MOF-BFN: Metal-Organic Frameworks Structure Prediction via Bayesian Flow Networks
Authors: Rui Jiao
Hanlin Wu
Wenbing Huang
Yuxuan Song
Yawen Ouyang
Yu Rong
Tingyang Xu
Pengju Wang
Hao Zhou
Wei-Ying Ma
Jingjing Liu
Yang Liu
Venue: NeurIPS 2025 poster ()
Keywords: Metal-Organic Frameworks
Bayesian Flow Networks
Material Design
Primary area: machine_learning_for_sciences
TL;DR: We propose MOF-BFN, a structure prediction model for Metal-Organic Frameworks (MOFs) via Bayesian Flow Networks (BFNs).

Abstract: Metal-Organic Frameworks (MOFs) have attracted considerable attention due to their unique properties including high surface area and tunable porosity, and promising applications in catalysis, gas storage, and drug delivery. Structure prediction for MOFs is a challenging task, as these frameworks are intrinsically periodic and hierarchically organized, where the entire structure is assembled from building blocks like metal nodes and organic linkers. To address this, we introduce MOF-BFN, a novel generative model for MOF structure prediction based on Bayesian Flow Networks (BFNs). Given the local geometry of building blocks, MOF-BFN jointly predicts the lattice parameters, as well as the positions and orientations of all building blocks within the unit cell. In particular, the positions are modelled in the fractional coordinate system to naturally incorporate the periodicity. Meanwhile, the orientations are modeled as unit quaternions sampled from learned Bingham distributions via the proposed Bingham BFN, enabling effective orientation generation on the 4D unit hypersphere. Experimental results demonstrate that MOF-BFN achieves state-of-the-art performance across multiple tasks, including structure prediction, geometric property evaluation, and de novo generation, offering a promising tool for designing complex MOF materials.

MOF-BFN: Metal-Organic Frameworks Structure Prediction via Bayesian Flow Networks
Rui Jiao 12, /2.t
Hanlin Wu
Wenhing Huang Yuxuan Song²
Yawen Ouyang²
Yu Rong Tingyang Xu
Pengju Wang
Hao Zhou
Wei- Wei-Ying Ma²
Jingjing Liu? Yang Liu
Dept. Comp. Sci. Tech., Institute for AI, Tsinghua University IInstitute of AI Industry Research (AIR), Tsinghua University Gaoling School of Artificial Intelligence, Renmin University of China Beijing Key Laboratory of Big Data Management and Analysis Methods, Beijing, China Alibaba DAMO Lab
Abstract
Metal-Organic Frameworks (MOFs) have attracted considerable attention due to their unique properties including high surface area and tunable porosity, and promising applications in catalysis, gas storage, and drug delivery. Structure prediction for MOFs is a challenging task, as these frameworks are intrinsically periodic and hierarchically organized, where the entire structure is assembled from building blocks like metal nodes and organic linkers. To address this, we introduce MOF-BEN, novel generative model for MOF structure prediction based on Bayesian Flow Networks (BFNs). Given the local geometry of building blocks. MOF-BFN jointly predicts the lattice parameters, as well as the positions and orientations of all building blocks within the unit cell. In particular, the positions are modelled in the fractional coordinate system to naturally incorporate the periodicity. Meanwhile, the orientations are modeled unit quaternions sampled from learned Bingham distributions via the proposed Bingham BFN, enabling effective orientation generation on the 4D unit hypersphere. Experimental results demonstrate that MOF-BFN achieves state-of-the-arr performance across multiple tasks, including structure prediction, geometric property evaluation, and de novo generation, offering a promising tool for designing complex MOF materials.
Introduction
Metal-Organic Frameworks (MOFs) have attracted significant interest in recent years due to their
unique structural properties and wide range of potential applications [1 12]. These materials are characterized by high surface areas, tunable porosities, and exceptional versatility, which make them ideal candidates for use in catalysis [9]. gas storage [16), drug delivery [14, and other fields [88 3) Structurally, MOFs composed of metal ions clusters coordinated with organic linkers, forming periodic structures that can be tailored for specific functions (21). This tunability has led to their
Wenning Huang, Hao Zhou and Yang Liu are corresponding authors. f indicates equal contribution.
This work is done when Rui Jiao works as intern in Alibaba Group.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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exploration in numerous scientific and industrial domains, sparking a need for more efficient and accurate methods for predicting MOF structures.
Predicting the structures of MOFs is a critical prob- lem. Traditional erystal structure prediction methods typically rely on ab initio calculations, using opti- mization algorithms to find local minima on energy landscapes defined by Density Functional Theory (DFT) or machine learning (ML) force fields [30]. However, such approaches are computationally inten- sive. To overeome this limitation, recent work has explored deep generative models that directly learn the data distribution, thereby bypassing explicit en- ergy optimization. While these models have shown promising results on small-scale inorganic crystals, they encounter challenges in addressing MOFs: the unit cells of MOFs often contain hundreds of atoms, making direct atomic-level generation highly complex and computationally expensive. To mitigate this, prior work has adopted a hierarchical modeling paradigm, where MOFs are decomposed into smaller, rigid building blocks 5 13, as shown in Figure [] These methods aim to predict the positions and orientations of building blocks to reconstruct the full structure, reducing the complexity of the problem. Particularly, MOFFlow [13] is representative approach that employs flow-mate fow-matching techniques to jointly generate both the unit cell parameters and the positions and orientations of cach block, offering a promising solution to the structural prediction problem.
Repesting Unit
Decompsetion
Unit Cell
Totiste MOP Struchurs
Building Blockd
Figure I: Decomposition of MOF,
Despite its success, MOFFlow represents the unit cell as finite 3D graph with global features such as lattice parameters, without explicitly modeling the periodicity of MOF structures. Periodicity is a key characteristic of crystalline materials, and its proper modeling is essential capturing multiple properties. To this end, recent works have adopted fractional coordinate systems, which models atomic positions on a 3D torus normalized by the unit cell, naturally incorporating periodic boundary conditions. Several generative methods based on diffusion [10] 17] or flow matching (20) have been developed to model these fractional coordinates, showcasing promising results in various crystal generation tasks.
However, diffusion- and flow-based models typically rely on continuous-timg stochastic differential equations (SDEs) or ordinary differential equations (ODEs), which would inevitably be discretized during sampling. This discretization introduces truncation errors, which can accumulate and degrade model performance [19 31 In contrast, Bayesian Flow Networks (BFNs) [6]. recent advancement in generative modeling, reformulate the generation process via iterative Bayesian updates. This paradigm bypasses the need for solving SDEs or ODEs, thus eliminating the discretization errors. Notably, CrysBEN (28), a recent work based on BFNs, has demonstrated strong performance in modeling fractional coordinates for inorganic crystal generation, highlighting the potential of BFN- based frameworks.
Nevertheless, extending BFNs to MOF structure prediction faces additional challenges. In hierarchical MOF representations, building blocks are not simply treated as point particles but as rigid bodies with local geometries [13.. Therefore, in addition to predicting the fractional coordinates of each block, it necessary to model its orientation in diree-iimessioal space. Specifically, this requires learning distributions over rotation matrices, or equivalently, over the special orthogonal group SO(3), which remains unsolved in prior BFN- methods.
To address this gap, we propose MOF-BFN, a novel generative framework for hierarchical MOF structure prediction based on Bayesian Flow Networks. MOF-BEN operates in the fractional coordi- nate system to preserve periodicity and models block orientations using unit quaternions sampled from Bingham BFNs-a a new generative module that extends BFNs to the hypersphere S for rota- tion modeling. Given the geometry of each building block, MOF-BFN jointly predicts the lattice parameters, fractional coordinates, and orientations of all building blocks within a unit cell.
In summary, our contributions are as follows:
We introduce MOF-BFN, the first hierarchical structure prediction framework that jointly models periodicity, position, and orientation using Bayesian Flow Networks.
We incorporate fractional coordinates to capture periodicity and employ Bingham distribu- tions to generate orientations in the unit quaternion space.

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We demonstrate the superior performance of MOF-BEN over existing methods in multiple tasks including structure prediction, geometric property evaluation and de novo generation.
2
Related Works
Generative Models for Crystalline Materials. Generative models for erystalline materials, includ- ing both inorganic crystals and ortal-orgamic frameworks (MOFs), have made notable advancements. CDVAE (29) integrates a diffusion decoder into a VAE to generate structures from predicted lat-
tices. DifCCSP [0] improves this by jointly diffusing lattice matrices and fractional coordinates.
FlowMM 20 and CrysBEN [2 further improve the generation performance via more advanced
generative models for the torus space. FlowLLM 226 initializes flow matching process with
LLM-based prior 77] TGDMat 4 innovatively introduces text conditions to the generative model.
DiffCSP++ [1] considers space group-based generation given specific Wyckoff position (WP) as- signments, and SymmCD [5] extends this by further enabling the generation of WPs. Specific for MOFs, MOFDiff [5] extends CDVAE into a hierarchical diffusion model using coarse-grained blocks. While MOFDiff requires post-processing to fully optimize block orientations, MOFFlow [I3 directly models the orientation of each building block in the flow matching framework, leading an end-to-end paradigm. However, MOFFlow does not explicitly model the periodicity of MOFs, which address by incorporating the fractional coordinate system in this work.
Bayesian Flow Networks. Bayesian Flow Networks (BFNs, Graves et al. (6) define a generative process that transitions from prior to posterior through Bayesian updates. Their smooth generation has been effective in 3D tasks like molecule generation [25] and structure-based drug design [23]. CrysBFN (28) extends BFN to torus space via the von Mises distribution. In this work, we further enhance BFN with a hierarchical MOF-specific representation, enabling accurate modeling both block positions and orientations.
3 Preliminaries
3.1 Representation of MOF'
General Representations of MOFs. An MOF structure can be described as an infinite periodic arrangement of atoms in 3D space, where the smallest repeating unit is called the unit cell. A unit cell containing N atoms can be formed by a triplet M = (L. X, A),, where L = 11,12, La] E 33x3 the lattice matrix determining the periodicity of the unit cell, X
[< E R is the Cartesian coordinate matrix specifying the positions of the atoms, and A = [a.] h-dimentional one-hot encoding matrix representing the atom types. To describe the shape of the parallelpiped L, a commonly-used alternate the invariant lattice parameter & = (a, (,,bb, c, a 3 3))) R x (0, T)3 that characterizes the lengths and pairwise angles of the three basis vectors. The entire periodic structure can be formulated by {(a,, lay = a, x = x; + Lt, Vt E 2 }, where the integer vector t indicates the arbitrary translation along the basis vectors according periodicity.
RANN the
Block-level Representations of MOFs. MOFs typically consist of hundreds to thousands of atoms per unit cell, making it challenging to directly design full-atom generative models. To overcome this problem, a hierarchical modeling approach is commonly employed, where atoms are grouped into building blocks such as metal clusters and organic linkers [5] [3.. Formally, a unit cell with K building blocks (K N) is then described as M (L, X R ,B), where X E R [c,]K contains the local 2 contains N, atoms, described by atom types [a, and local coordinates [x>],> The canonical local coordinates are determined by PCA [13). which detailed in Appendix C. To intrinsically model the periodicity, we further apply the fractional coordinate system, which takes the lattice matrix L as the coordinate basis and describes the relative position of each atom within the unit cell. Specifically, the coordinates are transformed
and R E SO(3)K< specify the center positions and orientations,
structures of blocks. Each block C, <<),,,
IIn practice, following Lin el. al. 17 to project the parameters in continuous space R to simplify modelling. Hereinafter, the notation & denotes this projected representation.
we use (log(a), log(b), log(e), tan(a- 3), () tan(t-nn ))
3

