Flexible MOF Generation with Torsion-Aware Flow Matching
Authors: Nayoung Kim
Seongsu Kim
Sungsoo Ahn
Venue: NeurIPS 2025 poster ()
Keywords: metal-organic frameworks
crystal generation
flow matching
Primary area: machine_learning_for_sciences
TL;DR: We propose a flexible two-stage MOF generation framework that combines a SMILES-based autoregressive model with a torsion-aware normalizing flow.

Abstract: Designing metal-organic frameworks (MOFs) with novel chemistries is a longstanding challenge due to their large combinatorial space and complex 3D arrangements of the building blocks. While recent deep generative models have enabled scalable MOF generation, they assume (1) a fixed set of building blocks and (2) known local 3D coordinates of building blocks. However, this limits their ability to (1) design novel MOFs and (2) generate the structure using novel building blocks. We propose a two-stage MOF generation framework that overcomes these limitations by modeling both chemical and geometric degrees of freedom. First, we train an SMILES-based autoregressive model to generate metal and organic building blocks, paired with a cheminformatics toolkit for 3D structure initialization. Second, we introduce a flow matching model that predicts translations, rotations, and torsional angles to assemble the blocks into valid 3D frameworks. Our experiments demonstrate improved reconstruction accuracy, the generation of valid, novel, and unique MOFs, and the ability to create novel building blocks. Our code is available at https://github.com/nayoung10/MOFFlow-2.

Flexible MOF Generation with Torsion-Aware Flow Matching
Nayoung Kim
Seongsu Kim
Sungsoo Ahn
Korea Advanced Institute of Science and Technology (KAIST) fnayoungkim, seongsu. kim, sungsoo.. akaiskaist. ac
Abstract
Designing metal-organic frameworks (MOFs) with novel chemistries is longstand- ing challenge due their large combinatorial space and complex 3D arrangements of the building blocks. While recent deep generative models have enabled scalable MOF generation, they assume (I) fixed set of building blocks and (2) known local 3D coordinates of building blocks. However, this limits their ability to (1) design novel MOFs and (2) generate the structure using novel building blocks. We propose two-stage MOF generation framework that overcomes these limitations by modeling both chemical and geometric degrees of freedom. First, we train SMILES-based autoregressive model to generate metal and organic building blocks, paired with cheminformatics toolkit for 3D structure initialization. Sec- ond, introduce flow matching model that predicts translations, rotations, and torsional angles to assemble the blocks into valid 3D frameworks. Our experiments demonstrate improved reconstruction accuracy, the generation of valid, novel, and unique MOFs, and the ability to create novel building blocks. Our code is available at /tttps://githie httg///hhu...mmmaaournroee cww/nayoungn 1 waaprnng0000FF...mo /MM0FF1o--22
Introduction
Metal-organic frameworks (MOFs) are highly erystalline materials known for their permanent porosity, structural versatility, and broad applications in fields such as gas storage and separations [1, 21, catalysis [3], and drug delivery [4]. MOFs designed by assembling chemical building blocks, iee. metal clusters and organic linkers; their properties (e.g., pore size) can be tuned by swapping or modifying these components. This tunability has enabled the synthesis of over 100,000 distinct MOF structures to date [5], yet the theoretical space of possible MOFs is vastly larger, on the order of millions of structures [6]. Exploring this large design space is grand challenge in materials science.
Computational modeling of MOFs, i.e., 3D structure prediction and generating new MOFs, poses significant challenges because of their structural complexity. Traditional methods, such ab initio random search [7], attempt to design new MOFs or predict their structures by iteratively proposing combinations of building blocks and evaluating their stability using energy-based criteria. However, MOFs typically contain hundreds of atoms per unit cell, rendering these iterative approaches computationally prohibitive. Even state-of-the-arr deep generative models developed for inorganic materials [8, 9] struggle to scale to the size and complexity of MOFs [10].
In response, researchers have proposed MOF-specific generative models such as MOFDiff [11] and MOFFlow [10]. While these methods represent significant progress, their design choices still constrain both chemical diversity and structural fidelity. Specifically, they (1) rely predefined building blocks and (2) assume these building blocks are rigid. As a result, the design space is limited to recombinations of known components with fixed 3D conformations, hindering the discovery of novel chemistries and overlooking the intrinsic flexibility of organic linkers [12].
fixed set of
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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-805>
<SEP>
[=]=e]jojeeeeccce of,
Building block generator
(0-00)) ec (0
<E08>
B
MOF sequence
Structure initialization
Structure prediction
Generated MDF
Figure 1: Overview of MOFFL.W-22. MOFFLOW-2 a two-stage generative framework for MOF generation and structure prediction. The first stage uses building block generator to generate an MOF sequence in SMILES representation, which initialized to 3D coordinates with the metal library and RDKit. In the second stage, our structure prediction model assembles these building blocks by modeling translation т, rotation q, torsion Φ, and lattice l.
Contribution. To address these limitations, we intro- duce MOFFLOW-2, two-stage generative framework capable of producing novel MOF chemistries and pre- dicting high-fidelity 3D structures (Figure 1). Unlike prior methods, MOFFL.O-22 does not rely on a fixed building block library or assume rigid conformations.
Table Comparison of MOF generative models. Trans refers to translation modeling, Ror to rotation modeling, Tor to torsion angle modeling. and NBB to the ability to generate novel building blocks.
Method
Trans
Rot
Tor
NBB
In the first stage, an autoregressive Transformer gener- ates a canonicalized sequence of MOF building blocks represented as SMILES [13], enabling the creation of entirely novel chemistries. These SMILES can be initialized to 3D structures using cheminformatics tools such as RDKit [14). In the second stage, flow-based model predicts the full 3D MOF structure by jointly modeling (1) rotations and transla- tions of building blocks, (2) lattice structure, and (3) torsion angle for rotatable bonds in the organic linkers. Crucially, by explicitly modeling the torsion angle, MOFFLOW-2 captures the conforma- tional flexibility of organic linkers and no longer depends on predefined and rigid 3D conformations. See Table for comparison to prior works.
MOFDit [11]
x
MOFFLOW [10]
MOFFLOW-2
We evaluate MOFFLOW-2 2 on both structure prediction and generative design tasks. For structure prediction, we the model without providing 3D coordinates of the building blocks realistic but difficult setting. MOFFLOW-2 outperforms MOFFlow, demonstrating its ability to model flexible conformations without relying on rigid- rigid-body assumptions. For the generation task, MOFFLOW-2 achieves higher validity, novelty, and uniqueness than MOFDiff, and generates MOFs with more diverse range of properties. Importantly, it can generate MOFs with building blocks not seen during training, going beyond fixed building block library. These results show that MOFF1.OW-2 is effective for both accurate structure prediction and discovering novel MOFs, providing a step towards more automated and flexible MOF design.
Related work
Generative models for MOFs. Unlike inorganic crystals, MOFs present unique challenges due to their large size, often containing hundreds of atoms per unit cell. MOFDiff [11] addresses this by generating coarse-grained structures via diffusion and decoding them using fixed set building blocks extracted from training data. Similarly, MOFFlow [10] assumes access to the 3D coordinates of building blocks and treats them as rigid bodies, learning block-level rotations and translations instead of predicting atom-level coordinates. In this work, we propose MOFFLOW-2, unified and flexible two-stage framework for both MOF generation and structure prediction. Unlike previous methods, MOFFLOw-2 does not rely fixed building block library require 3D coordinates as input, enabling greater chemical diversity and generality.
Equivariance and generative models. While SE(3) and -E3)veqruaa generative models have shown promising results on molecular tasks [15-18], recent studies question whether equivariance is essential [19]. Notably, AlphaFold3 [20], Boltz-1 [21], and Proteína [22] achieve state-of-the-art results modeling biomolecular complexes without using equivariant architectures. Similarly, Orb [23] and ADiT [24] employ non-equivariant neural networks to model interatomic potentials and

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generate molecule and crystal structures effectively. Following this trend, MOFFLOW-2 adopts non-equivariant Transformer architecture.
3 Preliminaries
3.1 Structural representation of MOFs
Crystal representation. A 3D crystal structure is defined by the periodic repetition of the unit cell. A unit cell containing atoms is represented by the tuple = (A, X,), where = {ai A the atom types. with A being the set of chemical elements; coordinates; and = (a. b, a, (, 7) E R x [0P, 1809| is the lattice parameters, where (a, b, c) are the edge lengths and (,,8877) are the angles between them. The lattice parameters l can be converted to the standard lattice matrix I. (4, 12, 13) E IR3x3 where 1; is lattice vector for each edge. Translating the unit cell along the lattice vectors defines an infinite crystal structure as {an, In + LR E [N] ز where k = (k1, k2, Ag)
{x,} E R R- is the atomic
E E is the periodic translation.
MOF representation. The modular structure of MOFs allows block-level representation based on their building blocks. For an MOF with N atoms, we write S - (BaD. q: T, T E), where BDD = { Jm=1 denotes set of M building blocks. Each building block Cim) = (A(m), (()) contains Nm atoms with atom types A(m) E Nm and local coordinates ( E R N,,,3 We represent the 3D structural degrees of freedom with rotations
{m( E 50(3) | translations (7(m) E M= and torsions o = (d((m E S0(2) MM 1m=1, where Pm is the number of rotatable bonds in block Cim). Following Jing et al. [16], we define a bond as rotatable severing the bond creates two substructures, each with at least two atoms. The global coordinates X can be reconstructed by applying these transformations to local building block coordinates as (m) = (g (m)) 7(m), ((m)) ) -Y (m) and concatenating them X = (o..at(("")..,,X((.)))
We also consider 2D molecular representations of MOF building blocks, denoted as BzD = (G(m) ) j Each graph Gim) = (V(m), E(m)) consists of nodes representing atoms and edges corresponding bonds in organic linkers. These bonds are inferred using OpenBabel [25] and RDKit [14] by analyzing the 3D structure ((m) of the building block.
3.2 Flow matching on Riemannian manifolds
Flow matching (FM) trains CNF by learning vector field that transports samples from arbitrary prior distribution to the target distribution Since MOFFLOw-2 operates on SO(2) and SO(3). we here introduce flow matching generalized Riemannian manifolds [26].
Flows on Riemannian manifolds. Here, we define CNF on Riemannian manifolds. Consider smooth connected Riemannian manifold M with metric 9, where each point E MM has an associated tangent space T&M with inner product (:))) g. A flow Φc(x) : M is defined by the ordinary differential equation (ODE) at (x) = uy (x)())), (z(x)) = x ~ Po where E [0, and (x(x) M is the corresponding time-dependent vector field. We say that u, generates probability paths Pe (x) if (x) = (n]=Po(x)) where # denotes the pushforward operator.
Riemannian flow matching. The goal of flow matching is to learn time-dependent vector field (z; 8) that transports an arbitrary initial distribution Po(r)) to (x)) that closely approximates the target distribution qz)). Given a reference vector field un (2) that transports Po to q. we train the vector field v, (z; 0) by regressing directly towards ux(x) with the flow matching objective:
(0) E.., [[]o; (z; 0) u (z)]
(1)
where (( 1) is the uniform distribution [0, 1), (x the probability distribution at time t. and || I1 is the norm induced by the Riemannian metric g.
While the vector field u. (x)) (r)'n is intractable, flow matching introduces conditional formulation that yields tractable and equivalent training objective [27]. Specifically, the probability paths p<(x) and vector field (x) can be expressed as marginalization over the conditional probability paths (r(()))) and conditional vector field un (x((a))
(x) = //v p((x)z1)((11) dvol(z1), ux(x)= / ue(x)211 Pe (c(x(11)((21) dvol(z1), (2)
(x)
3

