim
Seongsu Kim
Minsu Kim
Jinkyoo Park
Sungsoo Ahn
Venue: ICLR 2025 Poster (2025)
Keywords: metal-organic framework
material structure prediction
AI for science
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Metal-organic frameworks (MOFs) are a class of crystalline materials with promising applications in many areas such as carbon capture and drug delivery. In this work, we introduce MOFFlow, the first deep generative model tailored for MOF structure prediction. Existing approaches, including ab initio calculations and even deep generative models, struggle with the complexity of MOF structures due to the large number of atoms in the unit cells. To address this limitation, we propose a novel Riemannian flow matching framework that reduces the dimensionality of the problem by treating the metal nodes and organic linkers as rigid bodies, capitalizing on the inherent modularity of MOFs. By operating in the $SE(3)$ space, MOFFlow effectively captures the roto-translational dynamics of these rigid components in a scalable way. Our experiment demonstrates that MOFFlow accurately predicts MOF structures containing several hundred atoms, significantly outperforming conventional methods and state-of-the-art machine learning baselines while being much faster. Code available at https://github.com/nayoung10/MOFFlow.

Published as a conference paper at ICLR 2025
MOFFLOW: FLOW MATCHING FOR STRUCTURE PRE- DICTION OF METAL-ORGANIC FRAMEWORKS
Nayoung Kim Seongsu Kim? Minsu Kim Jinkyoo Park'
Sungsoo Ahn'
Korraa Advanced Institute of Science and Technology (KAIST)
Pohhngg University of Science and Technology (POSTECH) (nayoungkim, min-su, jinkyoo. park,, sungsoo.ahn ahnnakaist.tt........k acc.stttc........... kr jeeonsukkim@@postecc.. ac. kr
ABSTRACT
Metal-organic frameworks (MOFs) are class of crystalline materials with promising applications in many areas such as carbon capture and drug delivery. In this work, we introduce MOFFLOW, the first deep generative model tailored for MOF structure prediction. Existing approaches, including ab initio calcu- lations and even deep generative models, struggle with the complexity of MOF structures due to the large number atoms in the unit cells. To address this lim- itation, we propose novel Riemannian flow matching framework that reduces the dimensionality of the problem by treating the metal nodes and organic link- ers rigid bodies, capitalizing on the inherent modularity of MOFs. By oper- ating in the SE(3) space, MOFFLOW effectively captures the roto-translational dynamics of these rigid components in a scalable way. Our experiment demon- strates that MOFFLOW accurately predicts MOF structures containing several hundred atoms, significantly outperforming conventional methods and state-of- the-art machine learning baselines while being much faster. Code available at https: / / thub. com nayoungl 0 D/MOFFlcc
INTRODUCTION
Metal-organic frameworks (MOFs) are a class of crystalline materials that have recently received significant attention for their broad range of applications, including gas storage (Li al., 2018), gas separations (Qian et al., 2020), catalysis (Lee et al., 2009), drug delivery (Horcajada et al., 2012), sensing (Kreno et al., 2012), and water purification (Haque et al., 201 1). They are particularly valued for their permanent porosity, high stability, and remarkable versatility due to their tunable structures. In particular, MOFs are tunable by adjusting their building blocks, i.e., metal nodes and organic linkers, to modify pore size, shape, and chemical characteristics to suit specific applications (Wang et al., 2013). Consequently, there a growing interest developing automated approaches to designing and simulating MOFs using computational algorithms.
Crystal structure prediction (CSP) is a task central importance for automated MOF design and simulation. The importance of this task lies in the fact that the important functions of MOFs, such as pore size, surface area, and stability, are directly dependent on the crystal structure. The conven- tional approach to general CSP is based heavily on ab initio calculations using density functional theory (DFT; Kohn & Sham, 1965), often combined with optimization algorithms such as random search (Pickard & Needs, 2011) and Bayesian optimization (Yamashita et al., 2018) to iteratively explore the energy landscape. However, the reliance on DFT for such computations can be compu- tationally expensive, especially for large and complex systems such MOFs.
Deep generative models are promising solutions to accelerate the prediction of the MOF structure. Especially, diffusion models (Ho et al., 2020) and flow-based models (Lipman al., 2023) have shown success in similar molecular structure prediction problems, e.g., small molecules (Xu et al., 2022; Jing et al., 2022), folded proteins (Jing et al., 2024; Lin et al., 2023), protein-ligand complex (Corso et al., 2023), and general crystals without the building block constraint (Gebauer et al.,
"Wrk partially done researcher POSTECH.

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building blocks
(Metal nodes, organic linkers)
MOFFLOW
Assembled MDF structure
Figure 1: Overview of MOFFLOW. MOFFLOW is a continuous normalizing flow that exploits the modular nature of MOFs by modeling the building blocks (i.e., metal nodes and organic linkers) as rigid bodies. It learns the vector fields for rotation (q), translation (T), and the lattice (E) that assembles the building blocks into a complete MOF structure.
t 0
Figure 2: Inference trajectory of MOFFLOW. Visualization of the inference trajectory of MOF- FLOW from t = 0 to t = 1, showing the progressive assembly of building blocks. We wrap building block centroids inside the lattice for visual clarity.
2022: Xie al., 2022; Jiao et al.. 2024a; Miller al., 2024). These models iteratively denoise the random structure using neural networks that act similar to force fields guiding atom positions toward a minimum energy configuration.
Contribution. In this work, we introduce MOFFLOW, the first deep generative model tailored for MOF structure prediction. MOFFLOW leverages the modular nature of MOFs, which can be decomposed into metal nodes and organic linkers (Figures 1 and 2). This decomposition enables us design a generative model that predicts the roto-translation of these building blocks to match the ground truth structure. To achieve this, based on Riemannian flow matching (Chen & Lipman, 2024), we propose a new framework that generates rotations, translations, and lattice structure of the building blocks. We design the underlying neural network as composition of building block encoder parameterized by an equivariant graph neural network based on a new attention module for encoding roto toto-translations and lattice parameters of the MOF.
We note that our method competes with existing deep-generative models (Gebauer et al., 2022; Xie al., 2022; Jiao al., 2024a; Miller al., 2024) for general CSP that encompass MOFs as special members. However, our method is specialized for MOF structure prediction by exploiting the domain knowledge that the local structures of the MOF building blocks are shared across different MOF structures. This is particularly useful for reducing the large search space of MOF structures; We consider MOFs up 2,200 atoms per unit cell (Boyd al., 2019), whereas crystals for general CSP consist of up to 52 atoms per unit cell (Jain et al., 2013; Jiao et al.. 2024a). In fact, as confirmed in our experiments, the recent deep generative model (Jiao et al., 2024a) for general CSP fails to scale to the large system size of MOFs. This also aligns with how torsional diffusion (Jing et al., 2022) improved over existing molecular conformer generation algorithms (Xu et al., 2021; 2022) by climinating the redundant degree of freedom.
We benchmark our algorithm with the MOF dataset compiled by Boyd et al. (2019), consisting of 324,426 structures. We compare with conventional and deep tearning-based algorithms for crys- tal structure prediction. Notably, MOFFLOW achieves a match rate of 31.69% on unseen MOF structures, whereas existing methods, despite being more computationally expensive, barely match any. We also demonstrate that MOFFLOW captures key MOF properties and scales efficiently to structures containing hundreds of atoms.
