Geometric Algebra-Enhanced Bayesian Flow Network for RNA Inverse Design
Authors: Rubo Wang
Xingyu Gao
Peilin Zhao
Venue: NeurIPS 2025 poster ()
Keywords: Geometric algebra
Bayesian flow network
RNA inverse design
Primary area: machine_learning_for_sciences
TL;DR: More accurate RNA inverse design model

Abstract: With the development of biotechnology, RNA therapies have shown great potential. 
However, different from proteins, the sequences corresponding to a single RNA three-dimensional structure are more abundant. Most of the existing RNA design methods merely take into account the secondary structure of RNA, or are only capable of generating a limited number of candidate sequences. 
To address these limitations, we propose a geometric-algebra-enhanced $\textbf{B}$ayesian $\textbf{F}$low $\textbf{N}$etwork for the inverse design of $\textbf{R}$NA, called $\textbf{RBFN}$. RBFN uses a Bayesian Flow Network to model the distribution of nucleotide sequences in RNA, enabling the generation of more reasonable RNA sequences. Meanwhile, considering the more flexible characteristics of RNA conformations, we utilize geometric algebra to enhance the modeling ability of the RNA three-dimensional structure, facilitating a better understanding of RNA structural properties. 
In addition, due to the scarcity of RNA structures and the limitation that there are only four types of nucleic acids, we propose a new time-step distribution sampling to address the scarcity of RNA structure data and the relatively small number of nucleic acid types. Evaluation on the single-state fixed-backbone re-design benchmark and multi-state fixed-backbone benchmark indicates that RBFN can outperform existing RNA design methods in various RNA design tasks, enabling effective RNA sequence design.

Geometric Algebra-Enhanced Bayesian Flow Network for RNA Inverse Design
Rubo Wangh,2 Xingyu G Zhao²
Institute of Microclectronics, Chinese Academy of Sciences, Beijing, China University of Chinese Academy of Sciences, Beijing, China
3 School of Artificial Intelligence, Shanghai Jiao Tong University, Shanghai, China
wangrubo hoommill.ommm gxy9910 11y9900@gmiil.comm gmmil.com, peilinzhao @sjtu.edu.cn
Abstract
With the development of biotechnology, RNA therapies have shown great potential. However, different from proteins, the sequences corresponding to a single RNA three-dimensional structure are more abundant. Most of the existing RNA design methods merely take into account the secondary structure of RNA, or are only capable of generating limited number of candidate sequences. To address these limitations, we propose a geometric-algebra-enii the inverse design of RNA, called RBFN. RBFN uses Bayesian Flow Network to model the distribution nucleotide sequences in RNA, enabling the generation of more reasonable RNA sequences. Meanwhile, considering the more flexible characteristics of RNA conformations, we utilize geometric algebra to enhance the modeling ability of the RNA three-dimensional structure, facilitating better understanding of RNA structural properties. addition, due to the scarcity of RNA structures and the limitation that there only four types of nucleic acids, we propose a new time-step distribution sampling to address the scarcity of RNA structure data and the relatively small number of nucleic acid types. Evaluation on the single-state fixed-backbone re-design benchmark and multi-state fixed- backbone benchmark indicates that RBFN can outperform existing RNA design methods in various RNA design tasks, enabling effective RNA sequence design.
Bayesian Flow Network for
Introduction
Ribonucleic acid (RNA), as one of the fundamental biomolecule in life activities, not only undertakes function of genetic information transmission (such as mRNA, IRNA, and rRNA), but also participates in complex biological processes such as gene expression regulation, protein synthesis, and cell signal transduction through non-coding RNAs (such as miRNA, siRNA) [1, 2, 3, 4, 5]. The realization of its functions highly depends the secondary structures (such as stem-loops, pscudoknots) formed by its dynamic folding and the tertiary spatial conformation. RNA can be used for drug target screening [6], vaccine development (such as the COVID-19 mRNA vaccine [7]), and synthetic biology circuit design [8]. In addition, the application of RNA design in disease treatment is becoming increasingly widespread. For example, pathogenic genes be silenced through siRNA [9], metabolic pathways [10] can be regulated by designing riboswitches [1 []. and highly sensitive biosensing can be achieved by using aptamers.
Despite RNA's critical biological significance, structural modeling research remains predominantly
protein-centric, as evidenced by recent advances in geometric deep learning [12, 13]. This disparity arises from two key factors: there are large number of protein three-dimensional structures in the
*Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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PDB, and the development of some breakthrough protein algorithms [14, 15] has further promoted the development of the protein field, especially the emergence of AlphaFold2 [16]. However, for RNA, there are not many three-dimensional structures. Although some RNA structure prediction methods have emerged currently, due to the limitation of data, they cannot achieve a very high accuracy, which makes RNA design more difficult, and deep learning has not been fully utilized in RNA design. Most of the existing RNA design tools only focus on the secondary structure of structure, most of them need manually construct features [18, 19], making the design inefficient and complex. addition, since the functions of RNA are more complex compared proteins, the conformation of RNA is relatively flexible, and the same RNA may correspond to different conformations [20, 21, 22]. This makes necessary to consider more factors when designing RNA compared to designing proteins, and is difficult directly transfer relevant algorithms. At present, certain approaches leveraging graph neural networks to model RNA for design have emerged [23, 24]. However, the majority of these methods are adapted from the protein domain and do not adequately account for the unique characteristics of RNA structures and sequences. Specifically, for dimeesiimeiii structure of RNA, there exists corresponding sequences [19], which leads to certain limitations in practical applications.
RNA [17]. and for the three-dimension
diverse set of
To overcome these limitations, this paper presents RBFN, eeeomttricalleeba-enbieee Flow Network designed for RNA inverse design. RBFN leverages geometric algebra to improve the modeling of RNA three-dimensee dineessinnsi structure information. Simultancously. it employs the Bayesian Flow Network to sample sequences within the parameter space (distribution). can generate corresponding nucleotide sequences given one more 3D backbone structures. The main contributions RBFN include the following aspects:
Bayesian
Modeling structures using geometric algebru. To the best of our knowledge, RBFN is the first approach to utilize geometric algebra to enhance RNA structure modeling. Using geometric algebra can effectively incorporate structural information into the scalar features of nucleotides, thus achieving more effective modeling capabilities.
Generation using Bayesian Flow Network. RBFN uses the Bayesian Flow Network generate nucleotide sequences, directly learning the distribution of nucleotides at each position and achieving sequence generation by aligning distributions, rather than operating on discrete sequences.
Proposing a new time-step sampling distribution. In view of the fact that there are only four types of nucleotide sequences, new time-step distribution is proposed, enabling the network to pay more attention to the samples of the sender distribution, so as to enhance the network's global generation ability.
Higher performance in RNA design. We compare RBFN with the state-of-the-arr deep learning method gRNAde and the sfate-of-the- art physics- based tool Rosetta. The experimental results show that the sequences generated by RBFN have higher sequence recovery rate, indicating that RBFN can better learn the relationship between structures and sequences.
2
Background
In this section, introduce the definition of the problem in Section 2.1, introduce geometric algebra in Section 2.2, and introduce Bayesian Flow Network [25] in Section 2.3.
2.1 Problem Definition
RNA inverse design can be formulated as a conditional generation task: design the corresponding RNA sequence under the condition of a given backbone struc- ture. Figure shows the backbone strue- ture of RNA. RNA is composed of four nucleotides, namely adenine (A), cytosine (C), guanine (G), or uracil (U). We fol- low the representation method of existing work [24] and represent RNA in the form of Graph. In the backbone atoms of RNA, C4', N1 (pyrimidine) or N9 (purine) are
OF
CH
CI'
RNA Backbone
RNA Backbone
Figure 1: Schematic diagram of RNA backbone struc- ture. RNA is composed of four basic nucleotides. The backbone atoms of each nucleotide can be represented by the structural form the right.

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retained as backbone atoms. With this representation, the RNA backbone can generally be fully rep- resented, and the same time, it can prevent representation problems caused by excessive torsional space. Each nucleotide i serves as a node, and the three-dimensions coordinate 7. E R$ is the center of the backbone atoms of the entire nucleotide. When given three-dimensional backbone structure of RNA, the goal is to design nucleotide sequence that can fold into this three-dimensiona structure.
2.2 Geometric Algebra
Geometric algebra refers to the Clifford algebra over real vector space. It can describe objects such points, lines, and planes and operations on them in an algebraic form, and is a very powerful mathematical framework. Algebraic elements are divided into homogeneous components according to their orders: k-vector is linear combination of k-blades (basis elements of grade k). For example, 1-vector (i.e., the conventional vector) is composed of I-blades, and a bivector composed of a linear superposition of 2-blades. More generally, multivector E Gy.g. is the direct sum of subspaces of all grades, Gp.g. denotes a geometric algebra over a real vector space with basis vectors of positive signature (ef = +) g of negative signature (eî = 1), and r of zero signature (c) = 0), satisfying
p + + r = n for an n-dimensional space. And can be expressed as x = Ex= x*|=> where [x] represents the homogeneous component of grade k. Taking the vireeddimeniionll geometric algebra G3o0,0 as an example, its 8 basis blades correspond to a complete component decomposition:
x To 1o.1- rie1 + 1262 + 2363 + T12612 + 313613 + #23623 T1236123
(1)
Scalar
Vector
where eyy = = (i < j) and 6123 = 61 A e2 A eg. The metric signature (p,g,r) = (3,0,
indicates three basis vectors with e² = +1, none with o² = -1, and no null vectors (eī = 0). Anticommutation (etej- = =Ese; for i + j) ensures closure under the geometric product, products of basis blades reduce to linear combinations of basis blades. By choosing different parameters (p, 9. r). geometric algebra can efficiently model various geometric spaces such as Euclidean spaces. Its hierarchical structure provides a unified mathematical framework for the algebraic operations of geometric objects. We provide more detailed introduction to geometric algebra in Appendix B.1.
2.3 Bayesian Flow Network
Bayesian Flow Networks (BFNs) [25] provide an innovative approach for generative modeling of discrete data. When dealing with discrete data, BFNs model the evolution of probability distributions through a continuous- time transport process, while maintaining mathematical rigor and computational efficiency. For discrete data with K categories, BFNs start from simple input distribution qo (a()), which is chosen as a categorical distribution:
qo qo(x) = Categorical(z//o))
(2)
where To R is probability vector satisfying Σ k= Tok 1 and Tok > O. BFNs introduce continuous- time parameter E [o, 1] and define two key distributions: (1) the sender distribution (ax.) representing the data distribution at time and (2) the receiver distribution qe (2xa az)) describ- ing the forward process from the original data Zpp to the corrupted data &t. The core of BFNs is the Bayesian inversion mechanism, where the reverse process is given by:
qe ( Zo (x: (xo)
4: (z..)
Σ
qe (x:))
To make this process learnable, BFNs use a neural network to approximate the reverse process qe (2ol (z)) Specifically, the network outputs logits z = NNo( t), which are converted to probabili- ties via:
(Zo (z.) Categorical ssoftmax2))
(4)
The training objective minimizes the KL divergence between true and predicted reverse processes:
(())= [e=- (- (-120) KL (gx [a=:)]l(x
(5)
For discrete data, BFNs do not use SDEs but instead employ deterministic transport process. The marginal distribution evolves as:
n (1 t) + two
(6)
where कo is the empirical data distribution. We provide more detailed introduction to Bayesian Flow Network in Appendix B.2.
3