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where (z(22(11) has boundary conditions po(x)z1) px()) and Pı (z] = 8(x x1) and (z(=)r1)
generates Pe (z) Then we can train 24 (x; 0) with the conditional flow matching (CFM) objective:
LCFM (0) =
[1]u;<>; () - (x(r(=1)]))]
(3)
4 Autoregressive building block generation
In this section, we introduce our model for generating the MOF building blocks, which are used by the structure prediction model to generate the final 3D structure. Formally, we model the joint distribution pa (q. T, L, Φ BBD) - po (ap)pe T, L, (Bs))) where B&D denotes the 3D building block representations. To this end, we train a SMILES generative model po (B to generate the 2D graph structure of each building block. Then we use pre-built built metal building block library and RDKit [14] to obtain BBD the initial 3D coordinates of the building blocks.
4.1 Autoregressive SMILES generation
The building block generator po (Ba)) is an autoregressive model that generates MOF sequences, which contain metal clusters and organic linkers represented as SMILES [13]. This section outlines the definition of an MOF sequence, the training objective, and the implementation details.
Defining MOF sequence. Given MOF with M1 metal clusters m,,...,, MMM, and M2 or- ganic linkers 01,-.., M2 we impose a canonical ordering on the building blocks to ensure a consistent sequence representation. Specifically, (1) Metal building blocks precede organic ones, separated by a special token <SEP>, (2) Within each group (metal or organic), building blocks are sorted by ascending molecular weight and separated with the symbol
and (3) The full sequence is wrapped with start and end tokens <BOS> and <EOS>, The resulting sequence is BRD = [<BOS> m_1 1. .mm... 2. _1 <SEP> o_1.o_2. 2........ __M2 <EOS>], where m_i and o_j de- note the SMILES string of metal and organic building blocks, respectively.
Next, Given the canonical MOF sequence BRD, tokenize it into sequence of discrete symbols [bi...., by] by mapping substrings (e.g., atoms, bonds, or special tokens) into elements of a fixed vo- cabulary. We use SMMILES-awer regular expression [28] for tokenization, resulting in vocabulary of size 59. See Appendix A. for an example of the tokenization process.
Training and model architecture. We model the sequence using autoregressive language model pa(bi,..., bs) IIIs (bb (bq.) with each token b, conditioned the preceding tokens bęs. We train the model with the standard maximum log-likelihood objective.
4.2
3D coordinate initialization
Here, we describe our algorithm for initializing the building block 3D coordinates ByD from the 2D SMILES representation BzD. These 3D coordinates serve as input to the structure prediction model. We use different initialization strategies for metal clusters and organic linkers due to their distinct structural characteristics.
Due to their complex chemistry, only a limited set of metal clusters is commonly used in MOFs [12]. Based this observation, construct metal library of averaged 3DD coordinates from the training data. Specifically, we group metal clusters by their canonical SMILES, select random reference structure within each group, align all other structures to the reference using root mean square distance (RMSD) minimization, and compute the average of the aligned coordinates.
In contrast, organic linkers exhibit greater structural diversity and flexibility [12], making the template- based approach impractical. Therefore, we use a cheminformatics toolkit (i.e., RDKit [14]) to initialize standard bond lengths and angles by optimizing the MMFF94 force field [29]. However, the resulting structures are only approximate, as the torsion angles around the rotatable bonds remain highly variable [16]. We therefore treat these torsion angles as learnable variables and model them with our structure prediction module to capture the full conformational flexibility of organic linkers.
Appendix B confirms this dataset has only 7-8 metal types with low RMSD variability.

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MOF structure prediction with torsional degrees of freedom
Our MOF structure prediction model pa (g. T, E, Φ (3)) is a flow-based model that predicts the 3D structural degrees of freedom rotations translations т. torsion angles Φ, and lattice parameters & conditioned on the initialized 3D building block structure Byp- In this section, we present the training algorithm based on Riemannian flow matching, Transformer-ben based architecture for predicting each structural component, and a canonicalization procedure for rotations and torsions to ensure consistent targets during training.
Importantly, similar to MOFFI. [10], our structure prediction model MOFFL.W-22 can be used independently of the building block generator. In fact, MOFFI.OW-2 is more general, as it can predict structures directly from the 2D building block B2D, without requiring known 3D conformations. This is more practical since the 3D structures are often unavailable or difficult to obtain in the real world.
5.1 Training algorithm
Prior distribution. We define the priors over each structural component as follows. Rotations and torsions are sampled independently from uniform distributions on their respective manifolds: ((S(33)) and á ~ ((S) Translations are drawn from a standard normal distribution with the center of mass removed: т NOO Ias). For the lattice parameters, the lengths (a, c) are sampled independently from lormallma distributions: (a. b, c) ~ LogNormal σ), where 4 == Hbb Pe) and o (aa, Tb. σe) are estimated by maximum likelihood objective; the lattice angles (ar, B, 7) are sampled uniformly from the range [60", 120"] [9, 10].
Conditional flows, vector fields, and flow matching loss. Following Chen and I. Lipman [26], define conditional flows on each manifold as geodesic interpolations between a prior sample zo and target data 21, where 2 represents one of translation, rotation, torsion, or lattice parameter. Specifically, we define the flow as 2, = exp2, 2. (t log,, (z1)), where t E [0, 1] and log(-), exp(-) denote logarithm map and exponential map on respective manifolds (see Appendix C.I manifold- specific details). This allows parameterizing the vector field by (z f) log,, (21)/(1 t) where the neural network predicts the clean data & from intermediate state zt. Furthermore, the conditional vector field is given by 2n (z( = logz, (z1)/(1= - t).
Given the closed-form expression of the target conditional vector field, parameterize the model as
= A
X,,
(4)
where X, is the noisy coordinate generated by applying rotation Aq1-u, translation Tr, and torsion Φe to the clean X₁ (see Appendix C.2 for precise definition); Aqu-=1 := q:qi the rotation that aligns X, back to X1; and the hat symbol indicates the model's 's prediction of each variable. Under this formulation, we can rewrite the training objective as:
((0) A11/A/-11 Aq de T 112 Aol --11--112+ Aalle 1.
(5)
where \1, A2, As. A are scaling coefficients for rotation, translation, torsion, and lattice, respectively. We present detailed training and inference algorithms in Appendix C.3.
5.2 Model architecture
Our structure prediction model built on a non-quuivariant Transformer backbone. The architecture comprises an initialization module, an interaction module, and an output module (Figure 2)
Initialization module. The initial node embeddings ((!)) E IR for i-th atom is computed as h(") = =[o((a)) f(;)) z4, Lt, :(t)] + <kk), where (a,) is a learnable atom type embedding. fa,) is a binary feature vector encoding atom-level features such as aromaticity [15], and sp(-) is sinusoidal embedding function, is timestep, and & is the index of the building block that contains the i-th atom. The building block index resolves ambiguity between blocks with identical SMILES by lexicographic ordering similar to Section 4.1. Full details are provided in Appendix D.
Interaction module. We adopt the Transformer encoder architecture for the 3D molecular graph == == (V) where V represents atoms and consists of edges between atoms within a specified cutoff threshold. For each edge ji E, we define the edge feature eij [b( RBF([I: 2, ll2)],
5

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Intialization module
Interaction module
Output module
RMSNorm
Rotation M
MurtHHeadAttention
Translation M
Lattice
RMSNorm
Torsion
IntlalizcatommEmeedding
FeedForward
Figure 2: Structure prediction model architecture. The model consists of three main modules: an initialization module that encodes atom features; an interaction module based on a Transformer encoder; and an output module with four prediction heads for each structural component i.e., rotation (dimension M, the number of building blocks), translation (M), lattice parameters (1), and torsion angles (dimension P. the number of rotatable bonds). Notably, torsion angles predicted by first constructing rotatable bond features with four corresponding dihedral atoms and updating the feature with attention to nearby atoms.
where b(i,,) E {0, 1) is a one-hot encoding of the bond type (single, double, triple, aromatic), or all zeros if no bond exists [15], and RBF(-) is a radial basis function for interatomic distance.
To compute self-attention scores, first compute the query, key, and value for each target atom and its neighbor atom j as: q, <- Linearg (h+), ki + Linear x (ih,, exyl). Vij + Lineary ([h,, eG])- Then the attention score from to j is computed as aij = Softmax (q, k,5/VD), where D is the hidden dimension, and (() is the set of neighbors of atom i. Finally, we update the node feature by aggregating the value vectors from its neighbors: ıı; ΣjEN()) Viy. We provide additional architectural details and hyperparameters Appendix D.2.
Output module. The output module consists of four heads that predict rotation A/i-1, translation A, torsion क1, and lattice parameters L.
(I (Lttice parameter) To predict the lattice parameter, we mean-pool the node embeddings (hs i=1 and apply two-layer MLP with a GELU activation [30], i.e., set é MLP Σm h4).
(Translation) For the prediction of translation the nodes in each building block (mm) C V, followed by two-layer MLP, i.e., set fm Alu(klllcAtttentionn
[m }M m=11 apply block-wise attention pooling
Eiey(m> hz)).
(Rotation) Similar to translation, the rotation head applies block-wise attention pooling to node embeddings. followed by two-layer MLP. Importantly, to represent rotations, we use continuous 9D matrix representation based on singular value decomposition (3VD)3 We found this approach be more stable than common parameterizations such as Euler angles, axis-angle, or quaternions which often lead to unstable training due to their discontinuities [31]. Concretely, during training, we supervise the predicted (possibly non-orthogonal) raw matrix output AQ.-1 using a mean squared loss against the target rotation Ag.-1- During inference, our model projects AQ onto the nearest valid rotation matrix by solving the orthogonal Procrustes problem [31]:
4 = Procrustes( AQ = arg min R RESO(3)
= U det(1, 1, deV VT. (6)
where U, V are from SVD outputs - i.c., SVD(AQ, = U.VT.
(Torsion) To predict the torsion angles, first construct invariant features for each rotatable bond and refine them with an attention mechanism. Specifically, for each rotatable bond between atoms (j, k), we first define the dihedral angle using four atoms (i..j, k I). where atoms and / are selected according to a consistent canonicalization scheme described in Section 5.3. The corresponding rotatable bond feature hight is constructed by concatenating the atom features and applying linear transformation in both forward and reverse directions:
hijkI Linear (hv, h2, lu, hul) + Linear, lu, h2 hu])s
where using both orders ensures that the feature invariant to the direction of the dihedral. Each rotatable bond feature then attends to neighboring atoms within a 5A radius using attention, similar to other prediction heads. The result is passed through two-layer MLP to predict the torsion angle.