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2 RELATED WORK
Crystal structure prediction (CSP). Traditional approaches to CSP rely on density functional the- ory (DFT) to identify energetically stable structures. To generate candidate structures, heuristic techniques such as random sampling (Pickard & Needs, 2011) and simple substitution rules (Wang et al., 2021) have been employed, alongside more sophisticated optimization algorithms such as Bayesian optimization (Yamashita et al., 2018), genetic algorithms (Yamashita et al., 2022), and particle swarm optimization (Wang et al., 2010). To address the computational burden with DFT calculations, many studies used machine learning as surrogates for energy evaluation (Jacobsen et al., 2018; Podryabinkin et al., 2019; Cheng et al., 2022).
Recently, deep generative models have emerged as promising alternative optimization- methods (Court et al., 2020; Hoffmann et al.. 2019; Noh et al., 2019; Yang et al., 2021; Hu et al., 2020: 2021: Kim et al., 2020; Ren et al., 2022). Notably, Jiao al. (2024a) proposes an equivari- ant diffusion- based model to capture the periodic (33)-invariance of crystal structure distributions, while Lin et al. (2024) and Jiao et al. (2024b) additionally consider lattice permutations and space group constraints, respectively. Miller et al. (2024) uses Riemannian flow matching to generate high- quality samples with fewer integration steps. However, these methods face significant challenges in predicting the MOFs structures, which often consist of hundreds of atoms per unit cell.
MOF structure prediction. Unlike general CSP, where a variety of algorithms have been de- veloped, MOF structure prediction remains significant challenge. Conventional MOF structure prediction methods heavily rely on predefined topologies to connect MOF building blocks (Mar- leny Rodriguez-Albelo et al., 2009; Wu & Jiang, 2024), restricting the discovery of structures with new topologies. To address this limitation, Darby et al. (2020) proposes to combine ab initio ran- dom structure searching (AIRSS; Pickard & Needs, 2011) with the Wyckoff alignment of molecules (WAM) method; however, the reliance on AIRSS makes it computationally expensive. We also note that a recent work (Fu al., 2023) considered related, yet different problem of MOF generation based on a deep generative model which does not include the structure generation.
Flow matching. Flow matching flows (Lipman et al., 2023; Albergo & Vanden -iiudden,,,,en,,, 2023; Liu et al., 2023). Since its introduc- tion. various extensions have been proposed, such as generalization Riemannian manifolds (Chen & Lipman, 2024) and efficiency improvements through optimal transport (Tong et al., 2024; Poola- dian et al., 2023). Due to its flexibility and computational efficiency, flow matching has made notable progress in several related domains, including protein generation (Yim al., 2023a;b; 2024; Bose et al., 2024), molecular conformation generation (Song et al., 2024), and CSP (Miller et al., 2024).
simulation-free approach for training continuous normalizing
3 PRELIMINARIES
3.1 REPRESENTATION OF MOF STRUCTURES
MOF representation. The 3D crystal structure of a MOF can be represented with the periodic arrangement of the smallest repeating unit called the unit cell. A unit cell containing N atoms can be represented with the tuple S (X, a, E) where X == [ nates, a = [a,] E AA is the atom types with A denoting the set of possible elements, and == (a, b, c, a. 3,7) E R [o, 180 3 is the lattice parameter that describes the periodicity of the structure (Miller et al., 2024; Luo et al., 2024). In particular, the lattice parameter & can be trans- formed into a standard lattice matrix L (11, (2, 13) R3x3, which defines the infinite crystal structure:
Nx3 is the atom coordi-
(1 (d'h an, d'1 In + KL T x
(1)
where l: = (ki, k2, k3) is the set of integers representing the periodic translation of the unit cell.
Block-wise representation of MOFs. Here we introduce the blockwise representation of MOFs that decompose given unit cell into constituent building blocks, i.e., the metal nodes and organic linkers. The blockwise representation is tuple S (B, q, E) where B = {Cm JM corresponds to M building blocks and q = [gm]]=11 T [Tm] 1 corresponds to set of M building block
-ro-traansltionn (4m, Tm) SE(3) Moreover, each block C. (am; Y,,) has Nm, atoms with
atom types am.. Jm Nm and local coordinates Y.. N. E R. x3 (defined
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in Section 4.1). Our main assumption is that the building blocks can be composed by the roto- translations to form the MOF structure, i.e., the atomwise representation (X, a. E) can be expressed by the blockwise representation (q. B, X = Concat 1oncat(X,....... 1...... XM). 1.),
(9ms Tm) Yn. where X is the result of the roto-translated local coordinate represented by group action We express the global coordinate X the concatenation X without loss of generality
3.2 FLOW MATCHING ON RIEMANNIAN MANIFOLDS
Flow matching is method to train continuous normalizing flow (CNF; Chen et al., 2018) with- out expensive ordinary differential equation (ODE) simulations (Lipman et al., 2023). Here, we introduce the flow matching generalized to Riemannian manifolds (Chen & Lipman, 2024).
CNF in Riemannian manifold. We consider a smooth and connected Riemann manifold M with metric g where each point E .M is associated with tangent space T.-M and inner product (s 9g We consider learning a CNF d, .M + M defined with the ODE $ (x) u, (z)(())) where Φo(x) = x and (x) E T&M is the time-dependent smooth vector field. The vector field (x()) transforms prior distribution Po to Pt according the following push-forward equation:
Pr (x) [o))=P((x) po po)))'(x)) exp (-/0 f[v ue ue(ar)ds))
E [0, 1},
where V. is the divergence operator and 6,60,1 (x(0
Conditional flow matching on Riemannian manifold. The goal of CNF to learn a vector field ue(+)) that transforms simple prior distribution po so that pi closely approximates some target distri- bution q. Given a vector field (-)-) and the corresponding probability paths (ceoceoo.1) one can train a neural network v, [2;0) 0) with the flow matching objective (ra( = (x), 0) ue (x))))))) where t ~ 24(0, 1], x ~ Pe (x(),, and lI-1lg is the norm induced by the metric However, the flow matching objective lacks analytic form of ue (z) that transforms the prior po into the target g. The key insight of conditional flow matching objective is to instead learn a conditional vector field u, (z) (2) for a data point #1 defined as (CF
Pt.ps (x((a)) (x;
let (x; 0) un (arlar) ))1g), where
Po(x/x1)) Po (x) and pı (2//a1) 5(x x1) with 5(-) being the Dirac distribution. The key idea of conditional flow matching is that one can derive the conditional vector field uy (x|z,) that marginal- izes over data points 31 q accordingly
induce the vector field (x) transforming the prior po into the desired distribution q. To construct u, (x|2z) ). Chen & Lipman (2024) proposes defining conditional flow 24 ( from the geodesic path (minimum length curve) connecting two points To, E M by 2t = exPx, (tlog. (I1)) for E [0, 1]. where exp, and log,, are the expo- nential and logarithmic map at point x E M, respectively. Then the desired conditional vector field d can be derived as the time derivative, i.e., u(xa((1)) = dt th (x (1).11).
At optimum, Vy generates 1i Pt with starting point p Po and end point 1و P1. At inference, we sample from the prior P and propagate from 0 to using one the existing ODE solvers. Note that each training step is faster than the methods based adjoint sensitivity (Chen et al., 2018) since conditional flow matching does not require solving the ODE defined by the neural network.