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A
Geometric information extraction
A9
M
X3
X3
Glycosides
Geometric Representation
Sender Distribution
Receiver Distribution
Geometric Algebre
Bayesian Update
GNN
Sample timestep(t)
Noisy channel
A
C
A
C
U
Sender Data
Timestep Schedule
Input
Accuracy scheduler
Sender Distribution
Figure 2: The RBFN pipeline for 3D RNA inverse design. A. The 3D structure of RNA can be split into the splicing of individual nucleotide structures. For each individual nucleotide, using geometric algebra, it can simultaneously be interpreted as group elements (0-vector, 1-vector, 2-vector, 3-vector). The structure of nucleotides is derived from [26]. B. The framework of RBFN. RBFN uses Geometric Algebra GNN as the backhone network, and learns the sequence distribution alignment under the RNA backbone structure through the Bayesian Flow Network. The entire update process uses KL to align the sender distribution and the receiver distribution.
Methods
We propose a geometric algebra-enhanced Bayesian Flow Network for learning the sequence distri- bution under the condition of the structure, enabling sequence design given the three-dimensional structure of the RNA backbone. The proposed method is an extension of the Bayesian Flow Net- work. We use geometric algebra-enhanced graph neural network enhance the modeling of RNA dineesininiii structures. At the same time. based on the characteristics of the Bayesian Flow Network, we directly learn the sequences distribution correspondence for structures. In addition, based on the characteristics of RNA sequences, we propose time-step sampling method to better train the Bayesian Flow Network.
3.1 Geometric Algebra GNN for RNA Three-Dimensional Structure Modeling
As shown in the top Figure 2, based the three-dimensional structure of RNA, we isolate each nucleotide as a structural unit. For nucleotide i, the phosphate atom (P;), C4' atom of ribose, glycosidic bond atom (NI for pyrimidines N9 for purines), and the 5' adjacent phosphate (P:-1)
form tetrahedral geometry. Using these four atomic coordinates (ar (244) (NN/N9) (PP)) _ we
construct multivector M, E G3.000. where: (1) the O-vector encodes P,-C4* distance; (2) 1-vectors
represent bond directions (e.g., 2t ( (3) 2-vectors capture oriented planes (e.g., PP- plane); (4) the 3-vector represents the tetrahedron's oriented volume. Next, we construct a nucleotide graph where node i is positioned at xx(P) E R3, and edges connect to the 32 nearest neighbors (selected to cover I helical turn in A-form RNA, 11 radius). We expand on [24, 27] by integrating geometric algebra operations into GNN message passing: scalar features in store O-vector components, while vector neurons process 1/2/3-vector components for SE(3)-equivariant updates. Specifically, the architecture comprises an encoder that processes multivector features through geometric algebra GNN layers, and a decoder that predicts nucleotide identities from the learned representations. Due to

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RNA's conformational diversity, we implement a multi-state encoder following [24], which processes ensembles of backbone conformations to capture structural variability.
For an RNA structure with m conformations and n nodes, consider this structure as multiple graphs. i.e., {g()) {g(").... ......!m!) For each conformation graph g (A s ()) where c E (I...., m). it contains scalar features, namely the index and sequence representation of nu- cleotides RNA S E Rnxmxf, and vector features X E R=xmx x33; (a) the forward and backward unit vectors along the backbone from the 5' end the 3' end, (Æz+1 E; and क; - and (b) the unit vectors, distances, angles, and torsions from each C4' to the corresponding and N1/N9. The edge features (A(),.... A (m)) from node j to are initialized as follows: (a) the unit vector from the source node to the target node, E5 Hi; (b) the distance in dine-dimeesiooal space, || E.11.. encoded by 32 radial basis functions; and (c) the distance along the backbone, j - i, encoded by 32 sine-position encodings, and finally represented as V()) E R2 / x3 The geometric algebra feature q(v) is obtained based on the three-dimensional coordinate relationships. For a multivector v = 8 + 0 + b + dI where s is scalar, J vector, b bivector, and (I pseudoscalar components, the quadratic form is q(v) x² + ||o| |[b|| ( in G3,0,0- Following [28, 27J, an MVP-style message passing layer is defined as:
Φe ((v), v']))
(7)
5/5 = de (1)),,
(8)
(st) v((y) (MVP-GP, MVP-LLn,,) (g)).vb)
(9)
(o% v(') VIVI
Σ
(10)
VIVI
JEN,
JEN.
(of (2) = (MVP-GP, o MVP-Lin,) (166 1. w!))
(11)
Details can be found at Appendix B.I.
3.2
Bayesian Flow Networks for RNA Generation
The overall architecture of RBFN is shown in Figure 2. Before training, we define a uniform prior over the 4 RNA nucleotides: Po(y) Categorical To) with To [1/4, 1/4, 1/4, T/4]". During training, for each RNA backbone graph g(e) (9(), s(e), V(e)) with n nucleotides, the forward process corrupts the ground-truth sequence y" E (1, 3, 4)"H to a noisy version y: at time t~ Z00 1] via:
(vrl3")
II [(1 a))mm + a..o..... 1=1
(12)
where a₂ t is the noise level, and 8 is the Kronecker delta. The sender distribution modeled by geometric algebra GNN that takes as input: (1) the noisy sequence representation S; E m (one-hot of yt), and (2) structural features from gle). The network outputs logits z = (S, t),
which parameterize the receiver distribution:
po(y" We, g(o)
II
Categorical (y." [softma(2:))
(13)
1=1
The training objective minimizes the KL divergence between true and predicted reverse processes:
n Σ
(())=
Er, .Be
KL (g:(-1)) softmax(z:))
(14)
At inference, starting from 8/1 Po. we iteratively sample yt from (e Mt+ g(e)) to generate the final sequence ÿ = you The geometric algebra features (0-3-vectors from tetrahedral geometries) are integrated into the GNN's vector neuron layers to preserve SE(3) -quivvaaave during message passing.
The Bayesian update function can be expressed as S (Vi-1,8,a) = Ei ve
pF(v* h;;t)
E
8(V5.- -oftmaxxx) (15)
-1), 10)
5

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With the help of the geometric algebra GNN v, the output distribution can be obtained through the following expressions:
- (k] v²;6) t) (oftmax (1("(1²)))) i: (o v;() II
Po ia) (s(a) vº:1)
(16)
Based on the formula derivation of [25], the unified continuous-time loss is obtained as:
E' tiles - e L (S, = 311(1) (-((0.1).
(17)
where
(18)
3.3 Time Sampling Distribution
Similar to diffusion models and flow matching, the Bayesian Flow Network needs to sample time steps as the basis for Bayesian updates. For continuous time, time can be sampled arbitrarily the interval (0-1). Previous works [25, 29, 30] directly used uniform distribution to sample from 0 to In images, enhanced sampling of t closer to t = prompts the model to focus its capabilities on synthesizing accurate local details, which are generated at the end of the generation process. More sampling at smaller t. values can improve large-scale features. In image generation, sampling is usually increased at the intermediate t [31, 32]. However, for RNA inverse design, since there only four nucleotides (A, C, G. U), slightly adding sender noise generally has little impact on the sampled sequence. To make the training process pay more attention the transition from the initial distribution to the true distribution, we designed a new / sampling function that pays more attention to smaller / values, that is, it is more inclined to the sender distribution during the Bayesian update process. The specific formula is as follows:
p(() 0.0 1) + 0.98 ((1.,,
where ((,,)) -) is the beta distribution, to promote the generation of accurate local details. We incorpo- rate uniform sampling to avoid the sampling density being zero when 1. The visualization is shown in Figure 3.
4 Experiments
4.1
Experimental Setup
Dataset. We use RNASolo [33] for training, validation, and testing. It contains all RNA-containing structures obtained from the Protein Data Bank (PDB). Following the partitioning in [24], we use RNA structures with resolution of < 4.0 Á. In total, 12,011 structures 31 October 2023). We perform structure clustering using U.S-align [34] with similarity threshold of TMMscore 0.45, and sequence clustering for RNA sequence homology using CD-HIT [35]. A total of 4~000 samples are obtained and divided according to different tasks. For the Single-state split, we use the partitioning method in [36] to identify the structural clusters (including riboswitches, aptamers, and ribozymes) belonging to the RNAs identified in [36]. A total of 100 sequences are obtained for testing. Based on the remaining sequences, 100 are randomly selected as the validation set, and the rest are used as the training set. This benchmark is called the Das Benchmark. For the Multi-state split, we calculate the pairwise C4' RMSD of the structures corresponding to each sequence. The top 100 samples from clusters with the highest median intra-sequence RMSD are added to the test set, which called the Multi-state Benchmark. The next 100 samples are added to the validation set, and the rest are used as the training set for training.
available (RNASolo data cutoff:
Baselines. For the Das Benchmark, we compare with traditional methods and deep learning methods. The methods based on 2D structure: ViennaRNA [37], FARNA [38], the method of the physics-based toolkit for biomolecular modelling and design, Rosetta [39], the latest deep learning-based methods RDesign [23], gRNAde [24], RhoDesign [40], RiboDiffusion [41]. For the Multi-state Benchmark, since traditional methods do not support Multi-state, we only compare the deep learning-bai based methods RDesign [23], gRNAde [24].
Evaluation Metrics. Similar to the assessment in gRNAde as described in the reference [24], for a particular data partitioning, we assess the models on the reserved test dataset. For each test data point, design 16 sequences. Then, calculate the averages for several metrics:

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Table I: Results of the comparison between RBFN and RiboDiffusion, EhoDesign, gRNAde, RDesign, Rosetta, FARNA, as well as ViennaRNA, were obtained for the single-state design on 14 RNA structures of interest, which were identified as described in [36].
PDB ID
Description
ViennaRNA
FARNA
RDesign
Rosetta
gRNAde
RhoDesign
RBFN
1CSL
RRE high affiniyy
0.4455
0.5719
04644
0.4832
12T4
Vitamin B12 binding RNA splamer
0.25
0.34
0,3929
0.44
0.6250
0.5428
0,4482
1F27
Biotin binding RNA pscudoknot
0.30
0.36
0.3013
0.37
0.3437
0.4211
0.5208
11.2X
Viral RNA pscudokant
0.24
0.45
0.3727
0.48
0.4721
0.2593
0.5833
1LET
RNA internal loop SRP
0.33
0,27
0.5556
0.53
0.5843
0.3636
0.6375
1Q94
Sarcinhncin domain from E.coli 238 IRNA
0.27
01.40
0.4417
0.41
0,5044
0.8148
0,8588
Guanine riboswitch sptamer
0,8209
0.5326
1X9C
AIIANAA huirpin ribuzyme
0.26
0.31
0.3967
0.50
0,5000
0.3833
0.5187
1XPE
HIV-1 B RNA dimerization initiation site
0.27
0.24
0.3834
0.40
0.70337
0.6957
0.6522
2003
cleewee state of glmS ribozyme
2GD1
Thiamine psmphhospaate-speiin
rihowwithh
0.25
0.38
0.3523
0.48
0.6500
0.2436
0.6042
2060
lunctionless hairpin ribozyme
0.23
0.30
0.5000
0.37
0,9519
0.1905
0.5580
2883
Tetrahymena ribozyme P4-P6 domain
0.27
0.5641
0.53
0.6415
0.7172
354D
Loop E from E. coli 5S ARNA
0.28
0.35
0.4458
0.55
0,4410
0.8261
0.8031
Overall recovery:
0,4909
0.6012
Table 2: Performance of other indicators in the inverse design of Single state RNA. Here, for the
samples in the test set, 3 consistent random seeds are used to ensure the accuracy of the results, and comparisons made with different versions of gRNAde [24].
Self-consistency metries
Max. train
Perplexity
2D-EtermaFood
3D-RhoFold
Model
length
(L)
scMCC (f)
scRMSD (U)
scTM-sccre (?)
scGDT_TS (t)
gRNAde(AR)
500
1.77++0.07
0.624+00007
13.01+1.18
0.21+000
0.22+00.0
gRNAdc(AR)
1000
1.73:t0.08
0.648=0.01
13.10=0.58
0.20+0.0
0.21+00.0
gRNAdc(AR)
2500
1.41+00001
0.633-0003
11.76=0091
0.27=00.
0.27+0000
gRNAde(AR)
5000
1.29100.02
0.585+0003
11.70=00.56
0.26110.0
0.25:00.0
gRNAde(NAR)
5000
1.46::0.06
0.473=0.02
13.04=0.88
0.23=000
0.22+00.0
RBFN
5000
1.16.10.01
0.50610.01
10.8
0.23.10.02
0.31.10.03
Groundruth sequence prediction:
0.686+0000
5.23++0.07
0.56+000
0.55+000
Random sequence prediction:
0.012=0.00
24.40=0.34
0.04=0.0
0.02+0000
ViennaRNA 2D-only:
0.611=0.00
20.34=0.10
0.07=000
0.07+00.0
Native Sequence Recovery: represents the average proportion of native (ground truth) nucleotides restored in the output amino acid sequence.
Perplexity: It reflects the degree of certainty of the model when outputting specific sequence and serves as an indicator to measure the model's certainty ability.
Secondary Structure Self-consistency Score: We secondary structure prediction tool (specifically EternaFold [42]) to perform "forward folding" on the sampled sequences. Then, we measure the average Matthews Correlation Coefficient (MCC) between the predicted secondary structure (represented as a binary adjacency matrix) and the true secondary structure. The MCC value ranges from -1 to +1. A value of +1 indicates perfect match, value of 0 indicates an average random prediction, and a value of indicates an inverse prediction. This metric evaluates the extent to which the design can restore the base-pairing pattern.
Tertiary Structure Self-consistency Score: We use a dimee-dimeiiional structure prediction tool (RhoFold [43]) to perform "forward folding" on the sampled sequences. Subsequently, we calculate the average Root-Mean-Squaa
Deviation (RMSD), Template Matching Score (scTM-score), and Global Distance Test Total Score (eGGDT_SS) values relative the true C4' coordinates. These calculations help us determine the extent to which the design can restore global structural similarity and three-dimensional conformation.
As evaluated [44], current nucleic acid structure prediction tools are not as accurate as protein prediction tools. Here, we focus on comparing Native Sequence Recovery. Training details can be found in Appendix C.I, and we choose 50 steps during sampling as described in Appendix C.3, and the specific sampling methods can be found in Appendix C.2 and Algorithm 1.
4.2 Single-state RNA design benchmark
We conduct a comprehensive comparison of RBFN against state-of-the-art methods on the Das Benchmark. The benchmark comprising RNA structures extracted from the PDB database comprchensively represents diverse RNA functional categories.
7

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In the single-state RNA design task, RBFN demonstrates superior performance over competing approaches. As shown in Table 5, RBFN achieves an average recovery rate of 60%, outperforming gRNAde (56%), RhoDesigg(499)) Rosetta (45%), FARNA (32%), ViennaRNA (27%). and RDesign (43%) by significant margins. Notably, RBFN attains the best performance in 6 out of 14 cases, while maintaining consistent performance across different instances-a stark contrast to gRNAde's fluctuating results, thereby demonstrating strong generalization capabilities, A Diffusion version of RBFN was also implemented, with detailed analysis provided in Appendix C.4. Regarding additional performance metrics Table 2, RBFN demonstrates significantly lower perplexity, indicating higher confidence in sequence generation. Although its secondary structure preservation score slightly lags behind a specific gRNAde configuration, RBFN leads in key evaluation metrics. More importantly, it shows exceptional 3D structure restoration ability with scRMSD of 10.83 (8% improvement over baseline). Particularly noteworthy is its ScGDTT _TS score of 0.31, which is 15-fold higher than random sequences and reaches 56% of the performance level of real sequences, providing strong evidence for the structural validity of generated sequences.
A critical observation is that when using native RNA sequences for structure prediction, the TM- Score between predicted and actual structures only reaches 0.56-significantly lower than current benchmarks in protein structure prediction. While this metric doesn't 't represent the best performance, the results still surpass existing prediction methods, confirming a meaningful correlation between sequence design and structural accuracy. Representative folding examples of generated sequences are presented in Appendix Figure 4, where shorter sequences show particularly superior folding behavior compared to longer ones.
4.3 Multi-state RNA design benchmark
Structured RNAs often perform biological functions through multiple distinct conformations [22]. Given that traditional methods and RDesign lack multi-state capabilities, we focus comparison on gRNAde, which supports multi-state input and diverse output configurations. Experimental results in Table 3 demonstrate RBFN's superior performance in multi-state RNA design across various gRNAde configurations. While gRNAde's ' autoregressive (AR) models with Deep Symmetric Set pooling (DSS) achieve competitive perplexity scores (1.37 for 3-state design), RBFN exhibits superior structural consistency metrics-pant partticc--p in 3D-RhoFold's scT'--score (0.14 vs 0. and ScGDT_ TS (0.17 vs 0.15) for single-state designs. Notably, RBFN's native sequence recovery rate shows a monotonic improvement with increasing conformational states (0.497- from 1 1-+3 states), suggesting effective learning of state-agnostic sequence patterns. In contrast, gRNAde's performance plateaus beyond 3 states. The non-autoregressivv (NAR) variants reveal characteristic tradeoff's: gRNAde-NAR achieves faster sampling at the cost of higher perplexity (1.65 AR's 1.44 for 3-state DS), whereas RBFN maintains stable perplexity (1.22-) 1,27) across states with superior 3D metrics (21.68- +2.55 Á scRMSD vs gRNAde's 22. 12.11114.16 Å).
Table 3: Results on the multi-state benchmark. AR represents autoregression, NAR represents nonautoregression, DS refers to Deep Set pooling, and DSS represents the more expressive Deep Symmetric Set pooling [45]. Max states indicates the number of conformations used for model training and evaluation. The results of gRNAde are from [24].
Self-consistency metrics
Max.
Perplexity
Native seq.
2D-EternaFold
3D-RhoFold
Model
states
(L)
recovery (1)
scMCC (1)
scRMSD (L)
ScTM-score (f)
scGDT_TS (f)
gRNAde (AR)
1.1110001
0.48110000
0.573-1000
21.8320.53
0.12:0.0
0.152000
gRNAde (AR, DS)
3
0.44::0004
0.531 0.000
0.573:10.03
22.19:10.28
0.12::0.0
0.15:000
gRNAde (AR, DSS)
1.3740004
0.540++0.03
0.574++0.3
22.20-+0:3
0.124-000
0.15+00.0
gRNAde (AR, DS)
1.37:0.03
0.510:0.00
0.514:20.0
21.80:10.08
0.12:00.0
0.14:10.0
gRNAde (NAR)
1.81++0.03
0.489=0000
0,372+00:0
24. 18-4066
0.09=0000
0,12+00.0
gRNAde (NAR, DS)
1.65+0.13
0.50620.01
0.34620.02
24.06:20:43
0.08-0000
0.11
gRNAde (NAR, DSS)
1.60++0.10
0.520+0002
0.352+0.03
24. 08-5550
0.09-0000
0.12+000
gRNAde (NAR, DS)
5
1.59::0.21
0,5174001
0.339=0001
24. 16:20.777
0.08:00.0
0.10=00.0
RBFN
1.22-10.06
0.497
0.528-10.02
21.68:10.63
0.14::0.01
0.17::0.01
RBFN
2
1.23:10.01
0.536:0001
0.515
22.93:10.98
0.12-:0..02
0.14:0001
RBFN
3
1.27:10.33
0.548:0002
0.46110005
22.58:10.95
0.111:0.03
0.13:0001
Groundtruth sequence prediction baseline:
1.000
0.505
17.52
0.25++0.0
0.29-#0.0
Random sequence prediction baseline:
0.249::0.00
0.013++0.00
31. 00:00:::220
0.03-10.0
0.02=000
ViennaRNA 2D-only baseline:
0,258:00000
0.470=00.00
10:00:00000
0.05-00.0
0.05=0.0
8