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Data Space
Parameter Space
Sender Dist. Ps
Training Inference
DR
Receiver Dist. Ph
Bayesian Update
Function h
Lattice e
Goussian Parameters of
Po
Periodic Backbone
Receiver Dist.
0:
Boyesian Flow Pr
Periodic Backbone A
Fractional Coordinates
von Mises Parameters
Orientations Q (sty*
Bingham Parameters 0
Unit
Prior Os
Figure 2: Overview of MOF-BEN. To model the joint distribution over different manifolds in the data space, MOF-BEN defines a generative process in the corresponding parameter space. During training, the model takes 0,-1 from Bayesian flow distribution on the unit cell M the input, and outputs an approximation of M. The training objective is to minimize the discrepancy between the estimation and the ground truth. During inference, the process begins from uninformative prior Pg. and progressively updates the parameters via the trained model.
into F = [f
J=1 = LL -L-1= 1,, 1= E the block-level fractional coordinates.
[0, 1)3xK, Hereinafter, we simplify FB as F to represent
Represent Rotations as Unit Quaternions. Apart from the rotation matrices, unit quaternions
offer compact and efficient way represent 3D rotations. For q [w, y, 2] and || 1, the corresponding rotation matrix is given by
1 - 22y + 22)
2(zy - wz)
2(xz + wy)
R(g)=
2(xy + wz)
1 2(x² 22)
2 wx)
(I)
2(xz wyy
2(yz + wx)
- 2(x² + y²)
Task Definition. The task of MOF structure prediction is to predict the lattice matrix and detailed block arrangement given the block types. By representing lattices the invariant parameters & and encoding orientations as quaternions Q [g]]K< the structure prediction task is formulated as capturing the joint distribution plE, F. Q ()). Furthermore, enable de novo generation, extend the formulation to model p(g. F, B), thereby incorporating the block type generation into the structure prediction framework.
3.2 Bayesian Flow Networks
Bayesian Flow Networks (BFNs) 6 are a class of generative models that progressively refine the parameters of distribution set via Bayesian updates. Specifically, a series of sender distributions is constructed by perturbing data with noise levels corresponding to a predefined accuracy schedule [a.][_ resulting in ps xz; a.). In parallel, the receiver begins with uninformative prior distribution parameterized by 0g. and progressively updates its belief based on the observed noisy samples [y.]k- -. At each step i, the receiver treats its distribution from the previous step as the input distribution pi (= and updates it via Bayes' rule as
pr(x (.) x pr (0|0 1)s0000--1PsP (3 ax), which corresponds to the following update rule in the parameter space:
0, h h(0 a;),
where h is called the Bayesian update function. The corresponding Bayesian update distribution yielded by marginalizing yi
Po (0 0:00011.0 1. σ; os;) = E Spp (zn (x) 5(0; -h(0;-1. az)))

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At step i, the marginalized Bayesian flow distribution is accumulated from the prior dg as
pr (0,12, 1 = po (0: (0:100.2:00)) ... Epp (0:-1 - (0, 2;; o;;).
(5)
However, during generation, the sender distributions are actually unavailable they requires ground- truth data. To address this, a neural network & is introduced to approximate the sender distributions by producing output distributions that aim to recover the clean data. At each step i, the model takes the current parameter 0;-1 as the input and outputs an estimated distribution po (<t (,, usually selected as the Dirac distribution on the model prediction that
po (0)-,, 8( c(0,-1,44)).
(6)
The approximated sender distribution, so-called receiver distribution, is further given by
PR PR(yi a.) po (é o(e._...)))PS à oz).
The training objective is to minimize the transmission error, quantified the KL-diivegence between the sender and receiver distribution as
(
kL (ps( a;;
(8)
which is generally measured by the distance or similarity between the predicted and ground truth data. Overall, the construction of BFN requires the following key components: (1) a prior parameter Og for initialization; (2) Bayesian update function h for progressively refining the parameter; (3) Bayesian flow distribution pr accumulated via for training; and (4) a closed-form KL divergence as the training objective, which will be detailed in the following section.
4 MOF-BFN
As illustrated in Fig. 2 to model the joint distribution P(c, F, Q)),, we design MOF-BFN over the parameter space of g.M = {BE, fF, 0²). During training, the model samples discrete timestep i (1, T) and takes 0,4, as the input parameter, and minimize the training objective abstracted as Eq. 88 on each manifold. Initialized from the prior O,M, each inference step first yields the output distribution from the trained model to approximate the receiver distribution, and then refines the input parameters via Bayesian update. The final prediction drawn from the output distribution the last step T. In this section, we first briefly introduce the design of BFNs on continuous and torus space in 4.1 and detail the proposed Bingham BFN in $ 4.2 Finally, we provide the training schemes of MFF-BFN on the structure prediction task and the extended de novo generation task in $ 4.3
4.1 Bayesian Flow Networks on Common Manifolds
Lattice Parameters on Continuous Space g The input distribution of & follows the Gaussian form N(:: u (pG)=1 parameterized by of {uf, ). Following the standard BFN on contin- uous space (6), the prior distribution is selected as the standard Gaussian with of = {0, 1}. After observing sample from the Gaussian form sender distribution yf NN(E, (a I) at step the Bayesian update function is given by
( = p
+ a
pf_ + of
Leveraging the additive property of Gaussian distributions, given af o 22/7 (1 2 2/T) and or a predefined small variance, the Bayesian flow distribution can be analytically obtained as
(ufle, = -N
2:/ (1- ",
2:/T
(1-025/T)
(10)
with of oT -2/TT Since both the sender and receiver distributions are Gaussian with identical variance, their KL divergence simplifies to mean-squared error form as
ET
c E ( Ed--1) 2
|E de (e(
(11)
5

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Fractional Coordinates on Torus T x As established by Wu et al. [28, circular distributions are appropriate for modeling the periodicity of fractional coordinates, and one suitable choice is the von Mises distribution defined as Po(a;; m, k) x exp(x c0s(2 - m))) with m E [0, 1) denote the random variable and mean direction, and x: 2 0 controls the concentration. Parameterized as of {m, K. J. the input distribution is initialized by m (O, 1), N.j O. We can further describe the parameter in the complex form with of = =. e2xmo's where is the imaginary unit, implying that K. m depict the modulus and argument of c." Acquiring the sample y ~ (o(Po o with pre-defined of the Bayesian update function at step defined as
cf = - F az 2
(12)
resulting in the accumulated Bayesian flow distribution as
cc F, [aj Pté 12=1)
= PE (f Pa -5 Eofe
(13)
j=1
The training objective is the KL-divergence between von Mises distributions as
F T Iolaf) LF - E B Plaff122) a Ii(af)
cos (2T (2 (0F(0=1 ()
(14)
where lo(-), are the modified Bessel functions.
4.2 Bingham BEN for Quaternion Generation
In this section, detail the design of key components for implementing BFN in the quaternion parameter space 0 For simplicity, we omit the superscript Q and focus on a single quaternion S; the full design for Q = [q.) follows naturally by extension.
Bingham Distribution. To model variation 3D orientations, one often requires probability distribution defined over the space of rotations. Since each 3D rotation can be represented by a unit quaternion on 5d, it natural to consider distributions supported on this manifold. However, one key point according to Eq. (I) is that the mapping from unit quaternions rotation matrices is two-to-one, i.e. R(q)) R( q). Hence, any valid distribution must respect this antipodal symmetry. Fortunately, directional distribution on the hypersphere called Bingham distribution serves precisely this role. Mathematically, the Probability Density Function (PDF) of the Bingham distribution takes the form
1 pB(q: M. A) = Z(A) exp (q M AMg).
(15)
where Z(-) is the normalization term, and M AM is the eigendecoopposition of a symmetric matrix with orthogonal M E R2x4 as the cigenvectors, and diagonal A as the eigenvalues. Based on Eq. (15). one can directly derive the following properties of the Bingham distribution:
Proposition 1. The PDF of the Bingham distribution maintains the antipodal symmetry, i.c. (q:q: M, A) PB( (o((-q:)
q: A).
Proposition 2. When A 0, the PDF is reduced to a uniform distribution, i.c., pa M, 0) = 2
Proposition 3. Due to the normalization constraint on the hypersphere, any bias applied on the eigenvalues would not affect the distribution, i.c., PB (q; M, A + kf) = PD (n M, A), Vk E R.
According to proposition 3 A is typically reduced to A = diag A) = diag (0, R A2, A3) with
As S A proposition
X S Hence, Bingham distribution can also be parameterized by 0 = (M. }}. By 2 we initialize the parameters using the uniform prior 6g (Mo, 0), where My is an
arbitrary orthogonal matrix²)
Bayesian Update of Bingham Distribution. In Bayesian statistics, a conjugate prior refers to prior distribution that, when combined with a specific form of likelihood function, yields posterior
Theoretically, when the Bingham concentration matrix initialized as A = 0, the matrix A. M, AM becomes a zero matrix regardless of the choice of orthogonal matrix M, resulting in uniform distribution over the unit sphere. In practice we initialize M by performing torch. linalg. eigh on the zero matrix, which yields the identity matrix Mo I.

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distribution of the same functional form as the prior. This property is greatly essential for the Bayesian update process defined by Eq. 3). where the input distribution and sender distribution act as the conjugate prior and likelihood, respectively. Notably, the likelihood need not share the same form as the prior. For the Bingham distribution, admits particularly convenient conjugate relationship when the likelihood is modeled using the Watson distribution, which yields the PDF as
(w(q: :: μa K) Ca(K)) exp (K( q)²).
(16)
Here Ca(-) is the normalization term, μ S is the mean direction and K 0 determines the concentration. The squared inner product ensures antipodal symmetry, consistent with the Bingham distribution. Given - [M,-,, A } step and observation y, from the sender distribution following Pw (Wis q, at), the posterior is updated as
p(g/0,- Yi, on) pn(g; M, (); pw q,
(17)
() exp (q (M, ( + a,y.y, )g)
xx exp (q M, diag ( 1 a; q)²)
(18)
(19)
As Eq.
19 retains the form of Bingham distribution, the Bayesian update function 0; (M, X
h(0,-1. 1, Di, 0;j) is thus defined
M,, diag( (
+...y.))
(20)
A X max (X.).
(21)
The distribution of the updated parameters is finally acquired as
Pu (0: 0:-1. a.) ( ((0 h(0,-1, y/. a:)).
(22)
Efficient Sampling for Bayesian Flow Distribution. For BFNs applied to continuous spaces such as Gaussian distributions over IK", key additive property holds as
Pu 00" 0, q an + as)) Epu prr qu. ( Ors);
(23)
which allows the cumulative accuracy step to be expressed P: = Po Σj= 0g.. However, the property does not extend to the Bayesian updates in Eq. (22). the eigenvalues involved in the
Bingham updates are not additive. To address this, we follow the strategy similar to Eq. (13) on the
von Mises distribution, which also suffers from non-additive property. We simulate the posterior O, by accumulating the outer products in the exponential term. Given A, = M, diag (M) we have
pr (Aslq: [a;];=1) = Epw (wn
8 A,
Σ a,,,,,)
(24)
The samples [yr]] can be drawn independently and in parallel via rejection sampling from the angular central Gaussian (ACG) distribution, described in Appendix C.2. Finally, the schedule of accuracy [ax]]"> is computed numerically to approximate a linearly decreasing entropy, with further implementation details provided in Appendix C.3.
Training Objective. The training objective computed between the predicted and target Watson distributions, which is defined
4 E B.0T) { (q (o (MMM)))))))
(25)
where A-(-) is the second moment of the Watson distribution. Note that Eq. 25) measures the square of the cosine similarity between the prediction Φy (0,M, and the ground truth q, maintaining the antipodal symmetry.
4.3 Training Scheme
The entire training objective aggregates the losses defined on each manifold, resulting in the following composite loss function:
Lsp Vefe + TELF + tiafaa
(26)
7