4 METHODS
In this section, we introduce MOFFLOW, a novel approach for MOF structure prediction based on the rigid-body roto-translation of building blocks express the global atomic coordinates. To this end, using the Riemannian flow matching framework, we learn a CNF pa(g, т, {|B that predicts the blockwise roto-translations and the lattice parameter from the given building blocks. Compared to conventional CSP approaches defined on atomic coordinates (Jiao et al., 2024a:b; Lin et al., 2024: Miller et al., 2024), MOFFLOW enjoys the reduced search space, i.e. the dimensionality of blockwise roto-translation and the atomic coordinates are 6M and 3NN, respectively (6M & 3NN.
4.1 CONSTRUCTION OF LOCAL COORDINATES
To incorporate the MOF symmetries, we devise a scheme to consistently define the local coordinate Y regardless of the initial pose of the building block. Given building block (a, Y), our
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goal is to define global-to-loca function f A x N R Nx3 that defines consistent local coordinate system; that is, the function should satisfy
f(a, xQ + +xtt')
- f(a, X), VQ 5O(3),
(3)
where X is a random conformation of the building block. We can then define Y = f X). Such property can be satisfied by composition of (1) translation by subtracting the centroid and (2)
rotation by aligning with the principal component analysis (PCA) axes:
f(a X) C(X)R(a ((X)) C(X) X 1N
(4)
Here, C(X) denotes subtraction of the centroid and R(a, X) denotes rotation to align the building block with the PCA axes whose sign is fixed by a reference vector, following Gao & Günnemann (2022). See Appendix E for more details.
4.2 FLOW MATCHING FOR MOF STRUCTURE PREDICTION
In this section, we present our approach to training the generative model po 7, {|B) using the flow matching framework. We first explain how our method ensures SE(3)-invariance and introduce a metric for independent treatment of the components q. т, and L. Next, we outline the key elements for flow matching - i.e., the definition of priors, conditional flows, and the training objective.
To preserve crystal symmetries, design the framework such that the generative model is invari- ant to rotation, translation, and permutation of atoms and building blocks. Rotation invariance is guaranteed by using the rotation-invariint lattice parameter representation and canonicalizing the atomic coordinates based on the standard lattice matrix (Miller et al., 2024; Luo et al., 2024). Trans- lation invariance is achieved by operating on the mean- system where the building blocks are centered at the origin - ie., 1 Em= Tm = O. This is the only way to ensure translation invari- ance on (3) M as no R invariant probability measure exists (Yim et al., 2023b). Permutation invariance is addressed using equivariant graph neural networks (Satorras et al., 2021) and Trans- formers (Vaswani, 2017) as the backbone.
M
Metric for (3)).. Following Yim et al. (2023b), we treat SO(3) and R33 independently by defining an additive metric on SE(3) as ((g. (2', = (g. (')(((3))))))))) + (T) Here, (g. (0)))))))))))))))))))))))))))) and (T, T')aa are inner products defined as (4,9')soos
tr(gq" )/2 and (T, t',r))s = 7 =T'7' for 9.9 E (3) with 53) denoting the Lie algebra of SO(3) and 7, r' R
Priors. For each rotation q and translation т. the prior are chosen as the uniform distribution on SO(3) and standard normal distribution on RR3 respectively. For the lattice parameter f, we fol- low Miller et al. (2024) and use log-normal and uniform distributions. Specifically, for the lengths, let (a b, e) = IIxe maximun-likelihood objective (Appendix C). For the angles, Kunstleve et al., 2004) to constrain the distribution to the range (o 3) = (600 120).
(a.b.c) [a (ogNorrmll); μA, (x) σx) where the parameters are learned with the use Niggli reduction (Grosse-
Conditional flows. Following Chen & Lipman (2024), we train the model match the conditional flow defined along the geodesic path of the Riemannian manifold:
(2) exp, (t logg(0) ()) -(t)= = (1 t)r + (T)),, e e²) (1 - () + ()()) (5)
Here, exp, is the exponential map and log,, is the logarithmic map at point From this definition, the conditional vector fields are derived from the time derivatives:
(1) log.(e)) (Q)))
(1) (1)
((1) (())
We wx((()77))
u. (e(())))))
(6)
1
Training objective. Instead of directly modeling the vector fields, we leverage a closed-form ex- pression that enables re- parameterization of the network to predict the clean data qı, Tı, & from the intermediate MOF structure S(t) = (qt. Te, l, B). To achieve this, we train neural network to
approximate the clean data, expressed (q), 7, (i) F(S(t); 0) with regression on clean data:
((0) Ego (00200
1)2 (As [log](1)) (q1) logg (q1)
+12lA Ta 112c + As1le= 11 )]
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(++ xMM
Backbonslpdatte
NodeUpdate
H
EdgoUpdatt
2'
LutticuUpdate
= (C....,Cu))
Atom-level
Block lovel
Figure 3: Overview of our neural network architecture. Our architecture follows a hierarchi- cal structure, starting with atom-level update layers that encode building block representations into atomic-resolution embeddings. These are followed by block-level update layers, which iteratively refine the roto-translations (q. 7) block features H, pairwise features Z, and lattice parameters L. The final output is prediction of the clean data ( L1).
where D is the dataset, 2(00, 1) is the uniform distribution defined on interval [0, 1]. and Å1, A2, Aa are loss coefficients (see Appendix C).
4.3 MODEL ARCHITECTRRE
Here, we describe the architecture of our neural network F(S(t); used to predict clean data (q,,T,,()) We have two key modules: (1) the atom-level update layers to obtain the building block embeddings from atomic resolution, and (2) the building block-level update layers that aggre- gate and update information over MOF on building block resolution and predict Q1 T1, and & from the final building block embeddings. In what follows, we describe each module one-by-one.
Atom-level update layers. The atom-level update layers (Figure 3) process the building block representation Cm = (am, Yn) for m = ,,.....l, .M to output the building block embedding hm- They graph neural networks operating on an undirected graph Im (Vm, Em) constructed from adding edge between pair of atoms within the cutoff distance of 5A. It initializes the atom-wise features fey : k = atomic distances. Each layer updates the atom features {UA- k = 1, ,,...,,,, Nm as follows:
from atom types and the edge features (ex.x' {k, E E} from
v/K- = Uk + tv
Uhe
Σ MEN(R)
de (Uk, Uwi, Cx,
(8)
where {ur k = L..,,, is the set of updated atom features, (Kk) denotes the neighbor of atom the graph 9m. and Φv. de are multi-layer perceptrons (MLPs). Finally, the building block embedding hn is obtained from applying mean pooling of the node embeddings at the last layer followed by concatenation with sinusoidal time embedding of (Vaswani, 2017) and an MLP.
Block-level update layers. Each layer of the update module (Figure 3) iteratively updates a
its prediction of (q,7) E SE(3) and e along with the block features H = [/mm]= E MM
MMx,, and the pairwise features Z initialized by the intermediate flow matching output gt, T2, the node features are ini- tialized from the atom-level update layers, and the edge features are initialized as 2mm' = Φ2 (hm, Amm, dgram Tm 112), dl fm Il2)), where dgram computes distogram binning the pairwise distance into equally spaced intervals between OÅ and 20A. Finally, the block-level update module is defined as follows: H' = NodeUpdate(HH Z, q,T,)), Z' LatticeUpdato((. H').