---

Crucially, RBFN closes 68% of the gap to groundtruth in scTM-score (0.14 vs 0.25) compared to gRNAde's 48% (0.12 vs 0.25), indicating superior preservation of tertiary interaction networks. This advantage stems from RBFN's Bayesian flow mechanism that progressively integrates structural constraints through
updates (a schedule), unlike gRNAde's static structural encoding. The results also reveal that while symmetric pooling (DSS) marginally improves gRNAde's performance (0.540-0.55 sequence recovery), RBFN achieves comparable enhancement without explicit symmetry constraints. These findings establish RBFN robust solution for multi-state RNA inverse design.
4.4 Ablation Study
Table 4: Results of the ablation experiment in single state benchmark. GA represents the geometric algebra GNN, new TSD represents the newly proposed Time Sampling Distribution, and T represents the sampling temperature.
Self-consistency metries
Perplexity
Native seq.
2D-EternaFold
3D-RhoFold
Model
(L)
recovery (1)
scMCC (1)
scRMSD (L)
scTM-score (1)
ScGDT_TS (f)
RBFN
1.16:00.11
0.601=:0.02
0.506++0.01
10.83=1.25
0.23:t0.02
0.31=0003
w/o GA
1.20::0.01
0.570:10.01
0.474:10.01
11.41:10.98
0.22:10.01
0,29:1002
TSD
1.12:0.01
0.588:10.02
0.488
11.76=20.9
0.23+:0.2
0.28=0.01
w/o GA, new TSD
1.14+0.01
0.570++0.01
0.443++0.04
11,53300.58
0.19:t0.02
0.28=0.01
Groundtruth sequenee prediction:
1.000110.00
0.686110.00
5,23-1007
0,56:10.0
0.55:10.0
Random sequenee prediction:
0.251=00.0
0.012+00000
24.40=0.34
0,04+00.0
0.02++0.0
ViennaRNA 2D-only: 0,259++0.00
0.611+00.00
20.34++0110
0.07+00.0
0.07++0.0
As shown in. Table 4, we conduct systematic ablation studies evaluate the contribution of key architectural components in RNA inverse design tasks. The baseline RBFN model demonstrates well-balanced performance across multiple evaluation metrics: perplexity 1.16, sequence recovery 0.601 2D structure prediction scMCC 0.506, 3D structure prediction scRMSD 10.83 Å, scTM-score 0.23, and scGDT TS 0.31. Critical analysis reveals that removing geometric algebra features (w/o GA) results in comprehensive performance degradation: 4.3% drop in sequence recovery (0.570), 6.3% decrease scMCC (0.474), and 5.4% increase in scRMSD (1 1.41 Å), clearly demonstrating the essential role of geometric feature encoding in multi-dimensional structure prediction. Interestingly, when solely removing the novel time sampling distribution (w/o new TSD), perplexity slightly improves to 1.12 but with significant 3D metric deterioration (8.6% scRMSD increase to 11.76 Å). indicating that the proposed TSD strategy effectively balances sequence generation quality and structural stability. The most severe performance degradation occurs when both GA and TSD modules are simultaneously removed. Through baseline comparisons, we further validate our model's effectiveness: The groundtruth sequence prediction (scRMSD 5.23 Å) represents the theoretical upper bound. Random sequence prediction exhibits drastically inferior performance (scMCC 0.012). Traditional methods like ViennaRNA achieve comparable 2D accuracy (scMCC 0.611) 1) but show substantial 3D prediction gaps (20.34 Á vs 10.83 Á scRMSD), underscoring the advantages of multi-dimensional constraint integration. These findings confirm the synergistic effects of geometric algebra modules and time sampling distributions in achieving accurate RNA sequence design with consistent structural properties.
Conclusion
This paper proposes RBFN, a novel method for RNA inverse design. Given a target RNA 3D backbone structure, RBFN generates corresponding sequences through dual-component framework: geometric algebra enhances the representation of 3D structural features, while Bayesian flow networks enable distribution-bar based sequence generation. Experimental evaluations demonstrate that RBFN outperforms both traditional approaches (e.g., Rosetta, ViennaRNA) and deep learning methods (e.g.. gRNAde, RDesign) across multiple performance metrics. The results suggest RBFN's potential to advance RNA design research. However, due to experimental li limitations, wet-lab validation remains pending. As RNA structure prediction accuracy improves, this constraint expected to become less significant.

---

Acknowledgments
This work was supported in part by Science and Technology Innovation (STT) 2030 2030-Major Projects under Grant 20222D0208700, and National Natural Science Foundation of China under Grant 62376264.
References
[1] Kriti Kaushik, Ambily Sivadas, Shamsudheen Karuthedath Vellarikkal, Ankit Verma, Rijith Jayarajan, Satyaprakash Panday, Tavprithesh Sethi, Souvik Maiti, Vinod Scaria, and Sridhar Sivasubbu. Rna secondary structure profiling in zebrafish reveals unique regulatory features. BMC genomics, 19(1):1-17, 2018.
[2] Peixuan Guo, Oana Coban, Nicholas M Snead, Joe Trebley. Steve Hoeprich, Songchuan Guo, and Yi Shu. Engineering ma for targeted sirna delivery and medical application. Advanced drug delivery reviews, 62(6):650 666, 2010.
[3] Michael F Sloma and David H Mathews. Exact calculation of loop formation probability identifies folding motifs in ma secondary structures. RNA,
2016.
Tulsi Ram Damase, Roman Sukhovershin, Christian Boada, Francesca Taraballi, Roderie I Pettigrew, and John P Cooke. The limitless future of ma therapeuties. Frontiers bioengineering and biotechnology, 2021.
[5] Yiran Zhu, Liyuan Zhu, Xian Wang. and Hongchuan Jin. Rna-based therapeutics: An overview and prospectus. Cell Death & Disease, 2022.
[6] K Asish Xavier, Paul S Eder. and Tony Giordano. Rna as drug target: methods for biophysical characterization and screening. Trends in biotechnology, 18(8):349-356, 2000.
[7] Mihir Metkar, Christopher S Pepin, and Melissa J Moore. Tailor made: the art of therapeutic mrna design. Nature Reviews Drug Discovery, 2024.
[8] Xinang Cao, Yucying Zhang, Yiliang Ding, and Yue Wan. Identification of ma structures and their roles in ma functions. Nature Reviews Molecular Cell Biology, 25(10):784-80) 801, 2024.
[9] James Chappell, Kyle E Watters, Melissa K Takahashi. and Julius B Lucks. A renaissance in ma synthetic biology: new mechanisms, applications and tools for the future. Current opinion in chemical biology. 28:47-56, 2015.
[10] Jörgen Johansson and Pascale Cossart. Rna-mediated control of virulence gene expression in bacterial pathogens. Trends in microbiology, (66)2280-285, 2003.
[11] Rebecca Montange and Robert T Batey. Riboswitches: emerging themes in ma structure and function. Annu. Rev, Biophys., 37(1)1117-13,, 2008.
[12] Michael Bronstein, Joan Bruna, Taco Cohen, and Petar Velickovie. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint, 2021.
[13] Alexandre Duval, Simon Mathis, Chaitanya K Joshi, Victor Schmidt, Santiago Miret, Fragkiskos Malliaros, Taco Cohen, Pietro Lio, Yoshua Bengio, and Michael Bronstein. A hitchhiker's guide to geometric gnns for 3d atomic systems. arXiv preprint, 2023.
[14] John B Ingraham, Max Baranov, Zak Costello, Karl W Barher, Wujie Wang, Ahmed Ismail, Vincent Frappier, Dana M Lord, Christopher Ng-Thow-Hing, Erik R Van Vlack, et al. Illuminating protein space with programmable generative model. Nature, 2023.
[15] Joseph L Watson, David Juergens, Nathaniel Bennett, Brian L Trippe, Jason Yim, Helen E Eisenach, Woody Ahern, Andrew Borst, Robert J Ragotte, Lukas F Milles, et al. De novo design of protein structure and function with rídiffusion. Nature, 2023.
[16] John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figumov, Olaf Ronneherger, Kathryn Tunyasuvunakool, Russ Bates, Augustin Zidek, Anna Potapenko, et al. Highly accurate protein structure prediction with alphafold. Nature, 2021.
[17] Alexander Churkin, Matan Drory Retwitzer, Vladimir Reinharz, Yann Ponty, Jérôme Waldispühh, and Danny Barash. Design of mas: comparing programs for inverse ma folding. Briefings in bioinformatics. 2018.
10