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Table I: Structure prediction accuracy. Results for MOF structure prediction. indicates no match stol represents the site tolerance for matching criteria. Baseline results are from Kim et al. [3
stol 0.5
stol 1.0
# of samples
MR (%) t
RMSE
MR (%) f RMSE 1
RS
30
20
0.00
0.00
EA
30
20
0.00
0.00
0.09
0.3961
23.12
0.8294
DifCCSP
10
0.34
0.3848
38.94
0.7937
31.69
0.2820
87,46
0.5183
MOFFLOW 13
5
44.75
0.2694
100.0
0.4645
35.27
0.2735
92.99
0,5000
MOF-BFN
53.51
0.2498
98.37
0.4117
where each À balances the contribution of the corresponding component.
Backbone Model for MOF-BEN. The structure prediction backbone contains two main components. The first is the building block encoder ε, which maps the local structures of building blocks into invariant latent vectors. In our implementation. we adopt EGNN 24 as the encoder. The second is the grained structure predictor adapted from the periodic GNN designed in [TO), which takes the timestep i, the joint parameter O.M, and the encoded condition (B)) as the input. Importantly, it also incorporates the accuracy terms Kf and A, they indicates the current entropies of the non-additive distributions [28). The predictor outputs the estimated lattice parameters, quaternions, and fractional translations, which together define the output distributions at each step.
Novo Generation. We further extend our method to the novo generation task, where both the types and arrangements of building blocks are required for generation. To this end, we employ a Gaussian BFN trained on the continuous embedding space learned by the contrastive building block encoder from MOFDiff [ objective is extended as
Extension
to determine the identity of each block. The training
DNG = - + 7FLF + TaLq + TBLB,
(27)
where L8 optimizes the generation process on block embedding space and shares similar form as Eq. 34 More details are referred to Appendix D.
5
Experiments
In this section, evaluate MOF-BFN across variety of tasks. In 5.1 we show that MOF- BFN significantly outperforms existing full-atom and hierarchical approaches structure prediction accuracy. In 5.2 we demonstrate that the predicted structures exhibit strong agreement with ground-truth structural properties. $5.33] we further extend our method to the de novo generation task, where the identity of each block is also required to be determined. Finally, provide analyses on the efficiency of the fractional modelling and the BFN-based framework $5.4
5.1 Structure Prediction
Setup We use the BW-DB dataset of 324,426 MOFs from Boyd et al. 1. and decompose each structure into building blocks using the metal-oxo algorithm in MOFid (2) following Fu et al. (S). As suggested by Kim et al. [13], we remove structures with over 200 blocks and split the remaining data into training, validation, and test sets in an 8:1:1 ratio.
Baselines We compare against three types of baselines: (1) optimization- methods including
Random Search (RS) and Evolutionary Algorithm (EA) from CrySPY [0;; (2) DiffCSP [00,
full-atom generative model without block-level modeling; and (3) MOFFlow [3] hierarchical
generative model using flow matching on block representations.
Evaluation Metrics For each test structure, we generate & candidates and report match rate (MR) and root mean square error (RMSE). MR measures the proportion that least one of the generated structures matches the ground truth, while RMSE is calculated between the ground truth and the best

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matching, Matching is performed using StructureMatchee from pymatgen 122]. We follow Kim al. [13] and evaluate under two tolerance settings for site, length and angle tolerance criteria: (0.5, 0.3, 10.0) (strict) and (1.0, 0.3, 10.0) (relaxed).
Results Table reports the structure predic- tion results. As expected, optimization-bant methods (RS and EA) completely fail re- cover correct structures, achieving 0% match rates, which reflects the inefficiency of atom- level search for complex MOFs. DifICSP shows slightly better performance but still struggles due to its lack of block-level awareness, confirm- ing the limitations of full-atom generative mod- els for large systems. Compared to these base- lines, both MOFFlow and our proposed method, MOF-BFN, achieve significantly higher accu- racy. Notably, MOF-BFN consistently outper- forms MOFFlow under strict and relaxed matching criteria. Under the more realistic threshold of stol = 0.5, MOF-BFN achieves a higher MR (53.51% vs. 44.75%) and lower RMSE (0.2498 vs. 0.2694) when generating 5 samples. Even under the relaxed stol = 1.0 setting, MOF-BFF BFN maintains competitive accuracy with a lower RMSE. These results demonstrate that MOF-BEN improves predictive accuracy.
Table Geometric property evaluation. RMSE computed between the ground-truth and generated MOFs. Baseline results are from Kim et al. [13.
MOF-BFN
MOFFlow
DiffCSP
VSA (m²/em²)
232.8
264.5
796.9
GSA (m²/g)
247.5
331.6
1561.9
(A*)
315.4
530.5
3010.2
UCV (A*)
312.0
569.5
3183.4
0.0187
0.0285
0.2167
PLD (Å)
0,9072
1.0616
4.0581
LCD (A)
0.9257
1.1083
4.5180
DST (g/em")
0.0185
0.0442
0.3711
5.2 Geometric Property Evaluation
In this section, we assess whether the predicted MOF structures preserve essential structural properties that critical for downstream applications such gas storage and separation. To this end, evaluate eight commonly used structural descriptors: volumetric surface area (VSA), gravimetric surface (GSA), accessible volume (AV), unit cell volume (UCV), void fraction (VF), pore limiting diameter (PLD), largest cavity diameter (LCD), and density (DST). For each test case, we generate single structure and compute its properties using Zeo++ (27). The RMSE between the
predicted and ground-truth property values is then reported in Table 2
Results Table 2 reveals that MOF-BFN achieves the lowest property prediction error across all evaluated metrics, indicating that the model not only aligns structures accurately but also preserves the spatial characteristics and material properties vital for real-world applications.
5.3 De Novo Generation
For de novo generation, we generate 1,000 MOFs from both MOF-BFN and MOFDiff without any relaxation and evaluate them using two sets of metrics in Table 3 First, we assess connection point matching, where the number of metal and non-metal connection points, identified the bro- ken bonds between decomposed building blocks, should be equal. MOF-BFN achieves substan- tially higher matching rate (0.923 v.s. 0.723), indi- cating more accurate topological assembly. Sec- ond, we apply the MOFChecker tool [8] to the matched structures evaluate their chemical and structural validity. MOF-BFN consistently outper- forms MOFDiff across most criteria and a higher proportion of valid structures passing all checks (0.323 v.s. 107). These results demonstrate that MOF-BFN is more effective at generating chemi- cally plausible and topologically consistent MOF structures. More details are provided in Appendix
Table 3: Generation validity. Validation rates of 1,000 generated samples.
Validity Criteria
MOF-BFN
MOFDin
Connection Point Matching
matched
0.923
0.723
MOFChecker
has, carbon 1 has hydrugen 1 atomicc overlaps has owwrroocdiinttld c has www.ooooddeiieeeed..commooo Ln
1,000
1.000
0.975
0.989
0.115
0.259
0.193
0.365
0.049
0.147
overcooodinnted
0.150
0.342
unddenconddintted.......o c
0.248
has underroonddiatttd.......o
0.154
0.126
undercoondinatted _arrh__arh___
0.000
0.000
metal
1,000
1,000
has conecmmoocuule_
0.186
0.437
high charges__ee_
0.069
0.144
suspicions, terminal rxo
his undercoordinttod allail alkaline
0.027
0.001
geometrically.....mmmm eeposedd metal checks
0.304
0.350
Titol
total_valid
0.323
0.107

---

5.4 Analyses
Our method improves upon the strong baseline MOFFlow in two key aspects: (I) we replace Cartesian coordinates with fractional coordi- nates to better capture the inherent periodicity of MOFs; and (2) we redesign the generative mod- cling paradigm using BFNs in place of the orig- inal flow matching approach. To evaluate the individual contribution of each modification, construct an intermediate model named MOF- FracFlow. This variant also applies the flow matching framework but operates in the frac- tional coordinate space, using the same periodic backhone architecture as MOF-BFN. A compar- ison of these models is presented in Figure highlighting two key observations. First, MOF- FracFlow outperforms the original MOFFlow, particularly in terms of the match rate with 5 samples, underscoring the importance of incor- porating periodicity. Second, MOF-BFN consis- tently surpasses MOF-FracFlow, demonstrating that after introducing an additional generative target on the S3(3) manifold, BFNs continue to outperform flow matching counterparts, con- sistent with prior findings [8] Additional analyses about orientation representations and hyperparameters are provided in Appendix
MOF-PracFlow
MOF-8FN
Match Rate (MR)
40
35
30
25
RMSE
0.28
0.27
0.88
Sumple
Sample 5
Figure 3: Comparison among different variants. We include MOF-FracFlow an intermediate model using fractional coordinates system and flow matching for generation.
Conclusion
this work, we present MOF-BEN, a hierarchical generative framework for Metal-Organic Frame- works that integrates Bayesian Flow Networks with fractional coordinates and quaternion-baat orientation modeling. By operating in the fractional coordinate space and modeling orientation using Bingham distributions over unit quaternions, MOF-BFN jointly captures the periodicity, positions, and orientations of MOF building blocks in a unified generative process. Extensive experiments demonstrate that MOF-BFN achieves tate-of-theearr accuracy in structure prediction, property evaluation, and de novo generation, highlighting the effectiveness of MOF-BEN for MOF design.
Acknowledgments
This work is jointly supported by the National Science and Technology Major Project NNo.2022ZD0117) 17700111502), the National Key R&D Program of China (N..2022200160502)) the National Natural Science Foundation of China (No. 61925601, No. 62376276, No. 62276152), and Beijing Nova Program (20230484278).
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A Explanations of Key Notations
Table 4:
Key notations used in this paper.
Notation
Meaning/ Description
%
Prior parameter for initiaizzaion of Bayesian Flow Network (BFN)
Pt
Bayesian update for retining parameters iteratively
Bayesian flow distribttion accumulated via updates, used for training Latiice matrix
Le gaaa
R
Unconstrained lattice parameters, modeled Goussion distribation
da Pts))
Parameters (mean, precision) of Gaussian distribution for & at step Block-level fractional coordimates Parameters of von Mises distribution for F (mean direction and concentratonn) step Complex form representing fractional coerdinate parameters
Tock
space, modeled by Mises distribution
(mn o!)
Quatemion representation 3D orientations
E S
One single quaternion, modeled by Bingham distribution Bingham distribution parameters for quaternions with Loss functions
AM
normalized cigenvalues stop
Loss weights
B Proofs of Propositions
In this section, we prove the properties listed in $ 4.2.
The PDF of the Bingham distribution is rewritten as
1
pn(g: A)
Z(A) exp (q M AMq)
Proposition 1. The PDF of the Bingham distribution maintains the antipodal symmetry, i.c. PB PB(q: M, A) pBpppppp00 M.).
Proof.
PB(-qi A,
Z(A) exp DM (M
Z(A) exp (q M AMq)
PB(q:
Proposition 2. When A 0, the PDF reduced to a uniform distribution, i.c. pa(g: M, 2
Proof. When A = 0, the exponent becomes zero for all q E S3:
1 exp (q M (TMg)
1
Pa P 0)
Z(0)
Z(O)
That is, pa (q; M, 0) is constant over S indicating it is a uniform distribution. Since the surface area S 2x the normalized uniform density PB (g; (q;
M,
2=2
Proposition 3. Due the normalization constraint the hypersphere, any bias applied on the eigenvalues would not affect the distribution, i.c. PB pg(q; M, A + kI) = PB (q; M, A), Vk E R.
Proof.
PB M, M.A.KA A+k()
Z(A - KT) exp (q M k1+)))MM)
exp (q AT (A +kJ)Mg)
exp (q T M AMq + kg " Mq)
* exp (q M AMM)
o exp (q M AMq).