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5.3 Preprocessing with canonicalization and MOF matching
Canonicalization of rotation. MOF building blocks often have high symmetry point groups, where multiple rotations produce indistinguishable structures, As result, there exist multiple gground-truthi rotations áq--1 that align X, to X1, leading to unstable training (Figure 3a).
To resolve this ambiguity, we introduce a canoni- calization procedure that selects unique rotation target for each symmetric building block. Given a clean conformation Ci = (A, X1) and noised con- formation Co = (A, Xo), we first identify the set of symmetry--reserrvig rotations of Co. defined as R :- {g E 0(3) Co Co} n (O(3). We then apply each g E R to Xo to generate symmetrically equivalent conformations. Among these, we select the one with the lowest RMSD to X1 and recompute the rotation between the two structures with Kabsch alignment [34] (Algorithm 1). This yields the canon- ical rotation -gi- which we interpolate to obtain the final target rotation Δ for any & E [0, 1].
Doj
(a) Rotation
(b) Torsion
Figure 3: Canonicalization. MOF build-
ing blocks often exhibit high symmetry point groups and r-symmetry bonds, resulting in multiple valid rotations and torsion targets. This ambiguity can lead to unstable training. To resolve this, (a) we assign a unique rota- tion target by finding the closest rotation in terms of RMSD. (b) For torsions, we uniquely define the neighboring atoms for rotatable bond with canonical atom rankings of RDKit.
Canonicalization of torsion. MOF building blocks frequently contain π syymmetrry bonds associated with distinct rotations that results in an equivalent confor- mation [17. 35]. Therefore, representing torsion as a relative rotation can lead to multiple valid targets for a single noisy conformation. To address this, we adopt a torsion representation that explicitly specifies the four atoms of the dihedral, with the neighboring atoms selected consistently using RDKit's canonical atom rankings [14]. Concretely, given a rotatable bond between atoms j and k; we select atoms i and I as the lowest-ranked neighbors of and k, respectively (Figure 3b).
MOF matching. We introduce a preprocessing step to reduce the distributional shift between the training and inference structures. Namely, during training, the model receives DFT-relaxed 3D conformations as input, while inference, it receives coordinates initialized from the metal building block library and cheminformatics tools (see Section 4.2). These small discrep- ancies in bond lengths and angles leads to de-
Algorithm 1 Canonicalization of rotation targets
Input: Clean building block C₁ = (A,Xi), noisy building block Co = (A, Xo) where Xo
Agı-s X,
Output: Canonical rotation target Aoi
Identify G(Co) := {g O(3) Co Co}
2: Identify R := G(Co) n 80(3)
3: Solve g" + arg mingeR RMSD Xo, X1)
graded performance [16. 15]. To mitigate this,
4: Compute Avi Kabsh (X,, o* Xo)
we propose MOF matching, a preprocessing pro-
cedure adapted from conformer matching from Jing et al. [16]. Specifically, MOF matching replaces each metal building block with its template structure from the library and each organic linker with an RDKit-generater structure whose torsion angles are optimized to closely match the original conformation. The resulting matched coordinates are used for training to ensure consistency with inference-time input. See Appendix E for details.
6 Experiments
We evaluate MOFFLOw-2 on two key tasks, MOF structure prediction and MOF generation, to demonstrate its ability predict accurate 3D structures and design novel MOFs. We first describe the shared data preprocessing pipeline, then present the structure prediction task in Section 6. 1 and the MOF generation task in Section 6.2. Additional experimental details are provided in Appendix F.
Data preprocessing. We generally follow the preprocessing pipeline from prior work [11, 10]. Starting with the dataset from Boyd et al. [36], we apply metal-oxc decomposition algorithm from MOFid [37] and discard any structures containing more than 20 building blocks [11]. The resulting dataset is split into an 8:1:1 ratio for train/valid/test sets in the structure prediction task, and into 9.5:0.5 train/valid split for MOF generation [11, 10]. Since the dataset [36] consists of hypothetical structures, further filter out invalid MOFs using MOFChecker [38], and apply the MOF matching
7

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Table 2: Structure prediction accuracy. We compare the structure prediction performance of random search (RS), evolutionary algorithm (EA), DiffCSP, MOFFLOW, and MOFFLOw-2. MR the match rate, RMSE the root mean squared error, and indicates no match. stol is the site tolerance for matching criteria.
RS
EA
DifICSP [8]
MOFFLOW [10]
MOFFLOW-2
# of samples
20
20
1
5
1
5
MR (%) f
0.00
0.00
0.01
0.08
5.28
8.68
8.20
15.98
stol 0.3
RMSE (Å) 4
0.1554
0.1299
0.2036
0.2039
0.1894
0.1842
MR (%)
0.00
0.00
0.23
0.87
21.93
32.71
28.71
43.95
stol 0.5
RMSE (A)
0.3896
0.3982
0.3329
0.3290
0.3094
0.2925
procedure to address distributional shift at test time, discussed in Section 5.3. Preprocessing details and dataset statistics are available in Appendix F.
6.1 MOF structure prediction
Baselines. We compare our model against both classic optimization- and learning-based approaches. For the classic baselines, we consider random search (RS) and the evolutionary algorithm (EA) implemented in CrySPY [39]. Among the learning- baased baselines, we include DifiCSP [8], general crystal structure prediction model, and MOFFLOW [10], MOF-specific structure prediction model. To ensure a fair comparison, we assume that no models have access to ground -trth local coordinates. Accordingly, we retrain MOFFLOW with matched coordinates (Section 5.3) and evaluate it using the initialized structures described in Section 4.2.
Metrics. We use match rate (MR) and root mean square error (RMSE) computed with the StructureMatcher class from pymatgen [40]. StructureMatcher determines whether a pre- dicted structure matches the reference based on specified tolerances. The RMSE is calculated only over the matched structures. We report results under tolerance settings, (0.5, 0.3, 10.0) and (0.3, 0.3, 1.0), corresponding to site positions. fractional lengths, and lattice angle tolerances, respec- tively. The former is standard threshold in CSP tasks [8, 9]. while the latter stricter for fine-grained evaluation. Note that while MOFFLOW and MOFFLOW-2 are trained on matched coordinates, we evaluate their predicted structures against ground truth coordinates for fair comparison.
Results. Table 2 shows that MOFFLOW-2 outperforms all baselines across both tolerance settings. Consistent with prior work [10], observe that the optimization-basel methods (RS and EA) and the general CSP baseline (DiffCSP) achieve near-zero match rates, indicating their limited effectiveness on MOF structures. Compared MOFFLOW [10], our method achieves higher match rates and lower RMSE, underscoring the importance of explicitly modeling torsion angles for accurate structure prediction. We also provide visualizations of property distributions in Appendix G.I.
6.2 MOF generation
We evaluate the generative performance of MOFFLOw-2 by (1) measuring validity, novelty, and uniqueness (VNU), and (2) comparing the distribution of MOF properties against that of the training set. We benchmark against MOFDiff [11], a coarse gaaseegrained diffusion model for MOF generation. Both models generate 10,000 samples using their respective pipelines. To isolate the effect of the generative model, no force-field relaxation is applied to the generated structures. Additional results on conditional generation and energy-level evaluation are provided in Appendix G.2 and Appendix G.3.
Metrics. A generated MOF is considered valid it passes the MOFChecker test [38]. It is considered novel if its MDFid [37] does not appear in the training set. Uniqueness is computed as the proportion of distinct structures after removing duplicates from the generated set. We also report the percentage of novel building blocks (NBB), defined as the proportion of samples that are both valid and contain at least one building block not present in the training set. Additionally, we use Zeot+ [41] to evaluate MOF-specifir properties, such as surface area and density.
8

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16-3
LAB miVoo
LEEA
Le-4
LBC
denety
Goandimay
Volumetric Surface 1n3
Gravimetric Surface Arnale)
Accessible Volume
Lod
Unit Cell Volume
1m-1
1--3
pert
Prob. donety
Pore Limiting Diameter ln1
Largest Cavity Diameter inl
Density
Figure 4: Property distributions. We compare MOF property distributions of the ground-truth, MOFDiff, and MOFFI.OW-2 2. The distribution has been smoothed with kernel density estimation. Compared to MOFDiff, MOFF1.w-22 closely aligns with the ground-truth distribution and covers a broader range of values, demonstrating that MOFFLOW-2 can generate MOFs with diverse properties.
Figure 5: Samples from MOFFLOW-2. Visualizations of samples generated by MOFFLOW-2 that are valid, novel, and unique.
Results. Table 3 summarizes the generation performance of MOFFLOW-2 compared to MOFDiff. MOFFLOW- 2 achieves higher scores across all metrics. Notably, MOFFLOW-2 not only generates MOFs with novel combi- nations of known building blocks but also produces entirely new building blocks, demonstrating its potential for discover- ing previously unseen MOFs. In addition, MOFFL.Ww also faster than MOFDiff, which depends on an optimization- based self-assembly procedure for final structure construc- tion. As shown in Figure 4, the property distribution of MOFFLOW-2 aligns more closely with the ground-truth. Importantly, it spans a broader range of values than MOFD- iff, indicating that MOFFLOW-2 generates MOFs with more diverse physical properties. We provide visualizations of representative VNU samples generated by MOFFLOW-2 in Figure 5.
Table MOF generation results. MOFFLOw-2 outperforms MOFDiff in validity, novelty, uniqueness (VNU), and average sampling time. It can also generate MOFs with novel building blocks (NBB).
MOFDiff
MOFFLOW-2
Valid
(%)
10.13
38.84
VNU
(%)
7.95
31.35
NBB
(%) f
0.00
10.10
Time
(5)
3.19
1.82
7 Conclusion
We introduced MOFFLOW-2, a two-stage generative model for MOF design and structure prediction. In the first stage, the building block generator designs MOF sequences in SMILES, which are initialized into 3D structures using a predefined metal library and RDKit [14]. In the second stage, the structure prediction module assembles the complete MOF by jointly predicting rotations, translations, torsions, and lattice parameters. Experimental results show that MOFFLOw-2 outperforms existing models in both generative design and structure prediction.
Despite its strong performance, MOFFLOW-2 still has several limitations, including dependence on RDKit for conformer initialization, partial conditioning in property guiiddd generation (Appendix G.2), and approximate energy evaluation with machine learning interatomic potentials (Appendix G.3), We provide a detailed discussion of these limitations in Appendix H.