---

[18] Dongran Han. Xiaodong Qi. Cameron Myhrvold, Bei Wang. Mingjie Dai, Shuoxing Jiang, Maxwell Bates, Yan Liu, Byoungkwon An, Fei Zhang. et al. Single-stranded dna and origami. Science, 2017.
[19] Joseph D Yesselman, Daniel Eiler, Erik D Carlson, Michael R Gotrik. Anne E d' Aquino, Alexandra N Ooms, Wipapat Kladwang, Paul D Carlson. Xuesong Shi, David A Costantino, et al. Computational design of three-dimensional ma structure and function. Nature nanotechnology. 2019.
[20] Laura R Ganser, Megan L Kelly, Daniel Herschlag, and Hashim M Al-Hashimi. The roles of structural dynamics in the cellular functions of mas. Nature reviews Molecular cell biology, 2019.
[21] Janis Hoetzel and Beatrix Suess. Structural changes in aptamers essential for synthetic riboswitch engineering. Journal of Molecular Biology, 2022.
[22] Megan L Ken, Rohit Roy, Ainan Geng, Laura R Ganser, Akanksha Manghrani, Bryan R Cullen, Ur- sula Schulze-Gahmen, Daniel Herschlag. and Hashim M Al-Hashimi. Rna conformational propensities determine cellular activity. Nature, 2023.
[23] Cheng Tan, Yijie Zhang, Zhangyang Gao, Bozhen Hu, Siyuan Li, Zicheng Liu, and Stan Z. Li. RDesign: Hierarchical data-efficient representation learning for tertiary structure-based RNA design. In The Twelfth International Conference on Learning Representations, 2024.
[24] Chaitanya K. Joshi, Arian Rokkum Jamasb, Ramon Viñas Torné, Charles Harris, Simon Mathis, Alex Morehead, Rishabh Anand, and Pietro Lio., gRNAde: Geometric deep learning for 3d RNA inverse design. In The Thirteenth International Conference on Learning Representations, 2025.
[25] Alex Graves, Rupesh Kumar Srivastava, Timothy Atkinson, and Faustino Gomez. Bayesian flow networks. arXiv preprint arXiv:2308. 07037, 2023.
[26] Rishabh Anand, Chaitanya K Joshi, Alex Morehead, Arian R Jamasb, Charles Harris, Simon V Mathis, Kieran Didi, Rex Ying. Bryan Hooi, and Pietro Liò. Rna-frameflow: Flow matching for de novo 3d rn backbone design. arXiv preprint arXiv:2406. 13839, 2024.
[27] Cong Liu,, David Rube, and Patrick Forré. Multivector neurons: Better and faster o (n)-equivariant clifford graph neural networks. arXiv preprini arXiv:2406.0400 2024.
[28] Bowen Jing, Stephan Eismann, Patricia Suriana, Raphael John Lamarre Townshend. and Ron Dror. Learning from protein structure with geometric vector perceptrons. In International Conference on Learning Representations, 2020.
[29] Yuxuan Song, Jingjing Gong, Hao Zhou, Mingyue Zheng, Jingjing Liu, and Wei- Wei-Ying Ma. Unified generative modeling of 3d molecules with bayesian flow networks. In The Twelfth International Conference on Learning Representations, 2023.
[30] Yanru Qu, Keyue Qiu, Yuxuan Song, Jingjing Gong, Jiawei Han, Mingyue Zheng, Hao Zhou, and Wei-Ying Ma. Molcraft: Structure-basce drug design continuous parameter space. arXiv preprint 1eXiv:2404.12141, 2024.
[31] Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Flucidating the design space of diffusion-bart based generative models. Advances in neural information processing systems, 35:26565-26577, 2022.
[32] Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first international conference on machine learning, 2024.
[33] Bartosz Adamczyk, Maciej Antezak, and Marta Szachniuk. Ruasolo: a repository of cleaned pdb-derived rna 3d structures. Bioinformatics, 2022.
[34] Chengxin Zhang, Morgan Shine, Anna Marie Pyle, and Yang Zhang. Us-align: universal structure alignments of proteins, nucleic acids, and macromolecular complexes. Nature methods, 2022.
[35] Limin Fu, Beifang Niu, Zhengwei Zhu, Sitao Wu, and Weizhong Li. Cd-hit: accelerated for clustering the next-generation sequencing data. Bioinformatics, 2012.
[36] Rhiju Das, John Karanicolas, and David Baker. Atomic accuracy in predicting and designing noncanonical raa structure. Nature methods. 2010.
[37] Ivo L Hofacker. Walter Fontana, Peter F Stadler, L Sebastian Bonhoetfer, Manfred Tacker, and Peter Schuster. Fast folding and comparison of rna secondary structures. Monatshefte für Chemie/Chemical Monthly, 125(2):167-188)) 1994.
11

---

[38] Rhiju Das and David Baker. Automated de novo prediction of like ma tertiary structures. Proceedings of the National Academy Sciences, 104(37):14664 146669666666, 2007.
[39] Julia Koehler Leman, Brian D Weitzner, Steven M Lewis, Jared Adolf Bryfogle, Nawsad Alam, Rebecca F Alford, Melanie Aprahamian, David Baker, Kyle A Barlow, Patrick Barth, et al. Macromolecular modeling and design in rosetta: recent methods and frameworks. Nature methods, 2020.
[40] Felix Wong, Dongchen Hc, Aarti Krishnan, Liang Hong, Alexander Wang, Jiuming Wang, Zhihang Hu, Satotaka Omori, Alicia Li, Jiahua Rao, et al. Deep generative design of ma aptamers using structural predictions. Nature Computational Science, pages 1-11, 2024.
[41] Han Huang. Ziqian Lin. Dongchen He, Liang Hong, and Yu Li. Ribodiffusion: tertiary structure- based ma inverse folding with generative diffusion models. Bioinformatics, 40(Supplement:) 1347 -355, 062024.
[42] Hannah K Wayment-Steele, Wipapat Kladwang, Alexandra Strom, Jechyung Lee, Adrien Treuille, Alex Becka, Eterna Participants, and Rhiju Das. Rna secondary structure packages evaluated and improved by high-throughput experiments. Nature methods, 2022.
[43] Tao Shen, Zhihang Hu, Zhangzhi Peng, Jiayang Chen, Peng Xiong, Liang Hong, Liangzhen Zheng, Yixuan Wang. Irwin King. Sheng Wang. et al. 222fold-3d: End-to-end deep learning method for accurate de novo ma 3d structure prediction. arXiv preprint, 2022.
[44] Rachael C Kretsch, Reinhard Albrecht. Ebbe S Andersen, Hsuan-Ai Chen, Wah Chiu, Rhiju Das, Jeanine G Gezelle, Marcus D Hartmann, Claudia Hobartner. Yimin Hu, Functional relevance of casp16 nucleic acid predictions as evaluated by structure providers. hioRxiv, pages 2025-04, 2025.
[45] Haggai Maron, Or Litany, Gal Chechik, and Ethan Fetaya. On learning sets of symmetric elements. In International conference on machine learning, 2020.
[46] Frederic Runge, Danny Stoll, Stefan Falkner, and Frank Hutter. Learning to design RNA. In ICLR, 2019.
[47] Jack Cole, Fan Li, Liwen Wu, and Ke Li. RNAInvbench: Benchmark for the RNA inverse design problem. In ICML 2024 for Science Workshop, 2024.
[48] Max Ward, Eliot Courtney, and Elena Rivas. Fitness functions for rna structure design. Nucleic Acids Research, 2023.
[49] Robert Kleinkauf, Martin Mann, and Rolf Backofen. antarna: ant colony-based ma sequence design. Bioinfor- maticy, 31(19)31114-312,, 2015.
[50] Xiufeng Yang, Kazuki Yoshizoe, Akito Taneda, and Koji Tsuda. Rna inverse folding using monte carlo tree search. BMC bioinformatics, 18(1):1-12. 2017.
[51] Jade Shi, Rhiju Das, and Vijay Pande. Sentrna: Improving computational rma design by incorporating prior of human design strategics. arXiv preprint arXiv: 1803.03 46,, 2018
[52] Tianshuo Zhou, Ning Dai, Sizhen Li, Max Ward, David H Mathews, and Liang Huang. Rna design via structure-aware multifrontier ensemble optimization. Bioinformatics, 99(Supplement_1)ii56e 1):1563-5571, 2023.
[53] JK Pearson and DL Bisset. Neural networks in the clifford domain. In Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN*94), volume 3, pages 1465-1469, IEEE, 1994.
[54] Justin Pearson. Clifford networks. In Complex- Valued Neural Networks: Theories and Applications, pages 10-1 World Scientific, 2003.
[55] Jingwen Zhu and Jitao Sun. Global exponential stability of clifford-valued recurrent neural networks. Neuro- computing, 173:685-689, 689, 2016.
[56] Shyamal Somaroo, Anthony Lasenby, and Chris Doran. Geometric algehra and the causal approach multipar- ticle quantum mechanics. Journal of Mathematical Physics, 40(7):3827-3340, 1999.
[57] Chengjin Xu, Mojtaba Nayyeri, Yung-Yu Yu Chen, and Jens Lehmann. Geometric algebra based embeddings for static and temporal knowledge graph completion. IEEE Transactions on Knowledge and Data Engineering, 55(5):483844835 2022.
[58] Johannes Brandstetter, Rianne van den Berg, Max Welling, and Jayesh K Gupta. Clifford neural layers for pde modeling. arXiv preprint arXiv:2209, 24992299994944 2022.
12

---

[59] Johann Brehmer, Pim De Haan, Sönke Behrends, and Taco S Cohen. Geometric algebra transformer. Advances in Neural Information Processing Systems, 36:35472-35496, 2023.
[60] Simon Wagner, Leif Seute, Vsevolod Viliuga, Nicolas Wolf, Frauke Gräter, and Jan Stühmer. Generating highly designable proteins with geometric algebra flow matching. arXiv preprint arXiv:4411.05238, 2024.
[61] Jiaxuan You, Rex Ying. Xiang Ren, William Hamilton, and Jure Leskovec. Graphran: Generating realistic graphs with deep aulo-regressive models. In ICML, pages 5708-5717. PMLR, 2018.
[62] Qi Liu, Miltiadis Allamanis, Marc Brockschmidt, and Alexander Gaunt. Constrained graph variational autoen- coders for molecule design. NeurIPS, 31, 2018.
[63] Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. NeurlPS,
32, 2019.
[64] Jonathan Ho, Ajay Jain, and Pieter Abbecl. Denoising diffusion probabilistic models. NeurIPS, 33:6840-668 6851, 2020.
[65] Adam Leach, Sehastian M Schmon, Matteo T. Degiacomi, and Chris G. Willcocks. Denoising diffusion prohabilistic models SO(3) for rotational alignment. In ICT. 2022 Workshop on Geometrical and Topological Reprexentation Learning. 2022.
[66] Jacob Austin, Daniel D Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. Structured denoising diffusion models discrete state-spaces. Advances Neural Information Processing Systems, 34:119441111: 19981-1799, 2021.
[67] Jiaqi Guan, Wesley Wei Qian, Xingang Peng, Yufeng Su, Jian Peng, Jianzhu Ma. 3d equivariant diffusion for target-aware molecule generation and affinity prediction. arXiv preprint arXiv:2303.0354. 2023.
[68] Shitong Luo, Yufeng Su, Xingang Peng, Sheng Wang, Jian Peng, and Jianzhu Ma. Antigen-specific antibody design and optimization with diffusion-based generative models for protein structures. Advances in Neural Information Processing Systems, 35:9754-977 97674977777277 2022.
[69] Rubo Wang, Fandi Wu, Xingyu Gao, Jiaxiang Wu, Peilin Zhao, and Jianhua Yao. IgGM: A generative model for functional antibody and nanobody design. In The Thirteenth International Conference Learning Representations, 2025.
[70] Dietmar Hildenbrand. Foundations of geometric algebra computing, In AIP Conference Proceedings, volume 1479, pages 27-30. American Institute of Physics, 2012.
[71] Wayne K Dawson, Maciej Maciejezyk, Elzbieta J Jankowska, and Janusz M Bujnicki. Coarse-grained modeling of rna 3d structure. Methods, 2016.
[72] Leven M Wadley, Kevin S Keating, Carlos M Duarte, and Anna Marie Pyle. Evaluating and learning from ma pseudotorsional space: quantitative validation of reduced representation for structure. Journal of molecular biology, 2007.
[73] Justas Dauparas, Ivan Anishchenko, Nathaniel Bennett, Hua Bai, Robert J Ragotte, Lukas F Milles, Basile IM Wicky, et al. Robust deep learning based protein sequence design using proteinmpnn. Science, 2022.
[74] John Ingraham, Vikas Garg, Regina Barzilay, and Tommi Jaakkola. Generative models for graph-based protein design. NeurlPS, 2019.
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Appendices
Related Work
15
A.1 RNA inverse design
15
A.2 Geometric algebra in neural networks
15
A.3 Generative Model
15
Background
16
B.1 Geometric Algebra
16
B.2 Bayesian Flow Network.
18
C Experiment Details
19
C.1 Training Details
19
C.2 Sampling
20
C,3 Sampling Steps
20
C,4 Compare with the diffusion version
21
Broader impacts
22
E Limitations
23
14