---

Given the normalization constant (A), we have M,g:::MMMMM A+kI) Z(A) exp (q M AMq)
pa pa(q; A).
C Implementation Details for MOF-BEN
C.1 Determining Local Geometries for Building Blocks
To determine the orientations of building blocks, should first determine a reference frame for each
block. common-used solution Principle Component Analysis (PCA) 4 Given building block
Cj = (A,, X)), where A, |rr E RAxN, X, [ E R denote the atom types and coordinates within the block. The consistent local structure X, is defined
X, Rxx,
(28)
where X, | a..
No r= 1
N3 is the centered coordinates, and R c. ( R.
R = e2, e2, e3] is the eigenvector matrix of X, x and the signs = [e1, c22 cal is determined by
+ 2 0, 1,,, By
(29)
with vector v, determining the direction of the building block. Following (6, the vector is defined as vj arg m..... 1-
C.2 Rejection Sampling for Bingham Distribution
To sample from the Bingham distribution, we adopt a rejection sampling strategy using the Angular Central Gaussian (ACG) distribution as the proposal [5]. Given multivariate normal distribution ~ (0, E). the normalized direction q q/ q/||ql] follows an ACG distribution, whose density PACG (q) x (q ΣーーE-1 4) -4/2, where d the dimension of the hypersphere. To perform rejection sampling, we choose Σ = I- 2A such that the Bingham distribution is upper bounded by
scaled ACG distribution: ( S. A' (4) with A' related to the normalization term of each distribution. This bound leads to the condition
exp(a Aq) S A(q To (I-2A)q) -1/2
exp(q' T Aq)
A 1 (q [I 2A)q) 4/2
exp(g Aq) 2q Ag)
We define the function f(t = exp(t)( - 2// and maximize it over t R to obtain the tightest possible rejection bound. The maximum occurs f = ايا resulting in an optimal rejection constant Amin exp( 3=d)d"d )dd/2, Once sample is accepted. we apply linear transformation q' Mq if the Bingham distribution has eigendecomposition MAM This yields sample from the desired Bingham distribution.
C.3 Accuracy Scheduling for Bingham BFN
As the eigenvalue matrix A is diagonal, we denote the normalization term Z(A) Z(X) in this subsection. To determine a suitable value of a,, we consider the entropy of the Bingham distribution, which takes the form
H. = H(A,) log Z(A.) X log Z():),
Assuming that A; R 10,- - -3... we can approximate the entropy as a function of a single parameter Sr This assumption is an approximation that enforces isotropy around the principal axis, it is effective in practice for constructing a simple and stable entropy scheduler. Let 30 = 0 and 37 sufficiently large, lincarly interpolate the entropy H, (1- Ho + Hr, and numerically

---

solve for each intermediate value B₁ Consider the sender distribution [[,[[1,,000 0J. a;), second moment given by
its
VZ 0: -4;; (0, -a,, 04, aa;)
0: ass) + (1 (1- Σ
VZ(0. 0; -0, aa) 11.000000)
E yyy 1
diag
[0, -04, -0s, -05)
From this, the expected change in
B|3 - 3--1] = aq aq (Elyyy To Ely.y, 111)
= a (Z -0s
02g -0,10
Z(0, -a,, -a, -a,, -a;)
where the subscripts indicate the value the corresponding indices. And we can numerically solve for a given 34. Such scheduler gradually sharpens the Bingham distribution, increasing its concentration around the target direction, ensuring that the model starts with high-entropy prior and becomes progressively confident step increasing.
D Extension to De Novo Generation
D.1
Implementation Details
The utilized MOF datasets contains millions of building block conformations, making simple one-hot encoding for building blocks sparse and computationally inefficient. To obtain compact and continuous representations, a contrastive learning framework is employed in MOFDiff (.. Each building block is encoded into a latent vector using a SE(3) (E(3)--quvvariant message passing neural network, specifically GemNet-OC.
Let G denote the set of all building blocks obtained from training MOFs. For each building block C G, the encoder network fe maps its local structure continuous embedding ze fC( e R with latent dimension d = 32 in MOFDiff.
To ensure that the learned embeddings reflect chemical similarity, contrastive learning is performed using positive and negative pairs of building blocks. A positive pair (+) consists of two blocks sharing the same ECFP4 fingerprint, while negative pair (C,C indicates different block identities.
The contrastive learning objective is based the following loss:
Ejes+ exp (sim(z,, 2,)/T)
Loontrast
Σ log iES
(30)
Ejes exp (sim (Zi, 2T)/T)²
where S,S+ denote the batch and the positive subset of the batch, r > 0 is temperature hyperpa- rameter, and sim(.,) denotes the cosine similarity:
2 Z2
sim (Z), 22)
(31)
|2 ||
While the continuous encoding space enables efficient retrieval via KD-Trees, we empirically observe that the distribution of embeddings deviates significantly from a normal distribution, which poses challenges for training a BFN, To address this, we normalize the representations as
2, 2; mean
std
Similar to the BFN for lattice parameters, given the block embeddings Z [z]] of structure,
the input distribution is given by NiZ; (1 parameterized by 0 = {u pf and the prior distribution chosen as of (0, 1]. After acquiring a sample from the sender distribution v (Z, ( - at step i, the Bayesian update function is
{ 0 = -
1 + a v
15
+ a
(32)
+
a
1-1
3

---

The corresponding Bayesian flow distribution is accumulated as
f - NV oT 2z oT
2:/7
2:/3 25/T )))
OT
(33)
The training objective on the latent space is
a TT
LB = (.L.U 11 11
1 00.
(34)
And the entire training objective is extended as
LNN X XFLF + Aaha ABCB.
D.2 Results Relaxed Structures
Similar to MOFDiff, we further relax the generated structures via the UFF force field 77]. We refine both the lattice parameters and the all-atom positions by LAMMPS (9] and LAMMPS Interface 2 The numbers of valid structures before and after relaxation are reported in Table further demon- strating the generation quality of MOF-BFN.
Table 5: Generation validity. Number of structures that passed (1) failed (4) each criterion among 1,000 generated candidates.
Before Relaxation
After Relaxation
Validity Criteria
MOF-BEN MOFDiff
MOF-BFN
MOFDiff
Connection Point Matching
matched 1
923
723
923
723
UFF Relaxation
relaxed
849
662
MOFChecker
has cos_carronn f
923
723
849
662
has hos_hyyrgeen f
900
715
827
654
mas_atomic__a
____ttmmic_vvrlapps +
106
187
35
136
has aus_ovvrroordinated
178
264
17
has_ sss_overcoordinateed has jsserrcoordinttedd h
45
34
0
0
166
247
20
21
has ___nnerccorddnatteed has underrcoordinated has undercoordinated rare care_carth + f has has_lone_molecute + has biigh_ehargs_ has so__suppicioss_temieee _xoo 4
168
179
144
194
142
91
141
133
923
723
849
662
172
316
36
60
64
104
5
13
0
2
has undercoordinated alkali alkaline 4
1
has gwwmetrically_eppeeeee metal 4
281
253
8
24
Total
total, valid f
323
107
545
317
D.3 Novelty and Uniqueness
Evaluating novelty and uniqueness is also critical in de novo generation. To address this, we additionally conduct an additional analysis using MOFid. SMILES-style descriptor that captures both the building block identities and the topology of MOF structures. Specifically, we define novelty as the proportion of generated structures whose MOFid does not appear the training set, and uniqueness as the proportion of generated structures with MOFids that are distinct from all other generated samples. To ensure that novelty meaningful, we further define validity as structures that pass LAMMPS relaxation, satisfy all MOFChecker criteria, and are successfully processed by

---

MOFid. Table S the results. These results demonstrate that MOF-BFN attains higher overall V.N.U. rate. Notably, MOFDiff exhibits comparatively higher novelty and uniqueness ratio among valid samples, indicating stronger tendency to generate structures distinct from the training distribution. However, this advantage is offset by lower validity, as substantial portion of the generated candidates fail to satisfy basic structural or physical constraints. It is therefore important to contextualize novelty within the scope of validity, since novel structures lacking physical plausibility are unlikely to be of practical significance for downstream applications. By contrast, MOF-BFN yields a substantially larger number of valid structures while preserving a considerable degree of novelty and uniqueness. Consequently, it achieves the highest proportion of valid, novel, and unique (V.N.U.) samples. This highlights a more balanced trade-off between structural diversity and physical realism, which is essential for practical applications in MOF design.
Table 6: Generation VNU rates. Validity, Novelty and Uniqueness of 1,000 generated samples based on MOFid.
Model
Valid w/o MOFid
Valid w/ MOFid
Valid & Novel
Valid & Novel & Unique
MOFDiff MOF-BFN
323
284
281
28 1
545
450
424
407
D.4 Geometric Property Evaluation
We further evaluate the generated samples in terms of the Wasserstein distances against the training set on geometric properties listed in T As shown in Table the results indicate that the property distributions of MOF-BFN is closer the training data compared to MOFDiff.
Table 7:
Geometric property evaluation. Wasserstein distances computed between the training set and generated MOFs.
VSA (m²
GSA (m²/g)
AV (A²)
UCV (A²)
VF
PLD (A)
LCD (A)
DST (g/cm")
MOFDiff MOF-BEN
177.8
258.4
1039.7 885.2
4737.2
0.043
0.607
1.311
0.059
157.2
131.7
1626.3
0.022
0.665
1.177
0.043
E Extension to Conditional Generation
Enabling conditionl generation is also a valuable problem. To demonstrate this, we further conduct a preliminary experiment on void fraction (VF) via paassifer-guidees generation (CFG)-style approach 8 In this approach, the model was fine-tuned to accept an additional conditioning input representing the desired value of a target property. During generation, both an unconditional output x and a conditional output Pe were obtained. The final generated sample was then computed as expa (w log,: Æc), where exp and log denote the exponential and logarithmic maps defined on the specific manifold, including the lattice representation, fractional coordinate space, unit quaternion space, and block embedding space. The weighting factor w controls the degree of conditioning. Figure ssoos that the void fraction distribution of the generated structures shifts significantly toward the desired value under conditioning.
Figure 4: Void Fraction (VF) distribution of the original BW-DW dataset (left), the generated MOFs with w 1.0 (middle) and 2.0 (right). The distributions shift significantly toward the desired value under conditioning, and larger guidance weights lead to stronger control.
5