---

Acknowledgements
This work was supported by the NVIDIA Academic Grant Program. We thank NVIDIA for providing the GPU resources used in this work. This work was also supported by Basic Science Research Program through the National Research Foundation of (orea(NRF) funded by the Ministry of Edu- cation
147); Institute for Information & communications Technology Planning & (valuation(IITP) grant funded by the Korea govemmeen(MSIT) (RS-2019-1D) 001-22001111900755 Artificial Intel- ligence Graduate School Support Frogram(KAISTT) National Research Foundation of Korea(NRF) grant funded by the Ministry of Science and ICT(MSIT) (No. RS-2022-NR0721844 GRDC(Global
Research Development Center) Cooperative Hub Program through the National Research Foundation of Korea(NRFF) grant funded by the Ministry of Science and ICT(MSIT) (No. RS-2024-00036163 and Institute of Information & Communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (RS- 22S22025-02304967,,, AI Star Fellowship(KAISST).
We thank Vignesh Ram Somnath for his help with model scaling. We also thank Seonghyun Park, Hyomin Kim, and Junwoo Yoon for their help improving the manuscript.
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A Implementation details for building block generator
We provide additional details for implementing the building block generator (Section 4.1), including the tokenization process, batching strategy, and hyperparameter settings.
A.1 Example of MOF sequence tokenization.
We illustrate the tokenization process for canonical MOF sequence defined in Section 4.1, which transforms the string into sequence of discrete symbols (b.-...bs))
Consider the following canonicalized 2D MOF sequence B2D.
<BOS> [Cu+] [Cu+]
(=C(Cc)ccccc)))c))((( [0-])cc1 <EOS> <SEP> cicncen1. -= 2lcnccn1.0=C((----..
We apply the SMILES tokenization regex from Schwaller et al. [28] to split the string into the following tokens.
<BOS>, [Cu+], [Cu+], <SEP>, c, 1, c, n, c, c, n, 1,
0, =, C, C, [0-], ), c, 1, c, c, c, C C, 0, ) [0-], ) c, c, 1, <EOS>
then map each token to unique vocabulary index to get [b [b..... bs].
2, 7, 7, 4, 8, 9, 8, 10, 8, 8, 10, 9, 11, 12. 13, 14. 15, 16,
17, 8, 9, 8, 8, 8, 15, 14, 15, 13, 12, 17, 16, 17, 8, 8, 9,
A.2 Training details.
During training, we use dynamic batching [42] to prevent out-of-memory errors and reduce padding inefficiencies caused by highly variable sequence lengths. The key idea is (1) limit each batch to a fixed maximum number of tokens and (2) group sequences of similar lengths. Specifically, we add the samples heap-based buffer sorted by sequence length. Once the buffer exceeds a predefined capacity, we add the longest sequences to the current batch until reaching the token limit, then start new batch.
Batching,
Hyperparameters. Table 4, we present the hyperparameters for training the building block generator, including the batch configuration, model architecture, and optimizer settings.
Codebase. We implemented our sequence model with x -ttranfformer: [43]. We appreciate the author for the open-source implementation.
Hyperparameter
Value
Max tokens
8000
Number of layers
6
Hidden dimension
1024
Number of heads
8
Rotary positional embedding
True
Flash attention
True
Scale normalization
True
Optimizer
AdamW
Learning rate
3e-4
Betas
(0.9, 0.999)
Weight decay
0.0
Epochs
20
Table 4: Hyperparameters for training the building block generator.
21

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B Analysis and visualization of metal building blocks
Figure 6a and Figure 6b show the RMSD histograms of the extracted metal building blocks for the generation and structure prediction tasks, respectively. There are only 7 and 8 distinct metal building block types, despite the large sizes of the training datasets (>150k). The RMSD histograms further indicate that all metal types exhibit low structural variability. These observations justify our template-based strategy for initializing the metal structures, described in Section 4.2. We also provide visualizations of the metal building blocks in Figure?. 7.
Heaa
Heir CoLlo
Heal E2200
310--
11001
30-0-
3000-
Ster 6.10
2030
1080
17001
01:31
2080
0.6
0.R
9.5
010
ONEEEMEEEE
$1181
0100
(a) Generation
(b) Structure prediction
Figure Metal RMSD histograms. RMSD histograms the metal building blocks for (a)
generation and (b) structure prediction task, respectively. The low diversity and structural variability support our template-bar based approach for metal structure initialization.
[Cu]
[2n][Zn]
[o]][Cr][Cr][o]
[V]
[Ni]
[o][n]]22][Zn][zn]
[Ba]
[N]]
Figure Visualizations of metal building blocks extracted from the training dataset.
22

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C Algorithmic details for structure prediction model
C.1 Conditional flows and conditional vector fields
Here, we derive the conditional flow 2, = expx, (t log,(z))) and conditional vector field My (=(=) = log2, (21)/(1 for each structural component: rotations, translations, torsions, and lattice parame- ters.
Translation and lattice parameters. The translations E R RMxx and lattice parameters (a, b, at, () E IR4 x [0,, 180 lie on the Euclidean space, where the exponential and logarithmic maps are exp, + b and log, b a. Therefore, the conditional flow can be written as:
(1 To ITi, L, = (1 1 t) thl.
(8)
and the conditional vector fields as:
T1 T
(e(eee)) L1 l
ue (e(TilT)) 1
Torsions. Each torsion angle ó E |-n, πT) lies on a torus T, where the exponential and loga- rithmic maps defined as exp, = waap(a + b), log,, wrap(b a) with wrap(z) (x + 7) mod (2T) [44]. The corresponding conditional flow and conditional vector field are:
Φ: = wrap(1 wrap( 6o) + oo); ue(orl)))= = wrap( -(i)
(10)
1-t
Rotations. Each rotation q lie on SO(3), where we define log, b log(ba t) and exp, exp(b)a. Then, the conditional flow is given by
qe e ((1 t) log, (q1) = exp((1 t) log(go9; 1)]1,,
(11)
where exp: s3) - SO(3) and log SO(3) + s3) are the exponential and logarithmic maps for SO(3), respectively [45]. Since we target Aq. 9: (Section 5.1),
Aqr-1 exp((1 t) log (q)
Aqi-00 ((S((3))
(12)
Accordingly, the conditional vector field is given by u. (q))
log qı9.
log Aqu-1
(13)
1-t
where Aqu-1 AqI. = 91
C.2 Applying rotation, translation, and torsion
Rotations and translations. Given a MOF structure S = (A, X, e) with coordinates decomposed as ..... (M) 50(3) and translation
X = - x(11), we apply rotation Åq
= (7 (M)) IR xx3 independently to each building block:
(Q..T)
X (mm) (m)T + 1N... - (m)
(14)
Im=1
where 1... [1, T
Torsions. We adapt the definition from Jing et al. [16] to apply torsion angles to MOF structures. X XMM] and torsion angles Φ (0( E TPM]M_ J=
Given coordinates X = (1)
where P,, is
the number of rotatable bonds in the m-th building block, we update the coordinates around each rotatable bond (j, k) with current angle o and target angle ø' as:
Xb (Xvu) =K)) exp (d0'-o) (b)
②)) I2 23
(15)
[r.
rz
where V(j) denotes atoms on the side of atom and exp so(3) - SO(3). The other side remains unchanged X((Na Xy())-
23

---

C.3 Training and inference algorithu
Algorithm 2 and Algorithm 3 present the training and inference algorithms for the structure pre-
diction model, respectively. During training, the model outputs rotations _Q E RMx333 and torsions & RPx2 the Euclidean space, which are directly supervised with the tar- gets A and Φ1. At inference, these outputs are projected onto their respective manifolds Aq. 4-+11 -1 (rocrustesAAQu-il S0(3) and Φ1 / 12 E TT".
Algorithm 2 Training algorithm
Input: Training dataset {(Bnn, Si)} where Bnn is 2D building block representation and S₁ corresponding 3D MOF structure.
L1) X1,
for (BID, S1) do
2:
Sample time ~ ((0. 1).
3: Sample Aq To, Øo. to from prior distributions defined in Section 5.1.
4: Interpolate Aqu-st; Tr, Φr, l, according to Appendix C.I
5: Apply X, + (Aqi Tr, (.) X1 according to Appendix C.2.
6: Compute outputs (AQ ř1, 11. = FA X L.).
7: Optimize loss (()) Equation (5).
8: end for
Algorithm Inference algorithm
Input: 2D building block representation B2D, number of integration steps T Output: Predicted MOF structure (A, (i)
1: Initialize structure Xo + Initialize(Ban)) according to Section 4.2.
2: Sample Aqo. To, Φo. to from prior distributions defined in Section 5.1.
3: Apply transformation Xo- + (Aqo, To. Q0) Xo.
4: Set + 1/T.
5: for imri=0 =0......
do
6: Sett+i//T, Predict (AQ:-11 TI, 1 2n) = L.).
7:
8:
Project to manifold Ωq:-1
(rrocrustesAQQ--) Φ1 + 01/1
9: Take step Aq + (arrsteep(qq-1ittt At).
Take step dr EulerStep(yy. V1, Al) for y E (T, $.
11: Apply transformation X,+ +- (Aqi-st+ Ar, Tet 11 At. X,,
12: end for
24

---

D Model architecture for structure prediction
D.1
Initialization module
Building block index. We define the building block index & introduced in Section 5.2 to resolve ambiguity between building blocks of the same type-i.e.. those sharing the same SMILES representation. To assign consistent indices, we apply the following lexicographic ordering rules:
Metal building blocks are always indexed before organic building blocks.
2. Within each group (metal organic), blocks ordered by molecular weight.
3. Ties between building blocks of the same type resolved by sorting their centroid coordi- nates (z, y 2) in ascending order of æ, then y, then 2 [46].
D.2 Interaction module
The interaction module follows Transformer encoder architecture [47] with root mean square layer normalization [RMSNorm; 48] (Figure 2). Here, we provide further implementation details, including the model architecture (Algorithm 4) and associated hyperparameters (Table 5).
Algorithm 4 Interaction module (Transformer encoder)
Input: Initialized atom embeddings H [|4 E RD bond edges Evond, cutoff radius c E I.+, maximum number of neighbors Nmax-
Output: Updated atom embeddings H' = 11/ E R
1: # Construct edges and edge features [15]
2: Construct radius edges Eradius f- {(i.j) || - x2ll2 < cf with [W S Nmax Vi.
3: Combine edges E + Erniius U Ehond.
4: Construct edge features = (eig) with eij = [b RBF(( - x,yl=)]+-
5: # Update node features
6: + H + (HMSNooo H,)
Refer to Section 5.2 for details on MHA.
7: H' < H FFN(RMSNomm| ())
Hyperparameter
Value
Number of layers
10
Maximum radius c
50
Maximum neighbors Nmax
130
Node embedding dimension
1024
RBF embedding dimension
128
Number of heads
16
FFN embedding dimension
4096
Table 5: Hyperparameters for the interaction module.
25