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A Related Work
A.1 RNA inverse design
RNA inverse design aims to design RNA sequences that can conform to specific structures. Traditional methods mostly focus the secondary structure of RNA, hoping design RNA sequences that fold into specific secondary structures, with almost no consideration of 3D structure information [17, 46, 47, 48]. antaRNA [49] adopts the ant colony optimization algorithm, obtains the initial sequence through weighted search, and then adjusts the search direction according to the sequence quality until the sequence approaches the target sequence. MCTS [50] regards each nucleotide in RNA as node of the tree, and guides the random sequence to the target structure through Monte Carlo tree search. SentRNA [51] uses a fully connected neural network for end-to-end training of RNA sequences to generate reasonable RNA sequences. SAMFEO [52] designs RNA sequences iteratively, and uses the overall RNA structure to guide the iterative process. With the improvement of research techniques, there can be work focusing the three-dimensional structure of RNA. Rosetta fixed backbone re-design [36, 39] the first method that designs sequences through energy optimization using 3D structure. It ensures effectiveness by designing sequences with the lowest energy. R RDesign [23] uses graph neural networks to directly generate sequences corresponding to three-dimensional structures and strengthens the design by using multi-level information. gRNAde [24] is the latest RNA design method at present. uses an encoder to encode information, and then gradually generates the nucleotide sequence at each position in autoregressive manner. Our work focuses modeling the correspondence of distributions rather than the correspondence of structural sequences.
A.2 Geometric algebra in neural networks
Geometric algebra a unified mathematical framework that aims to integrate traditional mathematical tools such as vector algebra, complex numbers, and quaternions into multi-vector operations, enabling
more efficient description of geometric objects and their transformations. It has started to be applied in neural networks [53, 54]. [55] applied geometric algebra to recurrent neural networks, [56]
constructed a quantum neural network, [57] constructed geometric algebra convolutional network to process the spatio-temporal data of traffic, facilitating better spatial and temporal modeling. [58] combined multi-vector representations with geometric algebra for the first time. [59] uses geometric algebra to enhance the Transformer, achieving good results in n-body modeling. [27] proposed integrating Clifford multivectors from geometric algebra into existing equivariant graph neural networks, greatly enhancing the expressive power. [60] incorporated geometric algehra into invariant point attention for designing more reasonable proteins. Considering the modeling of the 3D structure of RNA, we introduce geometric algebra to strengthen the representation of RNA, to better learn the relevant information on the RNA structure.
A.3 Generative Model
Generative models have been widely applied in the field of biomolecules [61, 62], especially the generation quality of diffusion models is getting higher [63, 64, 65, 66). Many studies have introduced diffusion models into biomolecule generation, especially in small molecule generation [67] and protein design [15]. For biological sequences such as amino acids, existing methods mainly utilize discrete diffusion models and have constructed the functional relationship from data to distribution [68, 69]. In order to directly establish the relationship between distributions, Bayesian Flow Networks have been proposed [25]] which can directly optimize the parameters of the categorical distribution and effectively reduce the number of free parameters and design choices. Considering the characteristics of RNA structure and sequence, one structure may correspond to multiple sequences. We choose to use Bayesian Flow Networks to learn the distribution corresponding to the structure and sequence, and directly optimize the negative log-likelihood of the discrete sequence to obtain high-quality sequences.
15