---

F Additional Ablation Studies
Orientation Representation In addition to unit quaternions, another widely used formu- lation for modeling 3D orientations is the axis-angle representation, which characterizes rotation by a pair (u. f), where u E S denotes the rotation axis and E [0, TT) is the rotation
Table 8: Additional ablation studies on orientation representations.
Orientation Representation
Match Rate
RMSD
Axis-Angle
31 31.49
0.2770
Unit Quaternion
35.27
0.2735
angle. This representation is geometrically intuitive and has been extensively applied in rigid-body kinematics. To model this formulation within the proposed framework, we employ decomposed strategy, where the rotation axis is represented by a von Mises-Fisher (vMF) distribution on the unit 2-sphere, while the rotation angle is represented by von Mises distribution on the circle. This design preserves the intrinsic geometry of each component and enables independent Bayesian updates within the BFN framework.
The results on the structure prediction task are shown in Table Despite being theoretically valid representation, this alternative led to no- ticeably worse performance in structure predic-
Table weights.
Additional ablation studies on loss
%
78
Valid hefore relaxation
Valid after relaxation
0.2
10.0
323
54.5
tion. As modeling axis and angle separately breaks the unified structure of the SO(3) mani-
1.0
10.0
219
40.9
0.2
1.0
233
44.4
fold and complicates the joint learning of rota-
tion distributions. In contrast, the quaternion representation allows unified and conjugate-friendly update using the Bingham distribution. The additional results confirm that the use of unit quaternions and Bingham BFN is not only theoretically motivated but also empirically beneficial.
Loss weights For the DNG task, we further analyse the loss weights for the orientation 7 and the block embedding "ys The results in Table show that scaling either component improperly leads to drop in validity.
G Experimental Details
Hyperparameters for the structure prediction ($ 5.1) and de novo generation ($ 5.3) are provided in Table 10 Baseline results in Table and are from MOFFlow [6], and the samples for calculating MOFDiff results in Table are directly yielded from the official pre-trained checkpoint To ensure a fair comparison, we share the same split with MOFDiff, where 95% structures are used for training, and the remaining 5% are for validation. The structure prediction and de novo generation models trained on 8 GPUs with 80 GB memories, and the training procedures take 136 and 152 GPU hours, respectively.
H Baseline Selection
We provide the original scope of different models in Table I1 For the structure prediction (SP) and property evaluation task, follow the benchmark introduced in MOFFlow, and the selected baselines includes optimization-baset methods (RS and EA), the full-atom model DiffCSP and the block-level model MOFFlow. Optimization-bai based methods are excluded from property evaluation as they fail to produce meaningful structures in Table I For de novo generation (DNG) task. it requires generating both block identities and their structure. Among existing methods, only MOFDiff supports this setting specifically for MOF, and is thus selected as the baseline.
I Limitations
While MOF-BFN presents promising approach to hierarchical MOF structure prediction, several limitations remain that we leave to future work. First, our current framework treats each building block as a rigid body with a fixed local geometry. This rigid-body assumption simplifies the generative process but neglects the intrinsic conformational flexibility of many organic linkers and secondary
httpp:://gtaabb.ccc/ccrrett////p

---

Table 10: Hyperparameler settings for experiments.
Building Block Encoder
num Jane_s
node_ dim
edge dim
hidden dim
max Lax_radius
4
64
64
64
5
Structure Predictor
num layers
hidden dim
time dim
num freqs
6
512
128
64
BFN
.94
84.
34.
T
1000
1000
200
1000
50
7
7F
7q
7/8
1.0
1.0
0.2
10.0
Structure Prediction Training
Ir
min_ Ir
cadtau_ffctta plateau_patienm patience
Adam betas
5x10-4
1x10-4
0.6
30
[0.9, 0.98]
epochs
batch_size
gradient_clip val
weight_decay
1000
512
0.5
0.01
De Novo Generation Training
Ir
min_Ir
plateau_factor
plateau_ patience
Adam_
5x10-4
1x10-4
0.6
30
[0.9, 0.98]
epochs
batch_sizz
gradient_clln val
Seight_ddcayy
3000
512
0.5
0.01
Table 11: Model Scope. SP and DNG denote structure prediction and de novo generation, respec- tively.
Model
Atom-level SP
Block-level SP
Block-level DNG
RS & EA
DiffCSP MOFFlow
MOFDiff
MOF-BFN
building units. For instance, MOFDiff 3 reports that over million building block instances in the dataset correspond to only 242k unique molecular graphs, indicating that significant conformational diversity exists within each block type. Although our method supports the extension of building block vocabularies through a continuous embedding space, it does not yet account for conformation generation within each block. Integrating internal flexibility modeling into the current framework could further enhance the model. Second, our current work focuses on unconditional generation and structure prediction tasks, without explicitly incorporating guidance signals for specific downstream properties. In practice, many MOF design scenarios require property -orientd generation, such as optimizing for gas adsorption capacity or catalytic activity. However, techniques for guiding models toward desired property targets remain underexplored in the field of Bayesian Flow Networks. Developing conditional generation mechanisms within the BFN framework is important direction for future research to enable targeted material discovery.

---

J
Broader Impacts
This work contributes the development of MOF structure prediction and design. It may benefit applications in gas storage, separation, and catalysis by enabling more efficient exploration of the chemical design space. By improving structure prediction accuracy and generation validity, it can potentially accelerate material discovery in evaluated primarily on the BW-DB dataset [. which may contain inherent biases in block types, structural motifs, or chemical compositions. As result, the generalization ability of the model to underrepresented MOF types or application-speciffi domains could limited. Care should be taken when applying the model beyond the scope of the training data.
way. However, our model is trained and
K
Code Availablility
Our codes are available at abttpps//ggiaubl.co/jjaa///////
References
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[2] Peter G Boyd, Seyed Mohamad Moosavi, Matthew Witman, and Berend Smit. Force-field prediction of materials properties in metal-organic frameworks. The journal of physical chemistry letters, 8(2):357-363, 2017.
[3] Xiang Fu, Tian Xie, Andrew Scott Rosen, Tommi S Jaakkola, and Jake Allen Smith. Mofdiff: Coarse Coarse-grained diffusion for oetal-organic framework design. In The Twelfth International Conference on Learning Representations.
[4] Nicholas Gao and Stephan Günnemann. Ab-initio potential energy surfaces by pairing gnns with neural wave functions. In International Conference Learning Representations.
[5] John T Kent, Asaad M Ganeiber, and Kanti V Mardia. A new method to simulate the bing- ham and related distributions in directional data analysis with applications. arXiv preprint 13/0.811 2013.
[6] Nayoung Kim, Seongsu Kim, Minsu Kim, Jinkyoo Park, and Sungsoo Ahn. Mofflow: Flow matching for structure prediction of metal-organic frameworks. arXiv preprint arXiv:2410. 17270, 2024.
[7] Anthony K Rappé, Carla J Casewit, KS Colwell, William A Goddard III, and W Mason Skiff. Uff, full periodic table force field for molecular mechanics and molecular dynamics simulations. Journal of the American chemical society, 14025)::0024-10335 1992.
[8] Nianze Tao and Minori Abe. Bayesian flow network framework for chemistry tasks. Journal of Chemical Information and Modeling, 65(3):1178-11) 2025.
[9] Aidan P Thompson, H Metin Aktulga, Richard Berger, Dan S Bolintineanu, W Michael Brown, Paul Crozier, Pieter J In' 't Veld, Axel Kohlmeyer, Stan Moore, Trung Dac Nguyen, et al. Lammps-a flexible simulation tool for particle-based materials modeling the atomic, meso, and continuum scales. Computer physics communications, 271:108177 10171, 2022.
8