[2mm'l (mmm)
The predictions
BackboneUpdaaeeg..1 H'). where q','', H' z', are the updated predictions and the features. Importantly, NodeUpdate operator consists of the newly designed MOFAttention module followed by pre-layer nor- malization Transformers (Xiong et al., 2020) and MLP with residual connections in between. The EdgeUpdate, BackboneUpdate modules are implemented as in Yim et al. (2023b) and LatticeUpdate is identity function for the lattice parameter except for the last layer. At the fi- nal layer, the lattice parameter is predicted from mean pooling of the block features followed by an MLP. Complete details of each update module are in Appendix F.
EdgeUp Z, H'). (q' T')

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Algorithm 1 MOFAttention module
Input: Node features H [ edge features Z
[Zmm' Im, 1m.m'=3 rotations = [qm]m=x:
translations T [Tm]m m lattice parameter E, number of building blocks M, number of heads N,. number non-rotating channels N₁. and number of rotating channels N,.
Output: Updated node features H' = [R,, mmm1
for h {l..... ) do
2:
for m (T..... M) do
Query, key, values. D qmns Kimdy Umnh E RRN Imns kimhs Umb R3N,
3:
Gmns kinh, Umuh = Linearn (Nm)) Imshps kmhp: Umuhp = (Amr Tm) (
4:
5:
end for
6:
ln Linear, (f).
Lattice structure encoding 4 E IR
7:
Umm' Linear, ("man/ (Zmm') for m' {1....,M) M). )).
Attention bias bmmnh IR.
Th = Coottlus(Linnede 8:
D. Learnable coefficient T/h E IR
9:
Ci = -... and C2 = ON.
D. Coefficients C1, C2 E IR.
for m, m' {l,..... M] do
> Attention amm' / E
amm'h SoftMax 12: end for
(Cigmh kinth bmm'h + Ln
CaTa ll@mhp
kmchp I²))
13:
Omh - Em 4m and Omh = Em amm'h Umih for m E (I,..., M). Aggregate.
14: end for
15: Nm Linear AmnearConcatt.p(ong Dmhhp E)) for m E [I....,M} > Update the node features.
In particular, our MOFAttention module modification of the invariant point attention module proposed by Jumper et al. (2021) for processing protein frames. Our modification consists of adding the lattice parameter as input and simplification by removing the edge aggregation information. In particular, the lattice parameter is embedded using linear layer and added as offset for the attention matrix between the building blocks. We provide more details in Algorithm 1.
EXPERIMENTS
The goal of our experiments to address two questions. Accuracy: How does structure predic- tion accuracy of MOFFLOW compare to other approaches? Scalability: How does the performance of MOFFLOW vary with an increasing number of atoms and building blocks?
To address the first question, Section 5.1 compares the structure prediction accuracy of MOFFLOW against both conventional and deep Bearning-based methods. Furthermore, Section 5.2 evaluates whether MOFFLOW can capture essential MOF properties, further validating the accuracy of its predictions. The second question is answered Section 5.3, where we analyze the performance of MOFFLOW with increasing system size. Additionally in Section 5.4, compare MOFFLOW to the self-assembly algorithm (Fu et al., 2023) by integrating it with our approach.
5.1 STRUCTURE PREDICTION
Dataset. We use the dataset from Boyd et al. (2019), containing 324,426 MOF structures. Follow- ing Fu et al. (2023), we apply the metal-oxo decomposition of MOFid id (Bucior et al., 2019) to decompose each structure into building blocks. After filtering structures with fewer than 200 blocks, we split the data into train/valid/test sets in an 8:1:1 ratio. Full data statistics are in Appendix A.
Baselines. We compare our model with two types of methods: optimization-basee algorithms and CrySPY (Yamashita et al., 2021) to
deep learning- based methods. For traditional approach, implement the random search (RS) and evolutionary algorithm (EA). For deep learning, we bench- mark against DiffCSP (Jiao al., 2024a), which generates structures based atom types. MOF. specific methods are excluded due to lack of public availability.
Metrics. We evaluate using match rate (MR) and root square error (RMSE). We compare the samples to the ground truth using StrctureMat cher class of pymat (Ong et al., 2013). Two sets of threshold for stol, Itol, and angle tol are used: (0.5, 0.3, 10.0) in alignment with the

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Table I: Structure prediction accuracy. We compare optimization- methods (RS, EA), deep generative model (DiffCSP) and MOFFLOW. Due to computational constraints, RS and EA were tested on 100 and samples, respectively, while DiffCSP and MOFFLOW were evaluated on the full test set (30,880 structures). MR is the match rate and RMSE is the root mean squared error; indicates no match. stol is the site tolerance for matching criteria. The reported time is the average structure. MOFFLOW outperforms all baselines in MR, RMSE, and generation time.
stol 0.5
stol
# of samples
Avg. time (8).
MR (%) f RMSE
MR (%) 1
RMSE +
(Yamashita et al., 2021)
20
0.00
332
(Yamashita al., 2021)
20
0.00
1959
I
0.09
0.3961
23.12
0.8294
5.37
DifCCPP (Jiao et 2024a)
5
0.34
0.3848
38.94
0.7937
26.85
31.69
0,2820
87.46
0.5183
1.94
MOFFLOW (Ours)
5
44.75
0,2694
100.0
0,4645
5.69
1o Nn
C,NOF, Cu,H,.CuN.O, Zn.HaCuN,O,. Zn.Ho,C,Or
Ground Truth
Ours
DifCCPP
Figure 4: Visualization of the predicted MOF structures. We select structures from the 20 candi- dates with the lowest RMSE. The lattice is scaled to reflect the relative sizes. MOFFLOW accurately generates high-quality predictions with accurate atomic positions and lattice configuration.
general CSP literature (Jiao al., 2024a;b; Chen & Lipman, 2024) and (1.0, 0.3, 10.0) to account
for the difficulty of predicting large structures. MR is the proportion matched structures and RMSE is the root squared displacement normalized by the average free length per atom. We also measure the time required to generate k samples, averaged across all test sets.
Implementation. Both RS and EA use CHGnet (Deng et al., 2023) for structure optimization. We generate 20 samples for RS, while EA starts with 5 initial, 20 populations, and 20 generations. Due to the high computational cost for large crystals, generating more samples was feasible. For DiffCSP, we follow the hyperparameters from Jiao et al. (2024a) and train for 200 epochs. Our method uses an AdamW optimizer (Loshchilov, 2017) with learning rate of 10-4 and (0.9, 0.98). We use maximum batch size of 160 and run inference with 50 integration steps (see Appendix G for analysis on integration steps). We generate and samples for DiffCSP and our method. Implementation details are in Appendix C.
Results. Table presents the results, where MOFFLOW outperforms all baselines. Optimization- based methods yield zero MR, highlighting the challenge of using the conventional atom-based ap- proach for large systems. DiffCSP also performs poorly, underscoring the need to incorporate build- ing block information in MOF structure prediction. While achieve 100% MR at stol 1.0. this threshold is too lenient for practical application; however, we include for multi-level comparison. Visualizations comparing samples from MOFFL.ow and DiffCSP are shown in Figure
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28-3
mimom!
Ie-d
urb leSmviy
I0-4
1ed
Gmund-trath
Pdts Accessible Volume
Volumetric Surface
Gravimetric Surface
Unit Cell Volume
ure te3
1et
ure
censty
003
05
Limiting Diameter
Largest Cavity Diameter
Density
Figure 5: Property distributions. We compare the distributions of key MOF properties for ground- truth, MOFFLOW, and DifICSP. The distributions are histograms smoothed by kernel density esti- mation. Property units are displayed in the top-right corner of each plot. MOFFLON (red) closely aligns with the ground-truth distribution (blue), while DiffCSP (yellow) shows noticeable deviations. These results highlight that MOFFI,OW'S ability to accurately capture essential MOF properties.