---

D.3
Output module
We describe the architectures of the four prediction heads in the output module (Section 5.2). We begin with the block attention pooling module, which aggregates atom-level node embeddings from the interaction module into block-level embeddings (Algorithm 5). We then detail the architectures of the rotation (Algorithm 6), translation (Algorithm 7), lattice (Algorithm 8), and torsion heads (Algorithm 9). The corresponding hyperparameters are summarized in Table 6.
Algorithm 5 Block attention pooling module (BlockAtttentionPPoo))
Input: Atom embeddings from interaction module H = [h, E IR RPI
Output: Block-wise embeddings Hbb ob (m)
E R P mmm1 for each building block.
1: # Construct building block coordinates
pa Im=1 RM3 and features (m))
(m)) 1
[h( Im=1
R MMD
2: for vm = -,..., M do
(m) f Nm Eievim) Zi.
3: Compute building block centroid Libb
4: Compute averaged feature (p N. Σsev(m) h4-
5: end for
6: # Construct edges and edge features
7: Construct edges within the building blocks Enb {(m,j) [M] x V(m)
8: Construct edge distance features [emjl(m.j)Eu. where emy = RBF a (m)
- x2)12))
9: # Attention
10: for vm = -,..., M do
Compute query Qm f- Linearq (lip (m=)
11:
Compute key kmj + Linearx (m)) emil). Vj E V(m)
12:
13:
Compute value Umj- Lineary ((1 (()mm)) emsl)) Vj E V(m),
14: Compute attention score amz f- Softmax, (A kmjs /D).
Aggregate and update (m)
Lincar Eyevom) amg Umj))
end for
Algorithm 6 Rotation head
Input: Atom embeddings from interaction module H = [h; E IR DIN
Output: Rotation predictions for each block åg:-, E S0(3)
1: Block-wise attention pooling Hab + BlackAtteentionPooll/ff
2: Raw rotation output åQ Lincar Linear(GELU(Lnnenn
3: if is inference then
4: Project Aq. Procrustes(AQ) (AQ..1)).
5: end if
Algorithm 7 Translation head
Input: Atom embeddings from interaction module H = [h; IR PIN
Output: Translation predictions for each block E RRM33 ==1
1: Block-wise attention pooling Hab BlockkAttentionPPool(/)).
2: Update with MLP न. + Linear Linear(GELU (H)))((Lneer((H))))..
ΣM- ((m)
3: Remove mean
Algorithm 8 Lattice head
Input: Atom embeddings from interaction module H [h, E IR IRIN
Output: Lattice prediction & E IR3 x [0 180%..
1: Block-wise attention pooling Hab +- Block MlockAttentionPPool[1]1
2: Average and MLP & + MLP M
mm= A (mm)).
Two--ayer MLP with GELU activation.
3: Apply l < Sof)
26

---

Algorithm 9 Torsion head
Input: Atom embeddings from interaction module [
R rotatable bond index (pp: (o. p-= cutoff radius +. maximum number of neighbors Nmax-
Output: Torsion predictions Φ1 E TTP.
( 1: # Construct rotatable bond coordinates [o 1-=1 E Px3 and features [h I ERDx
P
2: for = l,.... P (i.e., each rotatable bond) do
3: For clarity let (i, k,1) + (pp: kp. 1p). (p)
4: Compute rotatable bond centroid Tyot + (x, + xx)/2.
5: Compute rotatable bond feature huu
(p) <- Linear (h+, hj, has hul) + Linear((Ps, hE, h2, hz)).
6: end for
7: # Construct edges and edge features
8: Construct edges E {(p.j) [P] [M] 2,112 c} with [W(p)] S Nmax= Vp.
9: Construct edge distance features (P.))EE where epj = RBF (1= xx 112)).
(p)
Atlention
11: for Vp 1, P do
Compute query 4p + Linearq(hm)
13:
Compute key kpj f- Linear x (Iime Epil), v(p.j) E Enn
14:
Compute value Upj f- Linearv ([h epi)), V(p.j) E Ejo
15:
Compute attention score apj + SeNtmax)eN(1) (1p kpps bni/VD)
16:
Aggregate and update h +- Linear Lincar(E (EjEN()) Ups
17: end for
Apply 21 + (HP(( where Hi [ 1==
Two-layer MLP with GELU activation.
19: if is inference then
Project Φi t 1 21l2.
21: end if'
Hyperparameter
Value
Node embedding dimension
1024
RBF embedding dimension
128
Number of heads for BlockAAteentionPool
16
Maximum radius (c) for TorsionHead
5
Maximum neighbors (Nmax) for TersionHead
24
Table 6: Hyperparameters for the output module.
27

---

E MOF matching
We describe the MOF matching procedure, a preprocessing step used to reduce distributional shift between training and inference, as outlined in Section 5.3. Given an MOF structure and its building blocks, we first determine whether each block contains metal elements classify as metal or organic. For metal blocks, we retrieve the corresponding template from the metal library and align it to the ground-truth structure. For organic blocks, we initialize the structure using RDKit, optimize torsion angles via differential evolution to minimize RMSD with the ground-truth [16], and then align the result. After processing blocks, we compute RMSD between the reconstructed and ground-truth truth structures. This procedure is repeated three times, and structures with final RMSD below 0.5À are retained. The full algorithm for a single matching iteration is shown in Algorithm 10.
Algorithm 10 An iteration of MOF matching
Input: DFT-relaxed MOF structure S (A, ,,), trial number n, base population size po, base maximum iteration lg-
Output: Matched MOF structure $ (A, x,e), L). RMSD(S, R4.
1: for vm = -,...,,M MM (i.e., each building block) do
2: if A (m)) contains metal element then
3:
# Metal building block
4:
Retrieve the corresponding template structure X (m) from metal library (Section 4.2).
5:
Align to original coordinates X(m) +- Align(X (m)), X(m)).
6:
else
7:
# Organic building block [16]
8:
Initialize structure X (m) with RDKit (Section 4.2).
9:
Set parameters p f- po + 10n, +- to + 10n.
Optimize torsion angles (m) <- DifferentialEvolution/ (8(m), ((m)) ,p, 1).
11:
Align to original coordinates (m)) + Align(X (mm), ((m)). end if
12:
13:
end for
14: Compute RMSD(S,
28

---

F Experimental details
Data preprocessing. We detail the data preprocessing steps described Section 6. First, we discard MOF structures with more than 20 building blocks since large structures that are too large may be difficult synthesize [11]. Next, we extract key features from each structure, including Cartesian co- ordinates, RDKit-derived atomic features, Niggli-reduced cells, symmetrically equivalent coordinates for each building block (Section 5.3), and canonical atoms defining the torsion angles (Section 5.3). We then construct the metal library (Section 4.2) and perform MOF matching (Appendix E) on the training dataset. Finally, since the dataset is synthetic [36], we filter out invalid MOFs using MOFChecker [38], which ensures the presence of key elements (e.g.. C. H. and a metal), no atomic overlaps or coordination issues, sufficient porosity, and the absence of highly charged fragments or isolated molecules.
Data statistics. We present the data statistics for structure prediction (Section 6.1) 1) in Tables and those for generation (Section 6.2) in Tables 10 and 11.
Property (number of samples 137,474)
Min
Mean
Max
number of species atoms
4 4/22 534.5
5.3 / 125.5 4415.5
8 8/1124
volume [
104490.5
density [atoms/A²]
0.1
0.7444
3.1651
lattice a, [Å]
6.86 8.27 8.56 59.98 / 59.98 / 59.99
12,94 15.61 19.34
47.00 /4.00/47.15 60.81
lattice 0, [']
91.31 / 91.35 90.90
120.01 120.01 / 120.02
Table 7: Statistics of the train split for structure prediction.
Property (number of samples 19, 603)
Min
Mean
Max
number of species atoms
22
5.3 125.9 4409 .4
1,404 105417.3
volume [
534.6
density [atoms/A*]
0.1074
0.7454
2.8989
lattice a, b, [Å]
6.86 / 8.43 / 8.57
12.94 15.64 19.30 /1.32/91.33 / 90.91
47.24 47 60.62
lattice 7.7
60.00 60.00 59.99
120.00 120.01 / 120.01
Table 8: Statistics of the validation split for structure prediction.
Property (number of samples 19, 792)
Min
Mean
Max
number of species atoms
4 24
5.3/ 124.2 4384.2
8/ 1012
volume [
536.4
123788.7
density [atoms/A
0. 1080
0.7444
3.0982
lattice a, 6, [A]
6.86 / 8.34 8.57
12.90 / 15.56 19.28
47.14 47, /4/ 60.95
lattice 5, [P]
60.00 / 60.00 60.00
91.20 / 91.20 / 90.86
120.00 / 120.00 / 120.02
Table 9: Statistics of the test split for structure prediction.
Property (number samples 187,047)
Min
Mean
Max
number of species/ atoms
4/ 22
5.3 125.4
8/ 1404
volume [
534.5
4410.2
123788.7
density [atoms/A*]
0.74
0.7445
3.0982
lattice a. b, [Å]
6,86 8.27 8856
12.94 / 15.61 19.33
47.24 / 60.95
lattice a, (
59.99 / 59.98 / 59.99
91.29/ / 91.33 / 90.88
120.01 120.01 120.02
Table 10: Statistics of the train split for generation.
Property (number of samples 103)43)
Min
Mean
Max
number of species atoms
4 / 26
5.3/ 127.6 4481.2
8/9
volume [Ao]
582.2
71167.7
density [atoms/A
0.1074
0.7454
2.8989
lattice a, b, [Á]
6.87 / 8.47 / 8.58
12.91 / 15.69 / 19.42
38.87
39.36 60.21
lattice a, 3,7 [*]
60.00 / 60.00/ 60.00
91.26 91.20 90.80
120.00/
119.99 120.00
Table 11: Statistics of the validation split for generation.
29