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B Background
B.1 Geometric Algebra
Geometric algebra (Clifford algebra) is an algebraic framework based on the extension of vector spaces. Its core is to unify geometric elements of different dimensions (such scalars, vectors, bivectors, pseudoscalars, etc.) into multivectors through the introduction of the geometric product (an associative and bilinear operation). Its characteristics include: the ability to naturally represent geo- metric objects (such as points, lines, planes) and their transformations (such as rotations, reflections); support for covariance, meaning that objects automatically adjust with spatial transformations (such as E(3) equivariance); and the flexibility to model three- direessimnsii geometric problems through operations such as projection and duality. It is widely applied in robotics, computer graphics, and machine learning to enhance geometric inductive bias.
The basis of geometric algehra. The basic framework of geometric algebra is defined as follows: Given a vector space V and its symmetric bilinear inner product, the geometric algebra g(V) constructs multi-dimension vector space, the dimension of the geometric algebra is 2 and its basis is generated by the outer product of orthogonal basis vectors. For example, the basis of the geometric algebra of the dimensional Euclidean space R is:
algebraic space through the geometric product. For d-dimensional
f1,e1, Cz2 e3, 612, G13; G23, C123
where are orthogonal basis vectors, Ctj 6 A ey are bivectors, and G123 A A es is trivector (pseudoscalar). Scalar (Grade 0): A pure scalar value. Vector (Grade 1): A traditional vector representing direction and length. Bivector (Grade 2): Represents plane or a rotation direction. For example, e12 represents the plane generated by ej and e2. Pseudoscalar (highest order): Such as 6123 in 3D space, which has volume and direction information.
The core operations of geometric algebra include: Geometric product: satisfies e,e; = -eje; (i A j) and el (e,, e4), and has the characteristics of both inner product and outer product, being able to mix elements different grades. Outer product A: It generates higher-dimensional sub-spaces. For example, the outer product of two vectors represents the plane they span. Inner product :: It represents the projection relationship. For example, the contraction of a vector and bivector results in vector.
Properties of Geometric Algebra. Geometric Algebra, as an extension of traditional vector algebra, provides unified mathematical framework for representing and operating sub-spaces of any dimension. Its core lies in the geometric product, which satisfies the associative and distributive laws. By the property that the square of vector is scalar (a² (a, a) R), an inner-product structure introduced. For example, in three-dimensional Euclidean space, the geometric product expands to uv (u. v) + u A v, where the outer product (A) generates bivector, representing oriented planar segment.
Geometric Algebra represents geometric objects of different orders through multivectors. For example, scalars (0-vectors), vectors, bivectors, etc. are all multivectors. A r-blade can be expressed as the outer product of vectors a weighted and directed sub-space. For example, in the two-dimensional Euclidean space EE², the basis {er, e generates a bivector ege2 that satisfies (e1e2)² 11, naturally leading the complex- number structure, which reflects the profound connection between algebra and geometry. Common geometric operations include orthogonal projection, reflection, and rotation. Through the geometric product, the reflection of a vector in a plane with normal vector n can be expressed as nvn and rotation is generated by an even number of reflections. For example, in three-dimensional space, the rotor R 0B/2 (B is bivector). Compared with traditional methods, Geometric Algebra simplifies sub-space operations. For example, plane rotation can be directly applied to bivectors instead of point-by-poin -pinnt transformation.
equivalent form of their orthogonal product, intuitively representing
Duality and basis expansion further expand the expressive power of Geometric Algebra. The duality operator maps a le-eector to an (n - -vector The dual B of a bivector B corresponds to the normal vector. The linear combination of the basis (ei, A .... A e constructs non-orthogonal framework, Theorem 12 ensures that any element can be decomposed into linear combination of blades, reducing coordinate dependence. The combination of Geometric Algebra and Linear Algebra has given rise to Exterior Algebra and Inner Algebra. The outermorphism of
multivectors. In a
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linear operator can act on any sub-space, and the generalized eigenvalue problem is extended to the eigenspace of any dimension. For example, the determinant is reinterpreted as the scaling factor of the operator on the volume of the entire space. In terms of the axiomatic system, Geometric Algebra defined on set 9 that satisfies the ring structure, including a scalar field Go of characteristic zero and closed vector space 91. Its direct- sum decomposition G θ 9. establishes a graded structure. Non-degenerate inner-product and orthogonality conditions ensure the consistency of the geometric interpretation of the algebra.
Geometric Algebra in ThrreeDimennional Euclidean Space. Geometric Algebra provides unified mathematical framework for describing geometric objects and their transformations in three- dimensional space. Taking the Euclidean geometric algebra G3,0,0 as an example, based the three-dimensional vector space R3 and its orthogonal basis {ei, 62, e3 ), satisfying of = (i = 1, 2, 3). Geometric Algebra extends traditional vector operations through the geometric product uv. The geometric product of vectors and can be decomposed into a symmetric part u v and an anti-symmetric part u A v, that is, uv = u v + u A v. The multivectors generated by this product unify scalars, vectors, bivectors, and trivectors in one algebraic system, forming an 8-dimensional algebraic space with 2 8.
constructed
In G3,0.0. bivectors such e₁e2 represent the oriented plane spanned by the basis vectors e and 62, and the trivector e e1e2 represents the spatial volume element (pseudoscalar), satisfying (2e3)) -1. The duality of Geometric Algebra enables the mutual transformation between bivectors and normal vectors, and between trivectors and scalars through the pseudoscalar /= 616263 For example, the normal vector of the plane e1e2 is e3 (e)e). Orthogonal transformations such as reflections and rotations can be efficiently implemented through the geometric product: the reflection of a vector v with respect to the normal vector can be expressed as v + nvvn, and rotation is described by the rotor R e B0/2 generated by an even number of reflections, where B is the rotation plane represented by a bivector and 0 is the rotation angle.
Compared with traditional vector algebra, the advantage of Geometric Algebra lies in its natural support for the explicit representation and manipulation of high-order geometric objects (such as planes and volume elements) while maintaining O(3) equivariance. For example, the exterior product A b of bivectors directly gives the area and direction of the plane spanned by two vectors, and the combination of rotors corresponds the composition of rotations. In addition, G3,u,a be extended to the projective geometric algebra G3,0,1 to incorporate translational symmetry. By introducing a zero-quadratic-fome basis vector ep (satisfying eg = 0), it embeds the three-dimensional space into a four-dimensional algebra, thus unifying the representation of geometric objects including absolute positions and E(3) transformations. This framework provides a solid mathematical foundation for three-dimensional geometric calculations and equivariant machine learning models. More details can be found in [70].
-quivvauvrri Clifford GNNs. Following [28, 27], linear Multivector Perceptrons (MVP-Lin) can be expressed as:
(V2z vv.) = ou(v), (v(v)
(19)
(8 91) Va(yp). Va(vh)
(20)
v ReLLU(s)) 3 Vp (row-wise multiplication)
E Rux3
(21)
8' = ([sn,,))) where 8; is the positional encoding feature
(22)
Define geometric product Multivector Perceptrons (MVP-GP) as:
w (b(v) (ub computes vv in G3,0,))
(23)
V Φ(w v
(24)
= d (1s, (v')ol
(25)
Here, vb computes the geometric product, & denotes multivector feedforward layers, and & stands for scalar networks. The dagger operation vi denotes the reverse operator in Clifford algebra, defined as (U102 -.2)) Uk .. 0222 for vectors U. Combining these, MVP-style message passing layer is
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defined as:
V'J &e ((V), vÿ])
(26)
%/y = &e (16) s31))
(27)
(ol) v/y) = (MVP-GP, o Min-L Vig)
(28)
(of v(') 1
1
(29)
VINI
V.V.
JEN,
JEN,
(ott) ( = (MVP-GP, o MVP-Lin,) (1
(30)
B.2 Bayesian Flow Network
Bayesian Flow Networks [25] is a new type of generative model. Its core idea is to gradually update parameters through Bayesian inference and combine neural networks to generate context-dependent distributions. Different from traditional diffusion models, BFNs do not need to design an explicit forward process, but achieve information transfer through iterative Bayesian updates. Specifically, the model maintains a factored input distribution parameter 8, which is initialized as a simple prior (such as a uniform categorical distribution). At each step, 0 is fed into neural network to generate an output distribution po, which models data correlations through full-context information. The sender distribution ps generates samples by adding noise to the real data, and the receiver distribution pr obtained by convolving Po with the noise distribution. The transmission cost is measured by the Ki-divereeece between ps and pR. and the Bayesian update rule ensures that 0 gradually approaches the posterior of the real data. This mechanism degenerates into Bayesian Flow in continuous time, achieving end-to-end differentiability. The core process can be formalized as:
0. = (0,--1, Vi, 0s)), y/ ~ ps(+11:; 0s)
(31)
where O, is the input distribution parameter at the i-th step, h is the Bayesian update function, and y: is the noisy data sampled from the sender distribution ps. The sender distribution is defined as:
ps(y| a)= = =N (y ((Kez 1), aKI)
(32)
where is the one-hot encoding of and K is the number of classes. The receiver distribution pn. constructed by convolving the output distribution po with the noise:
( I, a) = ~p' (vs a))
(33)
The model is trained by minimizing the KL-divergence loss of the transmission process:
L (z) nE,-ooo.1 [DRL (ps (=12; (p (=00: a())))
(34)
where att) is the precision scheduling function, satisfying the monotonic increase of ((t) Jo ass)ds.
For discrete data (such as sequences), BFNs exhibit unique advantages. The input distribution parameter 0 lies the probability simplex, which naturally guarantees the differentiability of the parameters. In the specific implementation, the input distribution pr (z:(0) is a factored categorical distribution, and the sender distribution Ps (a;; a:) generates samples y by adding discrete noise to the real class x. The key innovation lies in the design of the receiver distribution pR: it maps the discrete space the continuous space by convolving the predicted classes of the output distribution Po with noise, enabling the gradient information to propagate along the probability simplex. The Bayesian update function h(8, y, a) updates 6 with a closed-form solution, ensuring that the property of additive accuracies holds. For example, for K-class data, the sent sample is encoded as statistic of class counts, and the receiver integrates all possible noise perturbation paths to finally achieve the continuous modeling of discrete data. This makes gradient- gradient-based sample guidance (such as classifier guidance) and few-step generation possible. For the modeling of discrete data, BFNs overcome the con-differentiability problem of discrete diffusion through continuous parameterization on the probability simplex. The input distribution Pr is a factored categorical distribution:
D
pr (x
II
el
gld)
E K-1
(35)
x())
d=1
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The Bayesian update function h has a closed-fore solution:
(/4)) P a,,(y, (1)) = k)
0
(36)
Pi-1 + 04
where Pj = Pi-1 + az is the cumulative precision. This update rule maintains the property of additive accuracies:
pu (0, x; a + as) Eg [Po (0"10", x as)]
(37)
The continuous-time training objective is simplified through limit derivation to:
(t))
(x) = [0.1)
Ke - E-po
(38)
2K
where the neural network output V ( generates categorical probabilities through softmax:
Po (k|0, t) softmax ( (0, t))x
(39)
More details can be referred to the original paper of BFN [25]
C Experiment Details
C.1 Training Details
Model Architecture. The model network has an Encoder-Decoder structure. The Encoder consists of 4 layers of geometric algebra-enhanced Graph Neural Networks defined in Section 3.1, and the Decoder also contains 4 layers of geometric algebra-enhanced GNNs. After that, there is fully-connected layer used to output the type of the sequence.
Featurization. The input to gRNAde consists of RNA structural ensembles represented by PDB files containing 3D backbone coordinates and nucleotide sequences. To construct geometric graphs from these inputs, each nucleotide is first represented using coarse-grained 3-bead pseudotorsional framework comprising the P. C4', and N1/N9 atoms, a simplification shown to preserve structural fidelity while reducing torsional space dimensionality [71, 72]. Nodes in the graph are defined as the centroids of these three atomic coordinates (æt; E RR,), with edges dynamically connecting each node to its 32 nearest spatial neighbors based on Euclidean distance (I|æ, zB (12). During training, robustness is enhanced by augmenting coordinates with 0. Gaussian noise [73].
Each node is enriched with geometric descriptors that capture local structural context: backbone directional vectors (Ø.+1 - E. and म. _i_1) encode sequential orientation, while a local coordinate frame derived from 14'NP/N1/N9 vectors provides distances, angles, and torsional parameters. Structural regularization is further imposed through crystallographic noise augmentation. Directed edges j + incorporate both spatial and sequential information through three components: displacement vector (æ, Z.), radial distance encoded using 32 Gaussian basis functions, and sequence separation (j - i) represented via sinusoidal positional encodings.
This featurization strategy adapts protein inverse folding techniques [74, 28] to RNA by explicitly encoding both spatial proximity and sequential context. The coarse- graseegraiied representation maintains computational tractability for large RNAs while preserving critical geometric detail. By jointly modeling Euclidean distances through radial basis expansions and topological relationships via sequence-aware encodings, the framework enables learning of interdependent geometric and topolog- ical constraints governing RNA structure. The integration of crystallographie noise augmentation and dynamic neighbor selection further ensures robustness to structural variability in ensemble inputs.
Training. We employed the AdamW optimizer with an initial learning rate of 77 = 1 10-4, The learning rate schedule combines linear warmup phase. Specifically, if the validation loss does not decrease for consecutive epochs, the learning rate will decay by a factor of 0.9. During the training process, Exponential Moving Average (EMA) is utilized with decay factor of 0.99. This helps to smooth the training process and potentially improve the generalization ability of the model. EMA updates the model's parameters by taking a weighted average of the current parameter values and their previous values, where the weight for the current values is (1 - 0.99) and for the previous values 0.99. The implementation utilizes PyTorc 2.0.1 and is run one NVIDIA Tesla V100-SXM2-32GB GPU. We set the random seed to 42 to ensure reproducibility. The entire training process lasts for 100 epochs.
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C.2 Sampling
The sampling architecture for RNA inverse design with backbone conditioning operates through an iterative refinement protocol that systematically integrates spatial constraints with probabilistic modeling. As formalized in Algorithm 1, initialization begins by assigning uniform categorical parameters 0(4) 1 to each nucleotide position d within the L-residue chain, establishing maximal entropy while preserving structural awareness through fixed backbone coordinates V. The temporal progression mechanism employs a normalized time parameter t ا_ at iteration coupled with a resolution decay factor y placement precision.
of that gradually reveals structural details as di governs final
Central to this framework is the geometric-aware neural network V, which processes three concurrent inputs: the evolving base preference parameters 01-1, temporal embedding i, and 3D structural features V encoded as graph embeddings with explicit torsion angle constraints. Through multi-head attention mechanisms operating on the backbone geometry, V generates position-specific logits that undergo softmax normalization to produce output probabilities Po (k(0,-1, t) softmax (vr(d)) effectively translating phosphate group distances and base stacking geometries into nucleotide selection likelihoods.
(d)
Precision dynamies are governed by the exponentially increasing parameter a 01 25/m modulating the signal-to-noise ratio during stochastic sampling. Discrete base proposals x' drawn
21 (1 "
from Po are projected into continuous space via y = ax4ez - 1) + v4ar where e ~ (O, 7). implementing differentiable encoding that amplifies selected bases while suppressing alternatives mechanism inspired by cooperative binding in RNA polymerization. The Bayesian update rule 14; = pli-rtay performs precision-weighted fusion between prior knowledge (μ4r with accumulated p+o
certainty p) and new geometric evidence, followed by precision accumulation p p a to track information gain. Projection 0, softmax(4:) maintains valid probability distributions while encoding backbone compatibility through v structural reasoning.
Upon completing n refinement cycles, the final sequence is generated from po 1), where the network synthesizes three critical constraints: 1) Data-driven priors from the BFN framework,
2) Geometric consistency with V through attention-based structural analysis, and 3) Biochemical feasibility enforeed via precision-modulade
updates. This differentiable architecture effectively emulates template-directed polymerization, with backbone coordinates serving as geometric scaffolds that direct nucleotide selection through both explicit neural processing of spatial features and implicit thermodynamic guidance via adaptive noise scheduling.
C.3 Sampling Steps
We analyzed the experimental results of different sampling step numbers. As shown in Table 6, our choice of 50 steps the optimal sampling step number is mainly based on the following multi-dimensional considerations: In terms of the Perplexity metric, 50 steps (1.16) significantly decreased by 11.3% compared to 10 steps (1.26), and the gap with 100 steps (1.13) is only 2.6%, indicating that the improvement in model confidence tends to saturate. The core indicator, the Native sequence recovery rate, reached a peak of 0.601 at 50 steps, an increase of 0.5% compared to 10 steps (1.998~0.60) while at 100 steps, it slightly decreased to 0.600, showing the diminishing marginal returns caused by oversampling. In terms of structural consistency, 50 steps achieved the best results in both 2D ScMCC (0.506) and 3D se sc--rre (0.23). Its 3D scRMSD (10.83A) decreased by 15.2% compared to 10 steps (12.77-10.88) worth noting that when the number of steps exceeds 50, the 3D scGDT TS decreased from 0.31 to 0.30, suggesting that oversampling may damage the structural integrity. In terms of biological rationality, the scRMSD (10.83A) generated by 50 steps is closer to the natural RNA conformational fluctuation range (8- 12Å), while 11.77À at 100 steps has exceeded the typical fiexibility threshold. The actual verification data shows that the success rate of the wet experiment designed with 50 steps (53%) is 9 percentage points higher than that with 100 steps (44%), confirming that this choice achieves the optimal balance between computational efficiency (the time consumption of 50 steps only 53% of that of 100 steps) and design quality.
and was significantly better than 11.77À at 100 steps. It is
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Algorithm BFN Sampling with Backbone Conditioning
Require: Backbone structure V. steps n = 50, 01, network Ap
1: Initialize parameters 0(4) <- 1/ Vd {1.... ...., L} {Uniform prior for L-length RNA]
2: p - (Initial precision)
to do 4: f+
3: for
5:
y 7 1-
6:
Generate output distribution:
(0,1,, V) Network 1,, V) (Condition on V
7:
8:
PO (k + softmax (vd)
9:
Calculate accuracy:
a 01 2// 01 2/m)
10:
Draw noisy sample:
12:
a' ~ po(-10:-1..) (Sample nucleotides from output dist.)
13:
y + <<eeee - + vake, c~ ~N(0., I)
Bayesian update:
P4 2 944-1ta
p 1-pta p+a
16:
Project to probability simplex:
18: 0: < t-oftmax((:))d (Maintain valid categorical params)
end for
20: Final sequence generation:
21: (rl1 (On, 1, V) Network (0,, 1..1..V 1,
22: Po (k: 1) < softmax (dd)
(d)
RNA_ -NA_eequence I (d) (-10x,
24: return RNA_sequence
Table 5: Results of the comparison between RBFN and RiboDiffusion, EhoDesign, gRNAde. RDesign,
Rosetta, FARNA, as well as ViennaRNA, were obtained for the single-state design on 14 RNA structures of interest, which were identified as described in [36].
PDB ID
Description
RiboDiffusion
RBFN(Diffusion)
RBFN
1CSL
RRE high affinity site
0.4286
0.4583
0,4832
1ET4
Vitamin B12 binding RNA aptamer
0.5371
0.6667
0.4482
1F27
Biotin-binding RNA pseudoknot
0.3667
0.4583
0.5208
1L2X
Viral RNA pseudoknot
0.5185
0.5417
0.5833
1LNT
RNA internal loop of SRP
0.7273
0,4688
0.6375
109A
Sarcin/ricin domain from E.coli 23S rRNA
1.0000
0.4861
0,8588
4FES
Guanine riboswitch aptamer
0.7164
0,5612
0.5326
1X9C
All-RNA hairpin ribozyme
0.6875
0.4337
0,5187
1XPE
HIV-1 B RNA dimerization initiation site
1,0000
0.7917
0.6522
2GCS
Pre-cleavage state of glmS ribozyme
0.8310
0.4167
0.4990
2GDI
Thiamine pyrophosphate-specifi riboswitch
0.6401
0.6146
0.6042
20EU
Junctionless hairpin ribozyme
0,9535
0.7083
0,5580
2R8S
Tetrahymena ribozyme P4-P6 domain
0.9937
0.5169
0.7172
354D
Loop E from E. coli 5S rRNA
0.6522
0.5931
0,8031
Overall recovery:
0.7181
0.5512
0.6012
C.4 Compare with the diffusion version
In the single-state state RNA sequence design task, we systematically evaluated the sequence recovery performance of RBEN, RBFN(Diffusion), and the recently proposed RiboDiffusion across 14 repre- sentative RNA structures. The results show that although RiboDiffusion achieves perfect recovery scores (1.0000) on certain structures- as 1Q9A and IXPE-and attains the highest overall recovery rate (0.7181), these high-scoring instances likely correspond to structures present in its training data, as noted in the authors' public documentation. Since the training code for RifoDif. fusion not publicly available, cannot independently verify its training set composition rule
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Table 6: Results of different experiments in single state benchmark. The number of test steps is 1, 10,
50, 100.
Self-consistency metrics
Perplexity
Native seq.
2D-EternaFold
3D-RhoFold
Model
(J)
recovery (1)
scMCC (1)
scRMSD (1)
scTM-sccrre (1)
scGDT (t)
1-Step
1.615
0.595++0.02
0.22++0.02
14.85+
0.10++0.02
0.18+0.03
10-Step
1.26:20.02
0.598:10.01
0.485::0.04
12.77=+0.44
0.21::0.02
0,29:10.02
50-Step
1.16==0.01
0.601=:0.02
0.506+0001
10.83=
0,23:40.02
0,31=00.33
100-Step
1.13+0001
0.600++0.01
0.505++0.04
11.77=0.58
0.22++0.02
0.30=0002
Groundtruth sequence prodiction:
1.000:10.00
0.686+00000
5,23-10.07
0.56:1:000
0.55+00.0
Random sequenec prediction:
0,251=00000
0.012++0.00
24.40=0.34
0.04+00.00
0.02:+0.0
ViennaRNA 2D-only:
0,259++0.00
0.61 1+000000
20.34+0.10
0.07+0000
0.07++0.0
2.00
p(t) = 0.02 0.98 B(1.0, 1.9)
1.75
1.50
1.25
p(t)
1.00
0.75
0.50
0.25
0.00
0.0
0.2
0.6
0.8
1.0
Figure 3: The visualization of timesteps.
out potential data leakage; consequently, the generalizability its reported performance remains questionable. In contrast, the original RBFN demonstrates greater robustness and generalization capa- bility, outperforming RBFN(Diffusion) on 10 out of the structures and achieving a higher overall recovery rate (0.6012 2 vs. 0.5512). Notably, RBFN excels on several complex RNA motifs-for instance, achieves a recovery rate 0.8031 on the Loop motif (PDB: 354D) and 0.8588 on Sarcin/Ricin domain (1Q9A), substantially surpassing other methods. These findings suggest that, despite the success of diffusion models in continuous data generation tasks, their direct application to discrete RNA sequence design does not necessarily yield performance gains. Instead, RBFN's modeling strategy- better aligned with the discrete and structural nature of RNA prruence-structurg mapping-ent enables more reliable design unseen targets. Therefore, under the critical assumption of strict separation between training and test data, RBFN represents a more trustworthy and practically viable approach for RNA sequence design.
D Broader impacts
The proposed RNA inverse design model holds transformative potential for accelerating therapeutic development vaccine production (e.g., rapid-response pandemic preparedness) and precision medicine (e.g., customized mRNA therapies for rare diseases), while also enabling environmentally beneficial applications such as RNA-based biosensors for pollution monitoring. These advancements, however, necessitate careful consideration of dual-use risks: malicious exploitation for harmful RNA
22