5.2 PROPERTY EVALUATION
In this section, we demonstrate that MOFFLOW accurately captures the key properties of ground- truth MOF structures, offering and RMSE. These properties are crucial for various MOF applications, such as gas storage and catalysis. Specifically, we evaluate volumetric surface area (VSA), gravimetric surface area (GSA), largest cavity diameter (LCD), pore limiting diameter (PLD), void fraction (VF), density (DST), accessible volume (AV), and unit cell volume (UCV). Definitions and the details of each property are provided in Appendix B.
detailed assessment of prediction quality beyond match rate
We compare our model to DiffCSP as a representative general CSP approach. We exclude optimization-based baselines as they did not yield meaningful results. For each test structure in Section 5.1, we generate sin- gle sample with 50 integration steps, then evaluate the properties of predicted and ground-truth structures ing Zeo++ (Willems et al., 2012). We evaluate the models with RMSE and distributional differences. We do not filter our samples with the match criteria from Section 5.1.
Table 2: Property evaluation. Average RMSE computed between the ground- truth and generated structures for MOF- FLOW and DifICSP. MOFF1.Ww achieve lower error across all properties, demon- strating its ability to generate high-quality samples that accurately capture MOF properties.
RMSE
Table 2 shows that MOFFLOW consistently yields lower errors than DiffCSP across all evalu- ated properties. This demonstrates its ability to pro- duce high-quality predictions while preserving essen- tial MOF characteristics. Additionally, Figure 5 visual- izes the property distributions, where our model closely reproduces the ground-truth distributions and captures key characteristics. contrast, DiffCSP frequently re- duces volumetric surface area and void fraction to zero, highlighting the limitations conventional approaches in accurately modeling MOF properties.
Results.
MOFFLOW
DifCCSP
VSA (m²//m²²
264.5
796.9
GSA (m (m²/g)
331.6
1561.9
AV (A3)
530.5
3010.2
UCV (A3)
569.5
3183.4
VF
0.0285
0.2167
PLD (Å)
1.0616
4.0581
LCD (A)
1.1083
4.5180
DST (g/cm3)
0.0442
0.3711
5.3 SCALABILITY EVALUATION
Here, demonstrate that MOFF1.ww enables structure prediction for large systems, which is challenge for general CSP methods. To evaluate how performance scales with system size, we analyze match rates as a function of the number of atoms and building blocks. We compare our results with DiffCSP as the representative of the general CSP approach. We generate single samples for each test structure and use thresholds (1.0, 0.3, 10.0) for visibility.

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Results. Figure 6 presents our findings, with the r-axis representing the number of atoms, binned in ranges of (t, + 200). The final bin includes all atom counts beyond the last range, without an upper limit. MOFFlow consistently outperforms DiffCSP across all atom ranges. While our ap- proach shows only gradual performance degrada- tion as atom count increases, DiffCSP suffers sharp decline for systems with more than 100 atoms and fails to predict structures with over 200 atoms. In contrast, our method maintains a high match rate even for structures exceeding 1,000 atoms per unit cell, highlighting the effectiveness of lever- aging building block information for MOF struc- ture prediction. Additionally, Figure 6 shows how match rate scales with the number of build- ing blocks. The results show minimal performance degradation, demonstrating that our model effec- tively handles larger numbers of building blocks and efficiently scales large crystal structures.
MOFFlow DifCCP
100
80
80
Match rate 196)
60-
60
40-
40
20
20
200 400 600 800 8001000
3 4
8 121416 61.62000
Number atoms
Number of building blocks
Figure 6: Scalability evaluation. (left) Match rate comparison between MOFFLOW and DiffCSP by atom count. MOFFLOW pre- serves high match rates across all bins, while DiffCSP drops sharply beyond 200 atoms. (righr) Match rate of MOFFLOW by build- ing block count, with our method performing well even for complex structures with many blocks. These results highlight the scalability of our approach.
5.4 COMPARISON TO SELF-ASSEMBLE ALGORITHM
To make our evaluation more comprehensive, we also consider the self-lssembly algorithm used by Fu et al. (2023) baseline, although the performance is not directly comparable. The self-assembly (SA) algorithm is optimization-base based method that predicts the tation q by maximizing the overlap between building block connection points. Since the algorithm requires T, f, and C as input, it is not directly applicable for structure prediction on its own. Therefore, we conduct an ablation by combining the self-assembly algorithm with our predicted values of T and We note that the self-assembly algorithm defines the centroid as the cen- ter of mass of the connection points, and we account for this offset in implementation.
Table 3: Comparison with self-assembly (SA) algorithm. Since SA can only pre- dict rotations, we provide translations and lattice predicted by MOFFLOW for fair comparison. MOFFLOW alone achieves higher accuracy and faster inference times than SA.
MR (%)
RMSE
Time (s) 1
SA Ours
30.04
0.3084
4.75
31.69
0.2820
1.94
Results. Table shows that MOFFlow alone outperforms the combination with the self-assembly algorithm, indicating that learning the building block orientations leads to more accurate MOF struc- ture predictions than heuristic- based overlap optimization. Additionally, our method offers faster inference, further demonstrating its efficiency compared to optimization-baseed approaches. A com- prehensive comparison is provided in Appendix H.
6 DISCUSSIONS
We propose MOFFL,W, a building block-based approach for predicting the structure of metal- organic frameworks (MOFs). Our approach significantly outperforms general crystal structure pre- diction algorithms - both quality and efficiency - that fail to account for the modularity of MOFs. Additionally, MOFFLOW is scalable, successfully predicting structures composed of up to thou- sands of atoms.
Limitations. MOFFLOW was evaluated only on the hypothetical database (Boyd et al., 2019), high- lighting the need for evaluation on more challenging real-world datasets. Additionally, we assume that the local building block structures are known (i.e., the rigid body assumption), which may be impractical for rare building blocks whose structural information is missing from existing libraries (e.g., Gibaldi et al. (2022)) or is inaccurate. Finally, MOFFLOW is not invariant to periodic transfor- mations of the input; explicitly modeling periodic invariance could further improve its performance.
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REPRODUCIBILITY
We describe experimental details and hyperparameters in Appendix We provide our codes and model checkpoint in https: //githbb. com /wtmm//yyung1/////m 00fffflo.m
ACKNOWLEDGMENTS
This work was partly supported by Institute for Information communications Technology Tech- nology Planning & Evaluation(IITP) grant funded by the Korea govermeen((MSIT) (RS-2019- 1119005,, Artificial Intelligence Graduate School Support Program(KAIST) National Research Foundation of Korea(NRF) grant funded by the Ministry of Science and ICT(MSIT) (No. RS-2022- NR072184), and GRDC(Global Research Development Center) Cooperative Hub Program through the National Research Foundation of Korea(NRFF) grant funded by the Ministry of Science and (CS(M) (No. RS-2024-004361655,
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A DATA STATISTICS
In this section, we present the data statistics to represent the characteristics of the MOF dataset. We consider the MOF dataset from Boyd al. (2019) and the dataset is generated by the MOF- generating algorithms based on the topology from graph theory. The dataset is distributed on the MATERIALSCLOOUD. As mention in Section 5.1, we uses filtered strucutures with fewer than 200 blocks. The dataset is divided into train, valid and test in an 8:1:1 ratio. The statistic of data splits are represented in the Tables 4, 5 and 6.