---

F.1
Training details
Baselines. We follow Kim et al. [10] RS and EA, using CrySPY[49] with CHGNet[50] for energy-based optimization. RS employs symmetry-based structure generation. For EA, we start with random structures, select 4 parents via tournament selection, and generate oftspring using 10 crossovers, 4 permutations, 2 strains, and 2 elites, iterating up to 20 generations.
For DifICSP, we use a radius cutoff of 5A and a batch size of 64. The model was trained for 500 epochs 80GB NVIDIA A100 GPU, taking approximately days. Inference is performed with 1000 steps. All other settings follow the defaults in Jiao et al. [8]. FlowMM [9] is excluded from the baselines due to its high memory demands; based our estimates, training it for 500 epochs would take over 30 days on an 80GB A A100 GPU, which exceeds our resource constraints.
MOFFL.OW-2. We train our model on 8 80GB A100 GPUs for 200 epochs (about 4 days). To avoid out-of-memory issues caused by the large variation in MOF sizes, we use a dynamic batching strategy that limits each batch to maximum of 1500 atoms. Specifically, the samples are first added to a buffer and, once the buffer exceeds a predefined capacity, they are added to the batch in first-in-first-on manner until the maximum limit is reached. We use A optimizer [51] with learning rate of le-5, betas (0.9, 0.98), and no weight decay. Inference is performed in 50 steps.
G Additional experiments
G.1 Property evaluation for structure prediction
Figure 8 presents the property distributions for the structure prediction task (Section 6.1). All baselines closely align with the ground-truth, indicating that key MOF properties are well preserved.
ertuet
mig
prou dersly
2ITCS
0.5
0.9
0.2 Accessible Volume
0.4
0.8
6.8
0.0
5.0
0.2
0.4
100
Volumetric Surface Area
Gravimetric Surface Area*
Volume
Prob. aenatty
Vold Fraction
03
Pore Limiting Diameter
Largest Cavity Diametertri
Density
Figure 8: Key property distributions of MOFs generated by DiffCSP, MOFFLOW, and MOFFLOW-2 Distributions are smoothed using kernel density estimation. All baseline methods closely match the reference data distribution.
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(b) Energy per atom of generated MOFs
Figure 9: (a) CO2 working capacity distribution of MOFs generated by MOFFLOW-2 and MOFDiff, evaluated with GCMC simulations. (b) Energy per atom distribution (eV/atom) of generated MOFs, evaluated with UMA.
30

---

G.2 Conditional generation for high CO2 working capacity
We extend MOFFLOW-2 conditional generation, targeting MOFs with high CO2 working capacity. The building block generator is trained to rotss- to property embedding, = Linear(y) yV) Fourier(y), where y denotes the working capacity. inference, we sample 200 building blocks conditioned on large y values and evaluate them using grand canonical Monte Carlo (GCMC) simulations [11]. As shown in Figure 9a, MOFF1.OW-2 produces MOFs with higher working capacity than MOFDiff [11]. Note that MOFDiff relies on latent optimization with fixed target of y 15, while our approach conditions directly on the property.
G.3 Energy-level evaluation using MLIP
We further evaluate the energy levels of generated MOFs using UMA [52], state-of-the-arr machine- learned interatomic potential. Specifically, we generate 10,000 structures with both MOFFLOW-2 and MOFDiff and compute the energy per atom (eV using the uma-m-1. 1 model. As shown Figure 9b, the energy distribution of structures generated by MOFFLOW-22 aligns more closely with the training dataset than that of MOFDiff.
H
Limitations
Although MOFFLOW-2 demonstrates strong potential for MOF structure prediction and generation, several limitations remain. Firstly, because the pipeline relies on RDKit for initial conformer generation, it is challenging to predict structures whose organic building blocks are chemically invalid incomplete; for example, those lacking carboxylate groups, which are common in MOF decomposition schemes [37]. Second, for conditional generation in Appendix G.2, only the building block generator is conditioned the property, whereas the structure prediction module may also depend on it. Finally, our evaluation of energy levels with machine learning interatomic potentials is less accurate than that with density functional theory (DFT).
31