Reviews:
[{"review_id": "arBLuEBi7l", "reviewer": "Reviewer_7ek4", "summary": "This paper addresses Metal Organic Framework structure prediction and de novo generation. To tackle this, it leverages Bayesian Flow Networks that operate on a carefully selected MOF representation space, crucially, this space entails fractional coordinates, as well as orientations modeled as unit quaternions. These quaternions are parameterized via Bingham distributions.", "questions": "- Could the authors elaborate on the choice of unit quaternions and the Bingham distribution? It would be helpful to see empirical evidence, such as additional ablation studies, to justify the contributions.\n\n- I am wondering if all four distinct hyperparameters are necessary to balance the loss function. Typically, such balancing is used when optimizing disparate training signals, whereas here each loss component appears to correspond to a core aspect of the representation.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "### Strengths:\n- [Quality] The authors provide a theoretically sound approach to the representation of MOFs, which is further complemented by the selection of parameterization that contains key inductive biases connected to the application domain. The experimental results support the use of Bayesian Flow Networks, the proposed representations, and key parameterized distributions.\n- [Clarity] In general, the submission is well-structured, with precise and consistent notation that highlights the key design choices and theoretical motivations. The experimental setup is easy to follow, and the results are well-presented.\n- [Significance] MOF-BFN outperforms previous work, with selected representations that provide valuable insights for future work. Furthermore, it provides an overview of how to implement BFNs with the Bingham distribution. Therefore, it constitutes a meaningful achievement within the field\n- [Originality[ The authors cite relevant work, and their contributions are well-contextualized within the field of MOFs and broad material design. The contributions are novel.\n### Weaknesses:\n- [Quality] It is scientifically unclear which of the technical contributions leads to improved experimental results.\n- [Clarity] The large number of mathematical symbols makes reading and comprehension harder; Additionally, in the experimental section, it should be clear why only a subset of the external models are considered for each table/figure, please see my further remarks.\n- [Significance] I believe the paper would benefit from more extensive ablation studies. In particular, with respect to the selection of unit quaternions and the utilization of the Bingham distribution, instead of alternative representations and distributions", "was_revised": true, "final_justification": "The authors were able to answer my questions and provided convincing ablations during the rebuttal. Hence, I recommend acceptance for this paper.", "rebuttal": "We thank the reviewer for the valuable comments, and we answer the reviewer's questions as follows.\n\n> **W1: It is scientifically unclear which of the technical contributions leads to improved experimental results.**\n\nThank you for pointing this out. In the original submission, we provided an ablation study in Sec. 5.4 to isolate the contributions of the two key components. In particular, we compare MOFFlow, MOF-FracFlow, which incorporates fractional coordinates into the flow matching framework, and our MOF-BFN, which additionally uses Bayesian Flow Networks and quaternion orientation modeling. This progression helps attribute the performance gain. Using fractional coordinates leads to better handling of periodicity, and switching from flow matching to BFNs brings further gains in generative performance.\n\nIn addition, we conducted a further ablation study on orientation representation by **replacing unit quaternions with axis-angle representations**. This allows us to assess the importance of using a principled and symmetric representation for modeling orientations. The details of this comparison are provided in our later reply to W3.\n\n> **W2.1: The large number of mathematical symbols makes reading and comprehension harder**\n\nWe appreciate the reviewer’s suggestion and fully acknowledge the trade-off between technical precision and readability. To improve clarity, we will include a summary table in the appendix in the revised version to clarify the meaning of each key notation used in the equations.\n\n> **W2.2: In the experimental section, it should be clear why only a subset of the external models are considered for each table/figure.**\n> **In the related work or appendix, there should be a table that compares all external models and MOF-BFN with respect to their capabilities. This should be clear in the experimental section, since some of the tables/figures only display a subset of the comparable models.**\n\nWe thank this insightful advice. Regarding the baselines, each subset is selected based on the applicability of the corresponding method to the specific task.\n- Tables 1 and 2 follow the benchmarks introduced in MOFFlow [A], which focuses on evaluating **MOF structure prediction** in terms of both structural accuracy and property alignment. The selected baselines includes optimization-based methods (RS and EA), the full-atom model DiffCSP and the block-level model MOFFlow. Optimization-based methods are excluded from property evaluation (Table 2) as they fail to produce meaningful structures in Table 1.\n- Table 3 evaluates **de novo generation**, which requires generating both block identities and their structure. Among existing methods, only MOFDiff supports this setting, and is thus selected as the baseline.\n\nWe agree that a capability comparison table would help clarify the scope of each baseline, and we will add such a table in the revised version.\n\n> **W3: I believe the paper would benefit from more extensive ablation studies. In particular, with respect to the selection of unit quaternions and the utilization of the Bingham distribution, instead of alternative representations and distributions**\n> **Q1: Could the authors elaborate on the choice of unit quaternions and the Bingham distribution? It would be helpful to see empirical evidence, such as additional ablation studies, to justify the contributions.**\n\nAs suggested by the reviewer, we explored alternative representations for modeling 3D orientations. In addition to unit quaternions, one common formulation is **the axis-angle representation**, which describes a rotation as a pair $(u, \\theta)$, where $u\\in\\mathcal{S}^2$ is the rotation axis and $\\theta\\in[0,\\pi]$ is the rotation angle. This representation is geometrically intuitive and widely used in rigid-body kinematics. To model this representation, we adopted a decomposed approach, where the rotation axis was modeled by a von Mises–Fisher (vMF) distribution on the 2-sphere, and the rotation angle by a von Mises distribution on the circle. This design preserves the geometry of each component and enables separate Bayesian updates within the BFN framework. The results on structure prediction task are shown as follows:\n\n| Orientation Representation | Match Rate (%) | RMSD |\n| :-: | :-: | :-: |\n| Axis-Angle | 31.49 | 0.2770 |\n| Unit Quaternion | 35.27 | 0.2735 |\n\nDespite being a theoretically valid representation, this alternative led to noticeably worse performance in structure prediction. As modeling axis and angle separately breaks the unified structure of the SO(3) manifold and complicates the joint learning of rotation distributions. In contrast, the quaternion representation allows a unified and conjugate-friendly update using the Bingham distribution. The additional results confirm that our use of unit quaternions and Bingham BFN is not only theoretically motivated but also empirically beneficial. We will include them in the revised version.\n\nMoreover, we also considered directly modeling rotation matrices. However, defining efficient and closed-form Bayesian updates over SO(3) matrices remains underexplored in the context of generative modeling. We believe this is a promising direction for future research.\n\n> **I am wondering if all four distinct hyperparameters are necessary to balance the loss function. Typically, such balancing is used when optimizing disparate training signals, whereas here each loss component appears to correspond to a core aspect of the representation.**\n\nThanks for the insightful question. Although we did not perform extensive hyperparameter tuning, we analyse the weights for the orientation $\\gamma_q$ and the block embedding $\\gamma_\\mathcal{B}$. The results below show that scaling either component improperly leads to a drop in validity.\n\n| $\\gamma_q$ | $\\gamma_\\mathcal{B}$ | Validity before relaxation | Validity after relaxation |\n| :-: | :-: | :-: | :-: |\n| 0.2 | 10.0 | 32.3 % | 54.5 % |\n| 1.0 | 10.0 | 21.9 % | 40.9 % |\n| 0.2 | 1.0 | 23.3 % | 44.4 % |\n\n\n\n\n> **The relevant appendix sections should be cited across the main paper.**\n\nWe thank the reviewer's suggestion and will refer the appendices in the main paper.\n\n> **The hyperparameters in Eq.26 should not use lambda but preferably another letter. As this can lead to confusion with the lambda used for the eigenvalues represented before.**\n\nNice suggestion! We will replace the loss weight notations into $\\gamma$ to avoid misleading.\n\n> **“At each step $i$, the receiver treats its distribution from the previous step as the input” and Eq. 2 display notation that is not consistent with the original BFN work, as the input distribution is not used during sampling. Effectively, what gets updated is $p_R$ and not $p_S$. I find this confusing and would prefer if the notation was in line with the original work.**\n\nThanks for pointing this out. Our Eq. (2) was intended only as an intuitive illustration of the Bayesian update process. Indeed, the input distribution is not directly sampled during generation, but instead iteratively refined through parameter updates. Besides, as we stated in lines 145–149, since the sender distribution is unavailable during sampling, the receiver $p_R$ rather than $p_S$ is used. We will revise our notation and explanations to make this distinction clearer.\n\n> **It should be clear just by looking at Figure 3 that MOF-FracFlow is an ablation.**\n\nThanks! We will clarify this in the caption of Figure 3 to make the readers directly understand the figure."}, {"review_id": "of0i8tcN9g", "reviewer": "Reviewer_hsw3", "summary": "The authors introduce a novel way of parameterizing a joint lattice-position-orientiation distribution of hierarchical MOF structures. By simultaneously learning the bayesian flow distributions of the lattice parameters in R^6,  the fractional coordinates represented by elements of the torus T^3 (times the number of bodies), and the bingham distribution on the unit quaternions (again times the number of bodies), the authors demonstrate the ability to generate better structures compared to several baselines.", "questions": "-Do there exist families of distributions on other Lie groups such as $SE(3)$? The nice thing about using the unit quaternions, which are specific to SO(3), is that the Bingham distribution defined on them has a nice conjugate relationship with the Watson distribution.\n\n-Could the role of $\\mathcal{L}_{\\mathcal{B}}$ be clarified for doing de novo generation? \n\n-How does the block representation of MOFFlow compare to the representation proposed in the paper? If they are different, which (between the block representation being used or the learning algorithm itself) is the dominant factor in model performance? \n\n-In general, what else is different between MOF-BFN and MOFFlow, and why would we expect that change to make MOF-BFN superior?", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n-The paper is straightforward to read and understand, and offers a clearly novel idea for learning to sample hierarchical MOF structures. The results demonstrate clear improvement over existing approaches.\n\n-The authors use a clever conjugate relationship to enable bayesian flow distribution to be defined on the unit quaternions\n\nWeaknesses:\n-One thing i don’t fully understand is how widespread of a tool this would be for MOF researchers. I don’t fully understand the scope of problems this model can solve. Are there examples of this?\n\n-It seems like it would be eventually be useful to have some kind of MOF simulator (e.g. DFT) in the loop to provide effectively infinite data for the model to learn from. \n\n-Not sure how convincing the margin between MOFFlow and MOFBFN is.", "was_revised": true, "final_justification": "I had some questions about the original paper which the authors answered clearly and thoroughly, so I have decided to increase my score as a result.", "rebuttal": "We thank the reviewer for the valuable comments, and answer the questions as follows.\n\n> **W1: One thing I don’t fully understand is how widespread of a tool this would be for MOF researchers. I don’t fully understand the scope of problems this model can solve. Are there examples of this?**\n\nWe thank the reviewer for raising this important question. The proposed framework supports both structure prediction and de novo generation for MOFs, and enables several practical applications. Firstly, MOF-BFN can generate diverse candidates can be used to populate virtual databases for high-throughput screening pipelines. Secondly, as a generative model, MOF-BFN can serve as a structure initialization tool that can complement or accelerate DFT-based optimization pipelines. Last, by generating novel combinations of nodes and linkers, our model can help researchers identify unconventional MOFs with potentially superior performance, especially for applications like gas storage, separation, or catalysis. Overall, MOF-BFN provides a powerful and versatile generative framework for MOF design, we believe this makes it a valuable tool for the MOF community.\n\n> **W2: It seems like it would be eventually be useful to have some kind of MOF simulator (e.g. DFT) in the loop to provide effectively infinite data for the model to learn from.**\n\nWe agree that incorporating physics-based simulators such as DFT into the training loop is a promising future direction, but it is impractical to run them on the fly given the limited computational resouces. Hence, our current focus is on learning from existing structural data in a data-driven, simulator-free setting, in order to prove that our model has the capability to capture meaningful physical knowledge from the structural data. Looking ahead, we agree that simulators could be integrated into MOF-BFN in several post-training ways, such as fine-tuning the model on simulated properties, or incorporating reinforcement learning with DFT-based rewards. We consider this a highly promising extension and plan to explore it in future work.