Property (number of samples 247, 066)
Min
Mean
Max
number of species / atoms
3/
5.3/ 151.5
8 / 2208
working capacity (vacuum) [10 "mol/g]
-0.2618
0.4177
4.8355
working capacity (temperature) [10 [10-2mol/g]
-0.2712
0.2843
4.4044
volume (A1)
534.5
5496.7
193341.7
density [atoms/A*]
0.0737
0.7746
4.0966
lattice a, e [A]
6.13 8.56
13.81 16.47 20.39
/7.83/57..8 80/ 61.62
lattice [P)
59.76 59.99 59.97
91.08 99.05 90.75
120.29 / 120.01 120.02
Table 4: The statistics of the train split MOF dataset.
Property (number of samples 30,
Min
Mean
Max
numher of species atoms
3/ 16
5.3/ 152.4
8 2256
working capacity (vacuum) [10 [10-- mmol/g]
-0,2510
0,4163
5.1152
working capacity (temperature) [10- 1000mmol/g]
-0.1210
0.2834
4,4589
volume [A']
534.5
5538.1
118597.4
density [atoms/A**]
0.11
0.77
4,33
lattice a, b [Å]
6.86 / 8843/8.51
13.85 16.53 / 20.39
48.29/ / 55.1 60.97
lattice (²)
59.98 59.99 59.99
91.08 /9.00 90.73
120.11 / 120.01 120.02
Table 5: The statistics of the valid split of MOF dataset.
Property (number of samples 30, 880)
Mean 5.3 149.3
Max
/ working capacity (temperature) [10 mmolg]
number species / atoms
8 2368
working capacity (vacuum) [10 "mol/g]
-0.1999
0.4193
4.6545
-0.1318
0,2858
3,9931
volume [A*]
536.4
5401.1
124062.6
density [atoms/A1]]
0.108
0.777
4.074
lattice a, b, [å]
6.86 / 8.34 / 8.56
13.74 / 16.39 / 20.26
49.86 / /4.88 60.95
lattice a, [
59.91 60.00 59,99999 99
91.00 / 90.98 / 90.76
120.16 120.01 120.01
Table 6: The statistics of the test split of MOF dataset.
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Published as a conference paper at ICLR 2025
B GLOSSARY
Structural properties. In this section, we introduce the structural properties we measured. These properties were calculated using the Zeo++ software package developed by Willems et al. (2012). Zeo+ provides high-throughput geometry-based analysis of crystalline porous materials, calculat- ing critical features such as pore diameters, surface area, and accessible volume, all of which are essential for evaluating material performance in applications such as gas storage and catalysis.
Specifically, we calculated properties including volumetric surface area (VSA), the surface area per unit volume; gravimetric surface area (GSA), which represents the surface area per unit mass, the largest cavity diameter (LCD), which represents the diameter of the largest spherical cavity within the material; the pore limiting diameter (PLD), defined as the smallest passage through which molecules must pass to access internal voids; the void fraction (VF) (Martin & Haranczyk, 2014), which is the ratio of total pore volume in the structure to the total cell volume; the density (DST), which refers to the mass per unit volume of the material; the accessible volume (AV), indicating the volume available the center of given probe molecule within the pores; and the unit cell volume (UCV), representing the total volume of the repeating unit cell in the crystal structure. These param- eters provide critical insights into the MOF's porosity, surface area, and ability store and transport gases, and recent study shows the correlation with these properties with the bulk material (Krish- napriyan et al., 2020).
C IMPLEMENTATION DETAILS
Training details. We use the TimestepBatch algorithm (Yim et al., 2023b) to simplify batch construction. This method generates batch by applying multiple noise levels t [0, 1] to single data instance, ensuring uniform batch size. To manage memory constraints, we cap the batch size with N2, where N is the number of atoms.
Hyperparameters. Table 7 and Table shows model and training hyperparameters for MOFFLOW, respectively. In practice, generate q, k and from h independently, allowing each have distinct dimension. The non-rotating channels are represented as a tuple, with qk and v specified in that order. We set the log-normal distribution parameters for lattice lengths μ (2.55, 2.75, 2.96) and a = (0.3739, 0.3011, 0.3126), computed from the training data with the closed-form maximum likelihood estimation.
Baselines. Here, we provide the implementation details of the baselines. Unless specified otherwise, all hyperparameters follow their default settings.
DifCCSP (Jiao et al., 2024a): To address memory constraints, we replaced fully connected edge construction with radius graph (cutoff: 5Å) and used batch size of 8. The model was trained on a 24GB NVIDIA RTX 3090 GPU for days until convergence.
RS & EA: Both random search (RS) and the evolutionary algorithm (EA) were im- plemented with CRYSPY (Yamashita et al., 2021) and energy-based optimization with CHGNet (Deng et al., 2023). For RS, we generated 20 structures per sample with a symmetry-based search. EA began with 5 initial RS runs and performed up to 20 gen- erations with population size of 20, 10 crossovers, 4 permutations, 2 strains, and 2 elites. A tournament selection function with size of 4 was employed.
FlowMM (Miller et al., 2024) and EquiCSP (Lin et al., 2024) were excluded from our baselines due to their high resource demands when adapted for MOF structure prediction, requiring more than 20 days of training on an 80GB A100 GPU.
Computational resources. Table summarizes the computational resources required to train learning-base based models. Notably, the TimestepBatch implementation of MOFFLOW requires longer training times than DiffCSP in terms of GPU hours. To address this inefficiency, also release a refactored Batch version of MOFFLOW with details in Appendix D.
Codebase. Our implementation is built https: sesttubb.cco//ggatttrr///ggg MOFDi ff, and https: / / thub. wttp:ttt....cccccctteeeeeeeeeeee ctw.///cccrosoooo/ooteeeeeeeeeee frame- flow. We appre-
http:////tthaa..cccc///ss//////p https://githhb..cm http:/hwww..cooiccoosoooooooeooe
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Published as a conference paper at ICLR 2025
ciate the authors (Yim et al., 2023a;b; 2024; Fu et al., 2023; Corso et al., 2023; Satorras et al., 2021) for their contributions.
Table 7: Model hyperparameters of MOFFLOW
Hyperparameter
Value
Atom-level node dimension
64
Atom-level edge dimension
64
Atom-level cutoff radius
Atom-level maximum atoms
100
Atom-level update layers
Block-level node dimension
256
Block-level edge dimension
128
Block-level time dimension
128
Block-level update layers
MOFAttcntion number of heads
24
MOFAttention rotating channels
256
non-rotating channels
(8. 12)
Table 8: Training hyperparameters of MOFFLOW
Hyperparameter
Value
loss coefficient Å (q)
1.0
loss coefficient A2 (T)
2.0
loss coefficient X3 (E)
0.1
batch size
160
maximum 2
1,600,000
optimizer
AdamW
initial learning rate betas
0.0001
(0.9, 0.98)
learning rate scheduler learning rate patience
ReduceL.ROmPlateal
30 epochs
learning rate factor
0.6
Table 9: Computational resources. Comparison of batch size, GPU type (x number), GPU mem- ory utilization (GB), and training time (d: days, h: hours) for training learning-based models.