Reviews:
[{"review_id": "2nWrrJD4H3", "reviewer": "Reviewer_qUfU", "summary": "This paper introduces MOFFLOW-2, a two-stage generative framework designed for the prediction of metal-organic framework (MOF) structures and the creation of novel MOFs. The framework comprises two main components: (1) an autoregressive SMILES-based model for generating metal and organic building blocks, and (2) a flow-matching model for assembling these blocks into valid 3D structures by predicting translations, rotations, torsional angles, and lattice parameters. The approach addresses limitations of previous models by generating novel building blocks, predicting flexible 3D conformations, and not relying on a fixed set of building blocks or rigid conformations. Experimental results show that MOFFLOW-2 outperforms previous methods like MOFDiff and MOFFLOW in both structure prediction accuracy and the generation of novel, valid, and unique MOFs.", "questions": "See the weakeness.", "limitations": "Yes, the authors have included a limitation section in their paper.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n+ **Novel Approach**: The introduction of a two-stage framework that combines SMILES-based building block generation with flexible 3D structure prediction is innovative and allows for greater chemical diversity and structural flexibility.\n\n+ **Strong Experimental Results**: The model outperforms baseline methods in both structure prediction and MOF generation tasks, showing improved validity, novelty, and uniqueness in generated structures. This is particularly important for discovering new MOFs and expanding chemical space.\n\n+ **Clear Methodology and Comprehensive Architecture**: The authors present a clear and detailed explanation of their methodology, including model architecture, preprocessing steps, and the flow-matching process for predicting rotational, translational, and torsional transformations.\n\n\nWeaknesses:\n\n\n+ **Insufficient Ablation Studies**: A key concern is the absence of a clear breakdown of the contributions from individual model components (e.g., the SMILES generator and flow-matching) to overall performance. Since the generation process is factorized into two stages, it is crucial to independently analyze the contribution of each stage and identify potential bottlenecks. Is it possible to provide experiments that offer insight into this aspect?", "was_revised": true, "final_justification": "I thank the authors for the additional experiments. I will maintain my score, which inclines towards acceptance.", "rebuttal": "Dear reviewer qUfU,\n\nWe deeply appreciate your thorough review and constructive comments on our manuscript! We also appreciate your recognition of our work, including **the novelty of our approach**, **strong experimental results**, and **clear methodology/architecture.** Below, we address the reviewer's concern in detail.\n\n ---\n\n### W1. Insufficient ablation studies. Analyze the contribution of each component in the two-stage pipeline (e.g., the SMILES generator and flow matching) to the overall performance. \n\nThank you for your insightful comment. **While our comparison of MOFFlow-2 and MOFFlow-1 in Table 2 ablates the effect of torsion flexibility for structure prediction, we provide more studies to resolve your concern.** Our new results confirm that both (1) SMILES generation quality and (2) torsional flexibility modeling are important for MOF generation.\n\n**1. Varying SMILES generator quality:** We first ablate the effect of the SMILES generator on the MOF generation quality. To this end, we train the SMILES generator with a varying number of steps and evaluate its quality with SMILES-VNU* (left column). Each is then paired with the same flow matching module, and the full pipeline is evaluated using VNU (right column). Results show that improvements in the SMILES generator directly lead to higher overall MOF quality.\n\n**SMILES-VNU is the percentage of generated MOF sequences that are valid (RDKit sanity check), novel (not in training set), and unique (non-duplicated).*\n\n| SMILES quality (SMILES-VNU%) | Overall MOF quality (VNU %)|\n|--------------------------|---------------|\n| 45.9                     | 17.90         |\n| 58.6                     | 19.11         |\n| 69.0                     | 22.32         |\n| 84.6                     | 31.26         |\n| **85.3**                | **31.35**     |\n\n**2. Excluding torsional flexibility:** Next, we ablate the effect of torsional flexibility on the MOF generation quality. Replacing MOFFlow-2 with MOFFlow-1 (no torsion modeling) in the assembly stage clearly degrades the performance in MOF generation.\n\n|           | Valid (%) | VNU (%)   |\n|-----------|-----------|-----------|\n| SMILES gen + FM (w/o torsion) | 29.76     | 23.81     |\n| SMILES gen + FM (w/ torsion, ours) | **38.84** | **31.35** |\n\n---\n\nWe hope our experiments address your concerns! Please let us know if there are any further opportunities for clarification and improving the score."}, {"review_id": "OWDkKsJGoV", "reviewer": "Reviewer_EcT6", "summary": "This paper introduces MOFFLOW-2, a two-stage generative framework for metal–organic framework (MOF) design. The first stage leverages an autoregressive SMILES-based language model to generate novel metal and organic building blocks. The second stage employs a flow matching model to predict the 3D MOF structure by modeling translations, rotations, torsions, and lattice parameters. This approach addresses key limitations of prior work that relied on fixed, rigid building blocks.", "questions": "- How sensitive is the performance of MOFFLOW-2 to the initial RDKit geometries used for organic linkers?\n- How well does the model generalize to MOF chemistries that are entirely out-of-distribution from training?", "limitations": "- The physical plausibility of the generated MOFs is not validated using quantum or force-field level simulations.\n- The framework does not show results in property-conditioned or goal-directed generation, which may limit its applicability in practical design tasks.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n- Clear motivation and reasonable contribution: The paper addresses critical limitations of prior MOF generative models by enabling the generation of novel building blocks and incorporating torsional flexibility.\n- Substantial empirical improvements: On structure prediction, MOFFLOW-2 outperforms all tested baselines under strict matching criteria. On generative design, it shows higher validity, novelty, uniqueness, and property diversity.\n- Good writing and clarity: The paper is well-structured, with clear diagrams, descriptions, and motivation throughout.\n\n**Weaknesses:**\n- Lack of physical validation: The generated structures are not assessed using quantum mechanical (e.g., DFT) or molecular dynamics simulations, which limits confidence in their physical stability.\n- Limited task scope: The framework is not demonstrated in property-driven generation scenarios, which would be critical for targeted MOF design.", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer EcT6,\n\nWe deeply appreciate your thorough review and constructive comments on our manuscript! We also appreciate your recognition of our work, including **clear motivation and reasonable contribution**, **substantial empirical improvements**, and **good writing clarity.** Below, we address the reviewer's concerns and questions in detail.\n\n---\n### W1. Lack of physical validation.\n\n**To address your concern, we validate the energy of generated MOF structures using UMA [1], the state-of-the-art machine learning force field, which was verified to work on MOFs.** While DFT would be ideal, it is computationally infeasible for large MOF structures within the rebuttal period and our computational budget.\n\nSpecifically, we generate 10,000 structures with MOFFlow-2 and MOFDiff and compute energy per atom ($\\mathrm{eV/atom}$) with the `uma-m-1.1` model. As shown in the percentile table below, MOFFlow-2 consistently produces structures with lower energy levels compared to MOFDiff. We also plan to add histograms of energy levels in the future manuscript, which could not be included in the rebuttal due to NeurIPS rules. \n\n| Model/percentile  | 20%       | 40%       | 50%       | 60%       | 80%       |\n|--------------|-----------|-----------|-----------|-----------|-----------|\n| $\\textcolor{#bbbbbb}{\\text{Reference (data)}}$ | $\\textcolor{#bbbbbb}{\\text{-7.06}}$     | $\\textcolor{#bbbbbb}{\\text{-6.86}}$     | $\\textcolor{#bbbbbb}{\\text{-6.78}}$    | $\\textcolor{#bbbbbb}{\\text{-6.68}}$    | $\\textcolor{#bbbbbb}{\\text{-6.48}}$   |\n| MOFDiff      | -6.60     | -6.23     | -6.00     | -5.66     | -4.54     |\n| MOFFlow-2    | **-6.75** | **-6.45** | **-6.27** | **-5.97** | **-4.56** |\n\n&nbsp;\n\n### W2. The framework does not demonstrate property-driven generation scenarios.\n\nWe appreciate the reviewer’s insightful comment. **Our method can be easily extended to property-driven generation, and here we demonstrate this by generating MOFs to maximize CO2 working capacity.** To this end, we (1) train MOFFlow-2 conditioned on CO2 working capacity and (2) generate MOFs conditioned on sufficiently high values of this property.\n\nTo be specific, we re-train the building block generator to cross-attend to working capacity $y$ featurized as $c=\\text{Linear}(y) + \\text{Fourier}(y)$. During inference, we sample 200 building blocks conditioned on $y=4$ and compute the CO2 working capacity by running a grand canonical Monte Carlo (GCMC) simulation [2]. As shown in the table below, MOFFlow-2 successfully generates MOFs with high working capacity and outperforms MOFDiff. Note that MOFDiff applies latent optimization with a target of $y=15$. \n\n|                              | Working capacity vacuum swing (mol/kg) |\n|------------------------------|----------------------------------------|\n| Training data                | 0.4242                                 |\n| MOFDiff (w/ optimization)    | 1.1798                                 |\n| MOFFlow-2 (w/ conditioning)  | **2.1694**                             |\n\nWe also plan to update our code and add working capacity histograms in the future manuscript, which could not be included in the rebuttal due to NeurIPS rules.\n\n&nbsp;\n\n### Q1. How sensitive is the performance of MOFFlow-2 to the initial RDKit geometries used for organic linkers?\n\n**MOFFlow-2 performs robustly regardless of initial RDKit geometries, since it relies only on the bond lengths and angles, which RDKit reliably estimates.** While RDKit may be inaccurate for torsion angles, our model is insensitive to this since it randomly reinitializes them at $t=0$.\n\nTo empirically confirm this, we evaluate our model on four sets of initial conformers randomly generated by RDKit, given the same building blocks. As shown below, the performance remains consistent, with negligible variation in both validity and VNU metrics:\n\n|            | Valid (%)        | VNU (%)          |\n|------------|------------------|------------------|\n| Trial 1    | 37.6             | 31.0             |\n| Trial 2    | 37.4             | 30.6             |\n| Trial 3    | 37.3             | 30.5             |\n| Trial 4    | 37.5             | 30.8             |\n| **Mean ± Std** | **37.45 ± 0.11** | **30.72 ± 0.19** |\n\n&nbsp;\n\n### Q2. How well does the model generalize to MOF chemistries that are entirely out-of-distribution (OOD) from training?\n\n**Similar to any other machine learning algorithm, our method will struggle to generalize to data that is entirely out-of-distribution (OOD).** For example, MOFFlow-2 is not capable of generalizing to unseen element types. Also, its performance for MOFs with complex topologies and many building blocks may be lower than those common in the dataset. However, this can be alleviated by training our model on a more diverse set of MOF/crystal structures (e.g., datasets from [3] or the Materials Project [4]). \n\n---\n\nWe hope our experiments address your questions and concerns! Please let us know if there are any further opportunities for clarification and improving the score.\n\n### Reference\n[1] Wood, Brandon M., et al. \"UMA: A Family of Universal Models for Atoms.\" arXiv preprint arXiv:2506.23971 (2025).  \n[2] Fu, Xiang, et al. \"Mofdiff: Coarse-grained diffusion for metal-organic framework design.\" arXiv preprint arXiv:2310.10732 (2023).  \n[3] Lee, Sangwon, et al. \"Computational screening of trillions of metal–organic frameworks for high-performance methane storage.\" ACS Applied Materials & Interfaces 13.20 (2021): 23647-23654.  \n[4] Jain, Anubhav, et al. \"Commentary: The Materials Project: A materials genome approach to accelerating materials innovation.\" APL materials 1.1 (2013)."}, {"review_id": "UzVhZyW6RH", "reviewer": "Reviewer_Figq", "summary": "The paper introduced MOFFLOW-2, a two-stage generative framework for metal–organic frameworks that (1) generates metal clusters and organic linkers via an autoregressive SMILES model, and (2) assembles them into full 3D crystal structures by explicitly modelling translations, rotations, lattice, and torsional degrees of freedom using Riemannian flow matching.  The method has been evaluated on two tasks: structure prediction and generation.", "questions": "Regarding the initial 3D structure of linkers, did you investigate the model's sensitivity to the quality of the RDKit-generated conformer? Can you comment more on that?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- Two-stage framework: I think the idea of two-stage decomposition, first smiles generation, and then capturing 3D structure via flow matching is an effective solution for this task.\n- Empirical Evaluation: The authors evaluated their method on two tasks and the method seems to has better RMSE in structure prediction compared to previous baselines. \n\n**Weaknesses**\n\n- I think the evaluation is still limited, and the authors compared to classical methods and a few baselines like MOFDiff.\n- The method still consider the use of bioinformatics tools like RDKit, as the second stage relies on an initial 3D structure for the organic linkers generated by RDKit. Also, while the model learns to correct torsion angles, it still assumes the initial bond lengths and angles are accurate.", "was_revised": true, "final_justification": "I'm supportive in general but I still see some limitations such as the baselines used for comparison.", "rebuttal": "Dear reviewer Figq,\n\nWe deeply appreciate your thorough review and constructive comments on our manuscript! We also appreciate your recognition of our work, including **the effectiveness of our two-stage approach** and  **improved empirical results**. Below, we address the reviewer's concerns and questions in detail.\n\n---\n### W1. I think the evaluation is still limited, and the authors compared to classical methods and a few baselines like MOFDiff.\n\n**We consider our set of baselines to be comprehensive, given how the MOF generative model is an under-explored research area.** We note that MOFFlow-1 and MOFDiff are among the first deep learning models that can generate MOFs from scratch. MOFDiff itself had no learning-based baselines for similar reasons. \n\nWe also note that classical methods like random search (RS) and evolutionary algorithms (EA) are strong and widely used baselines in the crystal structure prediction (CSP) field, making them meaningful baselines for comparison.\n\nIf you have any suggestions for the baselines, we would be happy to incorporate them. \n\n&nbsp;\n\n### W2. Method relies on a bioinformatics tool like RDKit for initial structure generation, and assumes that the initial bond lengths and angles are accurate.\n\n**We do not think this is a weakness of our method, as the bond lengths and angles generated by cheminformatics toolkits are widely accepted to be accurate and accessible [1, 2].** Our algorithm re-initializes the remaining structure (torsion) and does not depend on highly variable information. Also, the reliance on RDKit for estimating the bond lengths and angles underpins many celebrated works on molecular generative models (e.g., Torsional diffusion [2], DiffDock [3], FlexDock [4]). \n\n&nbsp;\n\n### Q1. Did you investigate the model's sensitivity to the quality of RDKit-generated conformers?