\n\n> **W3: Not sure how convincing the margin between MOFFlow and MOFBFN is.**\n\nThanks for raising this concern. To better compare MOFFlow and MOF-BFN, we further report the mean, standard deviation, and the full range (min and max) of each metric across the five sampling runs used in Table 1. As shown below, MOF-BFN consistently outperforms MOFFlow in both match rate and RMSD across all runs, and the performance margin is statistically stable:\n\n|  |  | Match Rate (%) | RMSD |\n| :-: | :-: | :-: | :-: |\n| MOFFlow | Reported | 31.69 | 0.2820 |\n| MOF-BFN | Mean | 35.26 | 0.2751 |\n| | Std. | 0.10 | 0.0012 |\n| | Min. | 35.08 | 0.2735 |\n| | Max. | 35.38 | 0.2772 |\n\n> **Do there exist families of distributions on other Lie groups such as SE(3)?**\n\nWe appreciate the reviewer’s insightful question. In our work, we decompose the state of each building block into its coordinate and orientation, corresponding to a periodic vector in $\\mathcal{T}^3$ and a rotation in SO(3), respectively. This separation allows us to adopt conjugate-friendly distribution families that are well-suited for each component. Directly modeling probability distributions on SE(3) as a whole, and determining the Bayesian updates on it, remains an open research problem. Our current decomposition-based approach offers a practical alternative, and more unified treatments over SE(3) could be an interesting direction for future work.\n\n> **Could the role of $\\mathcal{L_B}$ be clarified for doing de novo generation?**\n\nWe thank the reviewer for the question. The role of $\\mathcal{L_B}$ in de novo generation is already discussed in Appendix C.2. Specifically, $\\mathcal{L_B}$ is the training objective of a continuous-space BFN, applied on the continuous latent representation learned by contrastive learning. This objectve enable the model to generate building blocks in the embedding space.\n\n> **How does the block representation of MOFFlow compare to the representation proposed in the paper? If they are different, which (between the block representation being used or the learning algorithm itself) is the dominant factor in model performance?**\n\nThanks for pointing this out. Our model uses exactly the same block representation module as MOFFlow, which is a 4-layer EGNN. This design allows us to fairly evaluate the impact of our proposed generative approach.\n\n> **In general, what else is different between MOF-BFN and MOFFlow, and why would we expect that change to make MOF-BFN superior?**\n\nWe appreciate the reviewer’s question. As noted earlier, both models share the exact same block representation model, ensuring a fair comparison. The key differences lie in the generative modeling framework. MOF-BFN introduces two major improvements over MOFFlow:\n\n- **Fractional coordinate modeling:** MOFFlow models building block positions in Cartesian coordinates, and MOF-BFN replaces this with a fractional coordinate system, enabling more accurate modeling of periodicity.\n- **Using of Bayesian Flow Network:** Instead of flow matching, MOF-BFN adopts the BFN framework, which defines explicit Bayesian updates over intermediate distributions. This leads to better generation performance.\n\nTo demonstrate the contribution of each modification, we provided a controlled ablation study in Section 5.4. We compared:\n\n- MOFFlow (Baseline)\n- MOF-FracFlow (Fractional coordinates + Flow Matching)\n- MOF-BFN (Fractional coordinates + BFN)\n\nThe results show a clear and consistent improvement at each stage."}, {"review_id": "HrRAdfxOSe", "reviewer": "Reviewer_Cxvg", "summary": "The paper introduces MOF-BFN, a generative model for predicting the structures of Metal-Organic Frameworks (MOFs) using Bayesian Flow Networks (BFNs). MOF-BFN leverages the local geometry of metal nodes and organic linkers to jointly predict lattice parameters, as well as the positions (in fractional coordinates) and orientations (via unit quaternions) of building blocks within a periodic unit cell. The model uses a Bingham BFN to effectively sample orientations on the 4D unit hypersphere. Experiments show MOF-BFN achieves strong performance in structure prediction, geometric property evaluation, and de novo MOF generation.", "questions": "1. How diverse are the generated MOF structures? Can you provide an analysis showing whether the generated topologies differ significantly from those in the training dataset?\n2. Have you evaluated whether the distribution of geometric and physical properties in the generated structures shifts relative to the training set? If so, can you discuss the implications of any observed shifts?\n3. Could you elaborate on the novelty aspect of your de novo generation results? How do you measure or ensure that newly generated structures represent novel topologies rather than variants of known ones?\n4. What are the main limitations of your approach, particularly regarding generation validity rates and scalability? How might these limitations affect the applicability of MOF-BFN in real-world MOF design tasks?", "limitations": "Section Missing", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. The paper tackles the complex task of MOF structure generation using Bayesian Flow Networks, extending the paradigm to a challenging problem.\n2. The paper has a sound modeling strategy and architecture design; the use of fractional coordinates and BFNs provides a novel alternative to conventional ODE and SDE-based models. The model appropriately incorporates relevant symmetries, improving its ability to handle the periodic nature of MOFs.\n3. The model is compared against relevant baselines and demonstrates strong performance across the reported metrics, highlighting its effectiveness relative to existing approaches.\n\nWeaknesses:\n\n1. The paper does not discuss the diversity or uniqueness of the generated MOF structures. It remains unclear whether the model generates topologies that are distinct from those in the training set.\n2. There is no analysis of whether the generated structures deviate in terms of geometric or physical properties (e.g., as shown in Table 2) from the training data. A meaningful distribution shift is often a desirable characteristic in generative models, as it indicates the ability to explore novel and diverse regions of the design space.\n3. The de novo generation results focus on structural validity rather than novelty. However, novelty is a central goal of de novo design and should be explicitly evaluated and discussed.\n4. The authors do not adequately address the limitations of their method. Given the complexity of MOF generation and the relatively modest validation rates, it is important to acknowledge the method’s current boundaries and potential areas for improvement.", "was_revised": true, "final_justification": "The authors have addressed all my concerns in the rebuttal", "rebuttal": "We thank the reviewer for the insightful comments, and address the concerns as follows.\n\n> **W1: The paper does not discuss the diversity or uniqueness of the generated MOF structures. It remains unclear whether the model generates topologies that are distinct from those in the training set.**\n> **W3: The de novo generation results focus on structural validity rather than novelty. However, novelty is a central goal of de novo design and should be explicitly evaluated and discussed.**\n> **Q1: How diverse are the generated MOF structures? Can you provide an analysis showing whether the generated topologies differ significantly from those in the training dataset?**\n> **Q2: Could you elaborate on the novelty aspect of your de novo generation results? How do you measure or ensure that newly generated structures represent novel topologies rather than variants of known ones?**\n\nWe thank the reviewers for raising the important question of novelty and uniqueness in de novo generation. To address this, we have futher conducted an additional analysis using MOFid [A], a SMILES-style descriptor that captures both the building block identities and the topology of MOF structures.\n\nWe define novelty as the proportion of generated structures whose MOFid does not appear in the training set, and uniqueness as the proportion of generated structures with MOFids that are distinct from all other generated samples. To ensure that novelty is meaningful, we further define validity as structures that pass LAMMPS relaxation, satisfy all MOFChecker criteria, and are successfully processed by MOFid. The following table summarizes the results.\n\n\n| Model | Valid | Valid & Novel | Valid & Novel & Unique |\n|-|:-:|:-:|:-:|\n| MOFDiff   | 28.4 %  | 28.1 %   |  28.1 % |\n| MOF-BFN   | 45.0 %   | 42.4 %   | 40.7 % |\n\nThese results show that MOF-BFN achieves a higher overall V.N.U. rate. It is worth noting that MOFDiff shows a relatively higher novelty and uniqueness ratio among valid samples. This suggests that it tends to generate structures that differ more from the training data. However, this comes with a lower overall validity, which means many generated samples do not meet basic structural or physical constraints. We believe that novelty should be considered in the context of validity, as novel structures that are not physically plausible may not be meaningful for downstream applications. In contrast, MOF-BFN achieves a significantly higher number of valid structures while maintaining a strong level of novelty and uniqueness. As a result, it achieves the highest proportion of valid, novel, and unique (V.N.U.) samples. We believe this reflects a more balanced trade-off between generating diverse structures and ensuring structural realism, which is essential for practical MOF design. \n\n> **W2: There is no analysis of whether the generated structures deviate in terms of geometric or physical properties (e.g., as shown in Table 2) from the training data. A meaningful distribution shift is often a desirable characteristic in generative models, as it indicates the ability to explore novel and diverse regions of the design space.**\n> **Q2: Have you evaluated whether the distribution of geometric and physical properties in the generated structures shifts relative to the training set? If so, can you discuss the implications of any observed shifts?**\n\nWe thank the reviewers for highlighting the importance of analyzing distributional shifts in the generated structures. We further evaluate the generated samples in terms of the Wasserstein distances against the training set on geometric properties listed in Table 2. As shown in the following table, the results indicate that the property distributions of MOF-BFN is closer to the training data compared to MOFDiff.\n\n| Model | PLD | LCD | GSA | VSA | UV | DST | AV | VF |\n|-|-|-|-|-|-|-|-|-|\n| MOFDiff   | 0.607 | 1.311 | 258.4 | 177.8 | 4737.2 | 0.059 | 1039.7 | 0.043\n| MOF-BFN   | 0.665 | 1.177 | 131.7 | 157.2 | 1626.3 | 0.043 | 885.2 | 0.022 |\n\nMoreover, we agree that enabling controlled deviation from the training distribution is also valuable. To demonstrate this, we conduct a preliminary experiment on void fraction (VF), where we implement a classifier-guided generation (CFG)-style approach [B]. Specifically, we fine-tune the model to accept an additional conditioning input $c$, where $c$ is the value of the desired value of VF. At generation time, we combine the unconditional output $x$ and the conditional output $x_c$ into $\\exp_x (w\\log_x x_c)$, where $\\exp$ and $\\log$ denote the exponential and logarithmic maps on the specific manifold (lattice representation, fractional coordinates, unit quaternions, and block embedding space), and $w$ is a weighting factor. We evaluate the guided samples based on their median VF and the Wasserstein distance from the training distribution.\n\n\n| w | Conditioned VF Value | Median | W. Dist |\n| :-: | :-: | :-: | :-: |\n| 1.0 | 0.1 | 0.091 | 0.258 |\n| | 0.4 | 0.371 | 0.076 |\n| | 0.7 | 0.629 | 0.241 |\n| 2.0 | 0.1 | 0.055 | 0.298 |\n| | 0.4 | 0.382 | 0.089 |\n| | 0.7 | 0.681 | 0.292 |\n\nAs shown above, the void fraction distribution of the generated structures shifts significantly toward the desired value under conditioning. It is also worth noting that all conditioned samples have larger Wasserstein distances than the unconditional model (0.022). and the distance grows with larger guidance weight $w$. These reflect a natural trade-off that, as we enforce stronger property control, the generated structures deviate further from the original data distribution. This suggests that MOF-BFN can not only reproduce training distributions but also be guided to explore specific property-conditioned structures.\n\n> **W4: The authors do not adequately address the limitations of their method. Given the complexity of MOF generation and the relatively modest validation rates, it is important to acknowledge the method’s current boundaries and potential areas for improvement.**\n> **Q4: What are the main limitations of your approach, particularly regarding generation validity rates and scalability? How might these limitations affect the applicability of MOF-BFN in real-world MOF design tasks?**\n\nWe appreciate the reviewer’s question. We have already provided a dedicated discussion in Appendix E, which outlines current limitations and future directions on two key aspects. First, our current framework models each MOF building block as a rigid body. This assumption simplifies the generative process and enables more stable orientation modeling, but it may neglects the intrinsic conformational flexibility of the MOF structures. Future works could extend the model to allow flexible generation of building blocks. Second, while MOF-BFN achieves strong unconditional generation performance, it does not yet fully explore property-guided generation, which is highly relevant for real-world inverse design. As a first step, we have conducted a pilot study as above (response to W2/Q2). This shows promising results, but a more systematic framework for property-guided generation remains an important avenue for future development.\n\n[A] 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\n[B] Tao, Nianze, and Minori Abe. \"Bayesian flow network framework for chemistry tasks.\" Journal of Chemical Information and Modeling 65.3 (2025): 1178-1187."}, {"review_id": "sVnWKyH2LP", "reviewer": "Reviewer_C95o", "summary": "This paper introduces MOF-BFN, the first framework for MOF structure prediction that jointly models periodicity, position, and orientation using a Bayesian Flow Network (BFN). It improves prior work by: \n1. Modeling periodicity explicitly using fractional coordinates\n2. Enhancing performance by leveraging Bayesian flow instead of flow matching \n\nIt also presents a technical contribution by extending BFN to model rotations. Experiments demonstrate superiority over existing approaches in both structure prediction and de novo generation tasks.", "questions": "1. Have the authors considered comparing the training and inference costs of MOF-FracFlow and MOF-BFN under consistent settings (e.g., batch size, GPU type, total epochs)?