Batch size
GPU Type
GPU memory (GB)
Training time
DiffCSP MOFFLOW (TimestepBatcc)
24GB 3090 (x1)
23.24 23.96
5d 2h
160
24GB 3090 (x8)
6.67 23.95
5d 15h
MOFFLOW (Batch)
160
24GB 3090 (x8)
21.62 23.80
1d 17h
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Published as a conference paper at ICLR 2025
D BATCH IMPLEMENTATION OF MOFFLOW
While the TimestepBatch algorithm (Yim et al., 2023b) simplifies implementation, slows con- vergence due to its effective batch size of (i.e., each batch contains noise-perturbed variants of a single instance). To address this limitation, we introduce the Batch implementation, which pro- cesses multiple data instances per batch, aligning with standard practice. As shown in Table 10, Batch achieves comparable performance to TimestepBatch while significantly reducing training time from 1087.25 to 332.74 74 GPU hours. also reduces inference time from 1.94 to 0.1932 seconds.
Table 10: Comparison of TimestepBatch and Batch implementations of MOFFLOW. The Batch implementation achieves comparable performance while significantly reducing training time (GPU hours) and inference time (seconds). Inference time is reported as the average per test instance, calculated by dividing the total elapsed time by the size of the test set.
Training time (h) Inference time (s) Match rate (%)
RMSE
TimestepBatch 1087.25
1.94
31.69
0,2820
Batch
332.74
0.19
32.73
0.2743
E DEFINING LOCAL COORDINATES OF BUILDING BLOCKS
Following, Gao & Günnemann (2022), principle component analysis (PCA) as backbone since it is S3(3) eqqivariaat to a sign. Specifically, if we denote PCA(X) = [e1, e2, ea], in the order of decreasing eigenvalues, VQ E SO(3),
PCA(XQ") = o (CCA(X), CE (-1,+111)
that is, the sign is not preserved upon rotation. To define a consistent direction, Gao Günnemann (2022) suggests the use of an equivariant vector function v(a, x) as
Ei
if v(a, X) e, 2
é
e2 otherwise.
Then, the final equivariant axes is defined as R = [ē,, E2, éa] where é3 = ēι X 62.
However, find that this definition is insufficient for application where some building blocks are 2-dimensional exhibit symmetry with respect to the origin and thus have v(a, X) = 0. For such cases, we define
arg min (in #0}
Vym
=
i.e., the vector from the centroid to the closest atom. Since the building blocks are symmetric, f still fulfills Equation (3) up to permutation, which handled by GNN and Transformers.
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Published as a conference paper at ICLR 2025
F MODEL ARCHITECTURE
Here, we provide the details of the NodeUpdate, EdgeUpdate, and BackboneUpdate modules. Our implementation follows Yim al. (2023b); Jumper al. (2021), with the exception of the MOFAttention module and the Transformers, where we use pre-layer normalized version (Xiong al., 2020). Each module is introduced with our notation. The function R(a, b, c, d) in Algorithm is defined as
(0")2-(000 (d"r))
dd 20"""""a""" (a")² c8"
28" d' c"
R(,,
20"e 42a"
(0"))"""""""-- 2d 6
2c d 20"
26
(a (b")² (dm))
(10)
Algorithm 2 NodeUpdate Module
Input: (q, T. H, e)
Output: H'
I: H + LayerNorm (MOFAtttnnM H Z, + H)
2: H <- Concat H, Linear(H
3: H <- (inear(Transformmrs/ (H) H (
4: H' ML)(H))
Algorithm EdgeUpdate Module
Input: (Z. H
Output: Z'
M M......M.... 2: hm f- Linear (H (e+1)
1: for m = 1,....
3:
for m' =,... M do
4:
2mm' f Concat hmmes Zmmm)
5:
end for
6:
Z [2mme JM
mmmm'=1
7: Z' + JayerNorm(MLP)
8: end for
Algorithm BackboneUpdaaee Module
Input: (g, T. H') Output: q',T'
1: for m = 1,...... M do
2: (b, Fm)) + Linear(Am))
3: (a,b, d) + (1,b, d.dl//vie
4:
Im +- b,
5:
(4m T'm) + (qm, Tm) 7q
6: end for
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G EFFECT OF INTEGRATION STEPS
Here, we investigate how the number of sampling integration steps affects MOFFLOW's perfor- mance. We randomly select 1,000 structures from the test set used Section 5.1 and evaluate the match rate, RMSE, and average sampling time for varying integration steps: (5, 10, 50, 100, 200).
For each structure, We generate single sample and set the thresholds for stol, Itol, and angle.tol to (0.5, 0.3, 10.0).
Results. Figure 7 presents our results. Notably, the performance peaks around 10 and 50 integration steps, with a slight decline observed for higher step counts. This aligns with the trends reported by Miller al. (2024). Based on these results, we use 50 integration steps in our main experiments, which yield the highest match rate of 31.1%, a low RMSE of 0.2821, and a fast sampling time of 1.267 seconds.
10-2
2.0
2.5
1.5
Match
3.0
Average
1.0
0.5
1.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1,0
Integration steps
Integration steps
Integration steps
Figure 7: Effect of integration steps on match rate, RMSE, and average sampling time. Perfor- mance is highest at 10 and 50 integration steps. We select 50 steps for the main experiments due to its optimal balance of the highest match rate, low RMSE, and efficient sampling time.
H COMPARISON WITH SELF-ASSEMBLY ALGORITHM
We compare MOFFLOW's scalability and sampling efficiency with the self self-assembly algorithm (Fu et al., 2023), highlighting their respective strengths and limitations.
Scalability, Since both methods operate at the building block level, we compare their match rates as function of the number of building blocks. The experimental settings follow Section 5.4. Figure 8a shows that while MOFFL.OW achieves higher overall match rate (31.69% vs. 27. 27.14%), the self- assembly algorithm scales better for structures with more building blocks.
Sampling efficiency. Figure 8b compares the assembly times the two methods. MOFFLOW (left) demonstrates significantly faster sampling, below 8 seconds per sample. In contrast, the self- assembly algorithm (right) often requires over 2000 seconds per sample. Additionally, the self- assembly algorithm's 's performance varies widely with initialization, with average assembly times ranging from 4.75 to 14.62 seconds per trial, reflecting its sensitivity to initial conditions.
MOFFlow
0.8
Match rate (%6
Self-assembly
Density
0.6
10-3
10
40
0.4
10-1
20
0.2
10-6
8 9 14
0.0
6
1000
2000
Number building blocks
Assembly time
Assembly
(a) Match rate by building block count
Distribution of assembly time
Figure& 8: Comparison of scalability and efficiency between MOFFLOW and the self-assembly algorithm. (a) Match rate across varying building block counts. MOFFLOW achieves higher match rates overall, but the self-assembly algorithm performs better for structures with a large number of building blocks. (b) Assembly time distributions for (left) MOFFLOWW and (right) the self-assembly algorithm, highlighting the significantly faster inference speed of MOFFLOW.