\n\n**Yes; MOFFlow-2 is robust to the quality of RDKit-generated conformers, since it relies only on the bond lengths and angles, which RDKit reliably estimates.** While RDKit may be inaccurate for torsion angles, our model is insensitive to this since it randomly reinitializes them at $t=0$.\n\nTo empirically confirm this, we evaluate our model on four sets of initial conformers randomly generated by RDKit, given the same building blocks. As shown below, the performance remains consistent, with negligible variation in both validity and VNU metrics:\n\n|            | Valid (%)        | VNU (%)          |\n|------------|------------|------------------|\n| Trial 1    | 37.6             | 31.0             |\n| Trial 2    | 37.4             | 30.6             |\n| Trial 3    | 37.3             | 30.5             |\n| Trial 4    | 37.5             | 30.8             |\n| **Mean ± Std** | **37.45 ± 0.11** | **30.72 ± 0.19** |\n\n---\n\nWe hope our experiments address your questions and concerns! Please let us know if there are any further opportunities for clarification and improving the score.\n\n### Reference\n[1] Axelrod, Simon, and Rafael Gomez-Bombarelli. \"GEOM, energy-annotated molecular conformations for property prediction and molecular generation.\" Scientific Data 9.1 (2022): 185.  \n[2] Jing, Bowen, et al. \"Torsional diffusion for molecular conformer generation.\" Advances in neural information processing systems 35 (2022): 24240-24253.  \n[3] Corso, Gabriele, et al. \"DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking.\" International Conference on Learning Representations (ICLR 2023). 2023.  \n[4] Corso, Gabriele, et al. \"Composing unbalanced flows for flexible docking and relaxation.\" The Thirteenth International Conference on Learning Representations. 2025."}, {"review_id": "kSju0tKV5A", "reviewer": "Reviewer_eFsz", "summary": "This paper introduces MOFFLOW-2, a flow-based generative model specifically designed for flexible Metal-Organic Framework (MOF, a family of materials)  structure prediction and design problem. The first stage leverages a SMILES-based autoregressive Transformer to generate novel chemical building blocks, while the second employs a torsion-aware flow matching model to assemble these blocks into 3D structures. MOFFLOW-2 incorporates rotations, translations, torsion angles, and lattice parameters into structure modeling. Experiments demonstrate that MOFFLOW-2 surpasses previous state-of-the-art methods in structure prediction accuracy and successfully generates novel, valid, and unique MOFs.", "questions": "1. The training of MOFFLOW-2 (as well as its baselines, MOFFLOW, MOFDIFF) seems to rely on a previous MOF dataset (Boyd et al.). Does the dataset itself enough for learning the distribution of MOF structure? In other words, do you think incorporating other data sources of large scale of molecule or material datasets for pretraining will help? If yes or no, why?\n2. In Table 2 of structure prediction task, under smaller tolerance threshold, why the MR and RMSE seems inconsistent by contrasting DiffCSP and MOFLOW-2? Can you explain this phenomenon?\n3. What is the major difference between MOFFlow-1 and -2? Which factor(s) cause the performance gain on the structure prediction task in Table 2?\n4. Regarding the design problem, how was the “valid, novel and unique” defined in Figure 5, table 3 and section 6.2? I could not find the definition of these metrics.\n5. By explicitly modeling the torsion angles, the modeled degree of freedom is augmented based on the model without it. Beyond that, what could be the potential “rigid” part in the MOF? In other words, which degrees of freedom are still not modeled under the setting of MOFFlow-2?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: \n- The introduction of torsion-aware flow matching significantly advances generative modeling capabilities for MOFs by addressing flexibility explicitly. \n- The paper provides extensive evaluations and comparisons with recent models, showcasing superior performance across critical metrics. \n- Demonstrates the ability to generate MOFs from novel building blocks, significantly extending the design space beyond existing methods (eg. MOFFlow-1)\n\nWeaknesses: \n- Although the method is effective, the computational cost and resource requirements for larger-scale and more complex MOF generation are insufficiently discussed (for example, the training cost, inference time, etc.), which could be relevant in real-world using scenario. It could be better to include these, more or less, in the main text.\n- The stability and real-world chemical feasibility of the generated (designed) MOFs, while briefly mentioned, lack comprehensive validation through computational chemistry methods (eg., DFT or ab initio calculations). That being said, it is okay to leave this for future work.\n- Given MOF is a specific family of molecules, there is limited exploration of how the method scales and performs with significantly different MOF types or more chemically diverse frameworks (more general family of molecules) as an ML for science work.", "was_revised": true, "final_justification": "I have read the rebuttal/response and discussed with the authors. I think this paper is in general in good quality, albeit building upon well-established works from the domain of molecular generation and conformer prediction. The authors have responded to my question in a good manner and they basically resolved my questions, in specific, the clearness, additional experiments and some discussion regarding a broader field application. In conclusion, I lean towards borderline acceptance, where the limitation lays in its significance for much broader fields in ML.", "rebuttal": "Dear reviewer eFsz,\n\nWe deeply appreciate your thorough review and constructive comments on our manuscript! We also appreciate your recognition of our work, including **advancing generative modeling capabilities with torsional flexibility**, **extensive evaluations**, and **expansion of the design space with novel building blocks.** Below, we address the reviewer's concerns and questions in detail.\n\n---\n### W1. Insufficient discussion of computational cost and resource requirements (e.g., training cost, inference time). Include in the main text.\n\n**The training cost is detailed in Appendix A.2 and F.1, and the inference time is reported in Table 3 of the main text.** For clarity, we summarize the key information below:\n\n|   | Batch size         | GPU type    | GPU memory usage (training) | Training epochs | Total training time | Inference time (s/sample) |\n|--------|----------|------|----------|-----|------|----|\n| Building block generator (Transformer Decoder) | ~45 (<8000 tokens) | 8xA100 80GB | 50.71GB - 78.29GB           | 20              | 56m 12s             | 0.59                      |\n| Structure prediction module (Flow matching)    | ~11 (<1500 atoms)  | 8xA100 80GB | 56.67GB - 78.42GB           | 200             | 4d 11h 33m          | 1.23                      |\n\nDue to space limitations, we follow common practice and have placed details on computational resources in the Appendix. However, to address your suggestion, we will refer to this information on line 294 (Section 6) of the main text.\n\n&nbsp;\n\n### W2. The stability of generated structures hasn't been evaluated with DFT or ab initio methods. Okay to leave for future work.\n\nWe appreciate your understanding! **Nevertheless, to address your concern, we validate the energy of generated MOF structures using UMA [2], the state-of-the-art machine learning force field, which was verified to work on MOFs.** While DFT would be ideal, it is computationally infeasible for large MOF structures within the rebuttal period and our computational budget.\n\nSpecifically, we generate 10,000 structures with MOFFlow-2 and MOFDiff and compute the energy per atom ($\\mathrm{eV/atom}$) with the `uma-m-1.1` model. As shown in the percentile table below, MOFFlow-2 consistently produces structures with lower energy levels compared to MOFDiff. We also plan to add histograms of energy levels in the future manuscript, which could not be included in the rebuttal due to NeurIPS rules. \n\n| Model/percentile  | 20%       | 40%       | 50%       | 60%       | 80%       |\n|------|-----|----|----|----|-----|\n| $\\textcolor{#bbbbbb}{\\text{Reference (data)}}$ | $\\textcolor{#bbbbbb}{\\text{-7.06}}$     | $\\textcolor{#bbbbbb}{\\text{-6.86}}$     | $\\textcolor{#bbbbbb}{\\text{-6.78}}$    | $\\textcolor{#bbbbbb}{\\text{-6.68}}$    | $\\textcolor{#bbbbbb}{\\text{-6.48}}$   |\n| MOFDiff      | -6.60     | -6.23     | -6.00     | -5.66     | -4.54     |\n| MOFFlow-2    | **-6.75** | **-6.45** | **-6.27** | **-5.97** | **-4.56** |\n\n&nbsp;\n\n### W3. MOF is a specific family of molecules. Limited exploration of how the method scales/performs for significantly different MOF types or more diverse frameworks as an ML for science work.\n\n**While MOF is indeed a specific class of materials, we still believe our work has huge implications.** MOFs are being investigated for many real-world problems, e.g., carbon capture, catalysis, or even drug delivery, and exhibit unique characteristics, e.g., modular nature, high-symmetry point groups, and the combination of organic and inorganic components, which require specialized algorithms. \n\nWe resonate with your point and consider generalization of our work to a broader domain, e.g., including new element types or applying our model to covalent organic frameworks, as a significant research direction. We note that, in principle, our work is generalizable to such new dataset without much modification.\n\n&nbsp;\n\n### Q1. Is the Boyd et al. dataset enough for learning the distribution of MOF structure? Will incorporating other data sources of large scale molecule/material datasets for pretraining help? Why or why not?\n\n**Since the Boyd et al. dataset [2] is a widely recognized MOF benchmark with diverse topologies and building blocks [3-5], we believe that it is a strong dataset for learning the distribution of MOFs.** However, it may not fully capture the edge cases, such as rare element types in the broader MOF space.\n\nTherefore, we believe that pretraining on large-scale molecular/material datasets can be beneficial, since the atoms in MOFs share the same fundamental interactions as in other molecules/materials. This strategy has proven effective in recent work (UMA [1]), and we expect similar benefits for our model.\n\n&nbsp;\n\n### Q2. In Table 2, under the smaller tolerance threshold (stol=0.3), why are the MR and RMSE between DiffCSP and MOFFlow-2 inconsistent?\n\n**This inconsistency arises because RMSE is calculated only on matched structures, meaning that DiffCSP performs worse than MOFFlow-2 when considering all the generated samples.** DiffCSP has a very low match rate (\\~0.08%), so its RMSE is based on a small number of trivial matches, resulting in low RMSE. In contrast, MOFFlow-2 has a much higher match rate (\\~15.98%), and its RMSE is computed over a broader and more complex set of structures, resulting in a higher value.\n\n&nbsp;\n\n### Q3. What are the major differences between MOFFlow-1 and MOFFlow-2? What causes the performance gain in the structure prediction task in Table 2?\n\n**MOFFlow-2 mainly improves over MOFFlow-1 through (1) removing the rigid body assumption and (2) adding the ability to generate MOFs from scratch.**\n\nWe make a one-by-one comparison as follows:\n1. **Rigid body assumption:** MOFFlow-1 assembles MOF structures by treating building blocks as rigid bodies and thus requires their ground-truth conformations as input. In contrast, MOFFlow-2 relaxes this assumption by explicitly modeling torsional flexibility, allowing the building blocks to adjust their conformations during assembly.\n2. **Generative capability:** MOFFlow-1 is designed solely for structure prediction and cannot generate new MOF structures. MOFFlow-2, on the other hand, supports both generation and prediction.\n\nThe performance improvement in the structure prediction task is mainly due to MOFFlow-2’s ability to model torsional flexibility, which accounts for conformational changes during the assembly. \n\n&nbsp;\n\n### Q4. How are validity, novelty, and uniqueness (in Figure 3, Table 3, Section 6.2) defined? I could not find the definition of these metrics.**\n\n**The definitions of validity, novelty, and uniqueness are defined in lines 331-333 of the main text.** For your convenience, we restate the definition here. \n- **Validity (%):** A generated MOF is valid if it passes the MOFChecker [6] test. This includes a basic sanity check for MOFs, such as the presence of metal and organic elements, porosity, no overlapping atoms, etc. \n- **Novelty (%):** A structure is considered novel if its MOFid [7] does not appear in the training set. This essentially means that a MOF is novel if either a novel building block or a novel combination of building blocks has been used. \n- **Uniqueness (%):** Uniqueness is the proportion of distinct structures after removing duplicates from the generated set. \n- **VNU (%)** is defined as the proportion of generated structures that are simultaneously valid, unique, and novel. \n\n&nbsp;\n\n### Q5. Which degrees of freedom have not been modeled under the setting of MOFFlow-2?\n\n**MOFFlow-2 does not model bond angles and lengths, since RDKit accurately generates them.** This is a widely accepted assumption [8, 9], which underpins many celebrated works on molecular generative models (e.g., Torsional diffusion [9], DiffDock [10], FlexDock [11]). While one could additionally model such degrees of freedom, we do not expect a meaningful improvement in accuracy despite the additional computational cost.\n\n---\n\nWe hope our experiments address your questions and concerns! Please let us know if there are any further opportunities for clarification and improving the score.\n\n### Reference\n[1] Wood, Brandon M., et al. \"UMA: A Family of Universal Models for Atoms.\" arXiv preprint arXiv:2506.23971 (2025).  \n[2] Boyd, Peter G., et al. \"Data-driven design of metal–organic frameworks for wet flue gas CO2 capture.\" Nature 576.7786 (2019): 253-256.  \n[3] Fu, Xiang, et al. \"Mofdiff: Coarse-grained diffusion for metal-organic framework design.\" arXiv preprint arXiv:2310.10732 (2023).  \n[4] Wu, Xiaoyu, and Jianwen Jiang. \"Precision-engineered metal–organic frameworks: fine-tuning reverse topological structure prediction and design.\" Chemical Science 15.40 (2024): 16467-16479.  \n[5] Liu, Yutao, et al. \"Porous framework materials for energy & environment relevant applications: A systematic review.\" Green Energy & Environment 9.2 (2024): 217-310.  \n[6] Jablonka, K. M. (2023). mofchecker (Version 1.0.0) [Computer software]. https://doi.org/10.5281/zenodo.1234  \n[7] Bucior, Benjamin J., et al. \"Identification schemes for metal–organic frameworks to enable rapid search and cheminformatics analysis.\" Crystal Growth & Design 19.11 (2019): 6682-6697.  \n[8] Axelrod, Simon, and Rafael Gomez-Bombarelli. \"GEOM, energy-annotated molecular conformations for property prediction and molecular generation.\" Scientific Data 9.1 (2022): 185.  \n[9] Jing, Bowen, et al. \"Torsional diffusion for molecular conformer generation.\" Advances in neural information processing systems 35 (2022): 24240-24253.  \n[10] Corso, Gabriele, et al. \"DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking.\" International Conference on Learning Representations (ICLR 2023). 2023.  \n[11] Corso, Gabriele, et al. \"Composing unbalanced flows for flexible docking and relaxation.\" The Thirteenth International Conference on Learning Representations. 2025."}]

Author rebuttal:

Meta-review: The paper presents a new 2-stage method for generating MOFs. All reviewers rated it as borderline accept, recognizing the technical novelty and strong empirical results while raising concerns about physical validation and property conditioning that were effectively addressed in a comprehensive rebuttal. The work makes a solid technical contribution to an important materials science problem, and I recommend acceptance.
Decision: Accept (poster)