\n2. Could the authors give a more detailed breakdown of the model architecture? Also, how is the number of building blocks $K$ determined during de novo generation?\n3. How does the model size of MOF-BFN compare to MOFFlow and MOFFrac-Flow (e.g., number of parameters)?\n4. Line 210-211 mentions that $M_0$ is an arbitrary orthogonal matrix. Could the authors elaborate on this?\n5. While the benefits of BFN are acknowledged, could the authors comment on the potential drawbacks of Bayesian flow (e.g., stability, scalability) compared to flow matching or diffusion? This question is asked purely out of curiosity.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "# Strengths\n- This is the first work to integrate SO(3) modeling within the BFN framework, offering a novel contribution by jointly modeling periodicity, lattice, and orientation for MOFs.\n- The method supports unconditional generation and explicitly handles periodicity, both of which are not well-addressed in previous work.\n\n# Weaknesses\n- Insufficient architectural detail: Section 4.3's \"Backbone Model for MOF-BFN\" provides only a high-level overview and lacks a self-contained explanation of the model architecture. Moreover, the method for determining the number of building blocks $K$ for de novo generation is not explained in Appendix C.\n- Potential computational inefficiency: While the method shows strong empirical performance, it likely incurs higher computational costs than flow-based methods due to the need for rejection sampling and posterior simulation over non-Euclidean spaces. A quantitative comparison of resource usage (e.g., training/inference time, memory, hardware) with baselines like MOF-FracFlow is necessary to assess trade-offs.\n\n## Typos and minor issues \n- Line 44: to jointly generates\n- Line 102: Represtations -> Representations \n- Footnote in page 3: represtation -> representation \n- Line 162: Incomplete sentence: “Initialized from the prior $\\mathbf{\\theta}_0^{\\mathcal{M}}$.”\n- Line 175: Bayesain -> Bayesian", "was_revised": true, "final_justification": "The authors have addressed my concerns, so I am retaining my acceptance score. I did not raise it to 6 due to existing limitations, such as the rigid body assumption and lack of conditional generation, as noted in the Appendix.", "rebuttal": "We are grateful for the reviewer’s thoughtful comments and provide our responses below.\n\n> **W1: Insufficient architectural detail: Section 4.3's \"Backbone Model for MOF-BFN\" provides only a high-level overview and lacks a self-contained explanation of the model architecture. Moreover, the method for determining the number of building blocks K for de novo generation is not explained in Appendix C.**\n> **Q2: Could the authors give a more detailed breakdown of the model architecture? Also, how is the number of building blocks K determined during de novo generation?**\n\n\nWe thank the reviewer for pointing this out. The backbone model contains two parts. The building block encoder is a strateforward implementation of EGNN without any changes. The coarse-grained structure predictor is adapted from CSPNet proposed in DiffCSP. Specifically, let $h\\_i^{(s)}$ denote the node representation of node $i$ at layer $s$. The input encoding is given by $h\\_i^{(0)}=\\rho(\\mathcal{E(C\\_i)}, M, \\kappa\\_i, \\lambda\\_i, f\\_{pos}(t))$, where $f\\_{pos}$ is the time encoder and $\\rho$ is an MLP to fuse all input features. The message passing process is followed by\n$$m\\_{ij}^{(s)}=\\varphi_m(h\\_i^{(s-1)}, h\\_j^{(s-1)}, \\xi, \\psi\\_{FT}(f\\_i-f\\_j))$$\n$$h\\_i^{(s)}=h\\_i^{(s-1)}+ \\varphi_h( h\\_i^{(s-1)}, \\sum_{j\\in\\mathcal{N}(i)}m\\_{ij}^{(s)}),$$\nwhere $\\varphi\\_m, \\varphi\\_h$ are MLPs and $\\psi\\_{FT}$ is the Fourier embedding to capture periodicity. The final outputs are given by\n\n$$\\hat{\\xi}=\\varphi\\_{\\xi}(\\sum\\_ih\\_i^{(S)})$$\n$$\\hat{f}\\_i=f\\_i+\\varphi\\_{f}(h\\_i^{(S)})$$\n$$\\hat{q}\\_i=\\text{Normalize}(\\varphi_{q}(h\\_i^{(S)})),$$\n\nwhere $\\varphi_{\\xi}, \\varphi_{f}, \\varphi_{q}$ are MLPs, $\\hat{\\xi}, \\hat{f}_i, \\hat{q}_i$ are the model output for each feature, and $S$ is the number of layers. We will provide more details in the appendix for better understanding.\n\nRegarding the choice of the number of building blocks K during de novo generation, we follow a standard approach used in molecule and crystal generative models, that is collecting the empirical distribution of K from the training set and sampling from it during generation. We will highlight this in the revised version.\n\n> **W2: Potential computational inefficiency: While the method shows strong empirical performance, it likely incurs higher computational costs than flow-based methods due to the need for rejection sampling and posterior simulation over non-Euclidean spaces. A quantitative comparison of resource usage (e.g., training/inference time, memory, hardware) with baselines like MOF-FracFlow is necessary to assess trade-offs.**\n> **Q1: Have the authors considered comparing the training and inference costs of MOF-FracFlow and MOF-BFN under consistent settings (e.g., batch size, GPU type, total epochs)?**\n\nThank you for raising this question! To assess the additional computational cost introduced by BFN, we have additionally conducted a direct comparison between MOF-FracFlow and MOF-BFN under the same training and inference settings. Both models are trained on 8 GPUs (each with 80 GB memory) using a batch size of 512 for 1000 epochs. Inference is performed on a single GPU, with a batch size of 512 and 50 sampling steps. The GPU hours consumed during training and inference are summarized below:\n\n| Model | Training GPU hours | Inference GPU hours |\n|-|-|-|\n| MOF-FracFlow   |  135.7  | 0.22   |\n| MOF-BFN   | 137.5   | 0.50   |\n\nThe additional cost of MOF-BFN arises primarily from posterior simulation on non-Euclidean manifolds. We further explore the efficiency of the rejection sampling detailed in Appendix B.2. Specifically, we sample 10,000 unit quaternions from a Bingham distribution with parameter matrix $\\Lambda=-diag(0,\\lambda, \\lambda, \\lambda)$ and measure the total GPU time as well as average number of rejection sampling iterations required per sample. The results are shown as follows.\n\n| $\\lambda$ | GPU seconds | Avg. Iteration |\n|-|-|-|\n| 1 | 0.023 | 1.41 |\n| 10 | 0.028 | 1.90 |\n| 100 | 0.030 | 2.20 |\n| 1000 | 0.032 | 2.24 |\n\nNotably, rejection sampling is parallelizable on GPU and does not involve any model forward, making it acceptable for training and inference.\n\n> **Q3: How does the model size of MOF-BFN compare to MOFFlow and MOFFrac-Flow (e.g., number of parameters)?**\n\nAll three models use the same EGNN-based block encoder, so differences in model size come primarily from the coarse-grained structure predictor. MOFFlow employs the OpenFold backbone, whereas MOF-FracFlow and MOF-BFN use the CSPNet backbone, which is coupled with the selections of the coordinate system. The number of parameters are detailed as follows:\n\n| Model | # of params (M) |\n|-|-|\n| MOFFlow | 22.5 |\n| MOF-FracFlow | 11.2 |\n| MOF-BFN | 11.2 |\n\nThis shows that MOF-BFN achieves better performance with fewer parameters compared to MOFFlow.\n\n> **Q4: Line 210-211 mentions that $M_0$ is an arbitrary orthogonal matrix. Could the authors elaborate on this?**\n\nThanks for the question. Theoretically, when the Bingham concentration matrix is initialized as $\\Lambda=0$, the matrix $A_0=M_0^\\top\\Lambda M$ becomes a zero matrix regardless of the choice of orthogonal matrix $M_0$, resulting in a uniform distribution over the unit sphere. In practice, we initialize $M_0$ by performing `torch.linalg.eigh` on the zero matrix, which yields the identity matrix $M_0=I$. We will clarify this implementation detail in the revised version to avoid confusion.\n\n> **Q5: While the benefits of BFN are acknowledged, could the authors comment on the potential drawbacks of Bayesian flow (e.g., stability, scalability) compared to flow matching or diffusion? This question is asked purely out of curiosity.**\n\nThank you for this insightful question. While Bayesian Flow Networks offer strong performance, we acknowledge that there are still open challenges and potential limitations compared to alternative generative models. One key drawback is that Bayesian updates must be explicitly designed for each target distribution, which can require nontrivial mathematical derivations, especially on non-Euclidean manifolds. There is currently no general framework for constructing BFN updates on arbitrary manifolds. We believe this is an exciting research direction for the future to broader the generalizability of BFNs.\n\n> **Typos and Miner Issues**\n\nThanks for pointing these out. We will correct these in the revised version."}, {"review_id": "POjWmijFhT", "reviewer": "Reviewer_7Smn", "summary": "The authors of this paper address the challenging task of structure prediction for Metal-Organic Frameworks (MOFs). They highlight the intrinsic complexities of the problem and the limitations of existing models such as CDVAE, DiffCSP, MOFFlow, and CrysBFN. To overcome these challenges, they propose Bayesian Flow Networks (BFNs) and introduce MOF-BFN, a novel generative framework for hierarchical MOF structure prediction based on this approach.", "questions": "Check weaknesses", "limitations": "Check weaknesses", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strength:**\n - The paper is well-written and easy to follow, with a clearly motivated problem and sufficient background information to support the study.\n- Overall methodology is novel. MOF-BFN operates in the fractional coordinate system to preserve periodicity. Also they models block orientations using unit quaternions sampled from Bingham BFNs—a novel generative module.\n- The experimental setup of the paper is solid, Authors have experimented with \tstate-of-the-art models on benchmark material generation tasks.\n- Source code is available\n\n**Weakness:**\n- Many relevant works on Crystal generations are missing FlowLLM, TGDMat, SymmSD, DiffCSP++\n- A recent ICLR work, TGDMat[1], has demonstrated promising performance on the crystal structure prediction (CSP) task using text guidance with a single sample in a diffusion model. Could you please evaluate your model's performance under similar text-guided settings, or alternatively, report results on the BW-DB dataset using TGDMat (Long) for a fair comparison?\n \t\t\t\t\t\t\n[1] https://arxiv.org/abs/2503.00522", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the innovative questions, and answer them as follows.\n\n> **W1: Many relevant works on Crystal generations are missing FlowLLM, TGDMat, SymmCD, DiffCSP++.**\n\nThanks a lot for pointing out the missing related works! We agree that these works explore promising directions for inorganic crystal generation. Specifically, FlowLLM initializes flow matching process with a LLM-based prior. TGDMat innovatively introduces text conditions to the generative model. DiffCSP++ considers space group-based generation given specific Wyckoff position (WP) assignments, and SymmCD extends this by further enabling the generation of WPs. However, MOF generation is slightly distinct from the generation of inorganic materials, where the hierarchical modeling and the orientation determination are the key problems, which are not directly addressed by the aforementioned methods. We will provide more discussions on these works in the revised version.\n\n> **W2: A recent ICLR work, TGDMat, has demonstrated promising performance on the crystal structure prediction (CSP) task using text guidance with a single sample in a diffusion model. Could you please evaluate your model's performance under similar text-guided settings, or alternatively, report results on the BW-DB dataset using TGDMat (Long) for a fair comparison?**\n\nThanks for the thoughtful suggestion. We agree that text-guided generation is an exciting direction, and TGDMat makes an important step toward integrating language models with crystal structure prediction. However, we would like to clarify that TGDMat is designed for full-atom generation of inorganic crystals, whereas our work focuses on hierarchical generation of MOFs, which presents very different challenges. In particular, results in Table 1 already include DiffCSP, a representative full-atom diffusion model. Its poor performance highlights the limitations of non-hierarchical methods when applied to complex systems like MOFs.\n\nMoreover, adapting text-guidance to MOF data faces practical challenges. TGDMat relies on RoboCrystallographer [A] to generate long descriptions. Applying this tool to MOFs is difficult in practice, as a single unit cell can contain hundreds of atoms, leading to long generation times (from tens of seconds to several minutes per sample) and resulting descriptions often exceed thousands of tokens. This makes it infeasible to apply to the full BW-DB dataset, which contains over 300,000 structures. In addition, current text representations do not fully capture the hierarchical nature of MOFs. \n\nWhile we believe text-guided MOF generation is a valuable and promising research direction, a systematic formulation of the description construction, benchmark design, and more reasonable evaluation metrics are required to fully tackle this problem. These are non-trivial and orthogonal to the main scope of this work, and we consider them important directions for future research.\n\n[A] Ganose, Alex M., and Anubhav Jain. \"Robocrystallographer: automated crystal structure text descriptions and analysis.\" MRS Communications 9.3 (2019): 874-881."}]

Author rebuttal:

Meta-review: The paper introduces MOF-BFN, a generative model for metal-organic framework (MOF) structure prediction based on Bayesian Flow Networks (BFNs). Its key contributions lie in representing block positions in fractional coordinates to capture periodicity and modeling orientations with unit quaternions based on proposing Bingham BFN. The method jointly predicts lattice parameters, positions, and orientations, achieving state-of-the-art performance in structure prediction, geometric property evaluation, and de novo generation. 

The main concerns raised were (a) limited ablation studies and (b) questions about the broader applicability and significance for MOF researchers. The authors have resolved these concerns in the rebuttal and the reviewers support acceptance of this paper. 

Overall, I recommend acceptance for this paper. It makes a clear methodological advancement and demonstrates strong experimental evidence.
Decision: Accept (poster)