21


Reviews:
[{"review_id": "UndlDB5tg3", "reviewer": "Reviewer_hN1H", "summary": "The authors tackle the problem of crystal structure prediction (CSP) of Metal-Organic Frameworks. Typically, CSP is performed using ab initio calculations using density functional theory (DFT). However, DFT is computationally expensive and scales poorly for large complex systems such as MOFs. MOFFLow presents a MOF-specific generative model for CSP that operates on secondary building units rather than isolated atoms for structure prediction.", "questions": "* Awkward notation in equation 1. L is 3x3. k is 3 x 1. Lk is therefore 3x1. x_n in line 143 is  1 x 3. So, x_n + Lk is not proper. \n* Line 156: building blocks contain a set of “local” coordinates. What is the reference frame used for these coordinates?\n* Section 5.2: Are the property evaluations done on predicted structures and compared with property evaluations on ground-truth structures?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "* Outperforms existing deep learning-based structure prediction models by a significant amount. \n* Highly scalable.", "weaknesses": "* Limited applicability on domains.\n* Crystal structures that cannot be decomposed into secondary structures cannot readily use these techniques", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AoPyqQWLLn", "reviewer": "Reviewer_zVZp", "summary": "The paper introduces MOFFlow, a frame-based flow matching approach to predict the structure of Metal Organic Frameworks (MOFs). MOFFlow treats the MOF building blocks (Metal nodes, organic links) as frames, and generates the 3D crystal structure by modeling the orientation and translation of the building blocks separately. A novel MOFAttention module is also introduced to model the hierarchical structure of MOFs. MOFFlow is benchmarked against DiffCSP, random structure search, and evolutionary algorithm baselines. The paper shows a significant improvement compared with DiffCSP in generating MOF structure, demonstrating the advantage of the hierarchical modeling approach in generating large MOF structures containing hundreds of atoms.", "questions": "- How did the authors obtain the initial structure of the individual building blocks? Do they use the conformation from the data? Or predict those with a specific algorithm? \n- Have the authors studied the number of steps required for the reverse process in flow matching? \n- Have the authors studied the scalability of MOFFlow and the self-assembly algorithm used in MOFDiff, similar to Fig. 6? \n- I would like to see more details on how Fig. 5 is produced. Since MOFDiff and MOFFlow models different types of problems, how do the authors perform a fair comparison in Fig. 5?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The hierarchical modeling approach to separate the translation and orientation clearly demonstrates an advantage in predicting the structure of MOFs. The advantage is most clearly demonstrated in Fig. 6, indicating MOFFlow’s ability to scale to large structures with more than 200 atoms.\n- The introduction of the MOFAttention module, inspired by the IPA module from AlphaFold 2, is interesting and novel. \n- The authors show an improvement over the self-assemply algorithm from the MOFDiff paper. This demonstrates an advantage of learned methods over heuristic based methods.", "weaknesses": "- The individual pieces of the paper are not particularly novel. The dataset, evaluation, and the MOF decomposition method are similar to the MOFDiff work. The flow matching approach is similar to FrameFlow. The PCA axis to denote the orientations of the building block is similar to Gao & Gunnemann (2022). \n- The paper lacks ablation study and/or more extensive benchmarking to illustrate the key component of their model that lead to the improvements. From DiffCSP to MOFFlow, there are many changes: 1) the adoption of a hierarchical modeling approach; 2) change from diffusion to flow matching; 3) change of the architecture of the score network. Which component leads to the biggest improvement? Although the likely answer is 1), more comparisons would greatly benefit the community.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2XmxDQmMXG", "reviewer": "Reviewer_6goZ", "summary": "The paper presents MOFFLOW, a deep generative model aimed at predicting the structures of metal-organic frameworks (MOFs), leveraging a novel Riemannian flow matching framework. This approach models the building blocks of MOFs as rigid bodies, significantly reducing the complexity of structure prediction. MOFFLOW operates in the SE(3) space, effectively capturing the roto-translational dynamics of the metal nodes and organic linkers. The authors claim that MOFFLOW outperforms both conventional crystal structure prediction (CSP) methods and state-of-the-art deep learning models in terms of accuracy and scalability, handling unit cells with up to thousands of atoms.", "questions": "1. I would suggest the authors add necessary baselines for a fair comparison, which include but are not limited to CDVAE, MOFDiff, FlowMM, DiffCSP++, etc.\n2. Providing insights into the relative contributions of different components (e.g., block-level update layers, MOFAttention module) would add clarity.\n3. Including a detailed comparison of computational resources (e.g., GPU hours, memory usage) against other models would be valuable.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The use of a Riemannian flow matching framework to handle the inherent complexity and modularity of MOFs is a novel aspect of the work. This approach allows MOFFLOW to reduce the dimensionality of the problem and improve scalability.\n2. MOFFLOW's ability to handle large structures with thousands of atoms and maintain high accuracy is a significant advantage, setting it apart from conventional methods that struggle beyond smaller systems.\n3. The authors provide detailed information on hyperparameters, training setup, and make their code and model checkpoints available, enhancing the paper's transparency and reproducibility.", "weaknesses": "1. While the paper compares MOFFLOW against CSP methods and the DiffCSP model, the absence of comparisons with more recent and potentially relevant MOF-specific deep learning models could limit the assessment of state-of-the-art competitiveness.\n2.  Although MOFFLOW incorporates multiple technical innovations (e.g., block-level embedding updates, MOFAttention), the impact of these individual components is not deeply dissected through ablation studies. Such an analysis would strengthen the understanding of the contributions of each module.\n3. While MOFFLOW is stated to be computationally efficient compared to baseline methods, a more detailed analysis of its training and inference costs relative to comparable models would help contextualize its practical deployment feasibility.\n4. (Minor point) The paper mainly focuses on properties relevant to structure prediction accuracy (e.g., match rate and RMSE). It would be beneficial to explore how MOFFLOW's predictions perform in real-world settings that are essential for the practical use of MOFs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7lZehIv1Vl", "reviewer": "Reviewer_1rvy", "summary": "This paper proposes MOFFlow, a flow matching framework for the structure prediction of Metal Organic Frameworks (MOFs). The approach decomposes MOFs into blocks and jointly generates the entire lattice, along with the rotation and translation of each block, using flow matching on different Riemann manifolds. The results demonstrate the superiority of the proposed hierarchical flow matching framework over previous baselines.", "questions": "1. How many different types of building blocks can be decomposed from the dataset?\n2. By generating only translations and rotations instead of fine-grained structures, the proposed method assume building blocks as rigid bodies. Could this assumption potentially limit the model's performance?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper addresses the relatively novel problem of MOF generation and is the first to design an end-to-end framework for jointly generating both the lattice and block-level translations and rotations.\n2. The authors construct benchmarks with easily calculable metrics for evaluation, and the results clearly showcase the model's effectiveness.", "weaknesses": "1. The discussion on periodicity in MOFs is insufficient. Periodicity is a crucial feature in crystallography. Apart from explicitly modelling the lattice parameters, previous works also adopt multi-edge connections [1] or the fractional coordinate system [2] to address this problem. Additional details on modeling periodicity would enhance the paper. For example:\n- In Line 235-237, the authors mention the mean-free system to maintain translation invariance. However, constructing mean-free or center-of-mass (CoM) free systems for periodic boundary conditions (PBCs) differs from non-periodic systems [3] and warrants further discussion.\n- The backbone model is based on a Cartesian coordinate system, which necessitates careful design of edge connections to capture intra-cell interactions.\n\n2. The design of the backbone model lacks clarity. It is unclear whether the atom-level and block-level modules are updated sequentially or alternatively. Including a figure of the overall architecture would be beneficial.\n\n[1] Fu, Xiang, et al. \"MOFDiff: Coarse-grained Diffusion for Metal-Organic Framework Design.\"\n\n[2] Jiao, Rui, et al. \"Crystal structure prediction by joint equivariant diffusion.\" \n\n[3] Lin, Peijia, et al. \"Equivariant Diffusion for Crystal Structure Prediction.\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)