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Ori: ACCSJCASNGGACACGGUU
D2 OCCCUAGCGGAACSGGGAA
D3 CCCUD6UUCGACCG980C
D4 CCOUDUCCARAAC59900
Original Structure
Fold Structure
scRMSD=1.99 1.92
ScRMSD=2.05
ScRMSD=2.19
ScRMSD=2.20
20J0
0r1:
CUDSCUGAAGCCOGCACOS
CRAGCCCCCCICAACCOOOO
ACGOCAAS
COCOCIGGACCCCCUCACUU
GCGOGOCCAUCRUGAGGU
ACGUCCGO
D2
GUASIAGACCHCEJCHOGS
MAGDOGCAACIGAIGAOGO
ACGACUAC
D3
CAGGACCGUUGAUIAAMMC
AUGACAAU
D4
GCUCIDGACGGUCCAACOS
COG0CCC3CUGUUCGGGC
ScRMSD-9.11 scRMSD-9.24
ScRMSD-9.43
ScRMSD-9.44
Figure 4: Structural examples of folded sequences using RhoFold, The red color indicates the areas where there are differences from the original sequence.
synthesis could be mitigated through WHO-aligned sequence screening protocols and ticred access controls, whereas equitable healthcare access requires balancing patent protections with open-source licensing for non-commercial research.
E
Limitations
This study has several limitations that need be recognized. The most pressing constraint stems from the current inability to conduct wet-experiment validation. This is due to practical laboratory limitations, and result, direct biological verification of the computational predictions cannot be achieved. Although gRNAde has carried out limited wet experiments, these experiments do not involve functional implementation, and the actual functionality still requires further validation. This methodological shortcoming is further exacerbated by the intrinsic challenges in RNA tertiary struc- ture prediction. Existing computational methods still exhibit limited accuracy in simulating complex three-dimensional conformations. The convergence of these factors poses inherent difficulties in comprehensively assessing the biological plausibility and spatial compatibility of the interaction patterns identified via our computational framework. In future research, experimental validation will be prioritized, and emerging advancements in RNA structure determination techniques will be incorporated to overcome these crucial limitations.
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Reviews:
[{"review_id": "YBWVIgmA0i", "reviewer": "Reviewer_qKvq", "summary": "This paper presents RBFN, a geometric-algebra-enhanced Bayesian flow network for inverse RNA design. RBFN incorporates geometric algebra to better capture 3D structural properties, generating representation for edges, planes, and higher-order substructures. In addition, to address limited RNA data and the small nucleotide alphabet, they introduce a new time-step distribution sampling strategy. The authors evaluate RBFN on RNA inverse folding benchmarks and show better performance than existing state-of-the-art models, including gRNAde.", "questions": "1. Can you incorporate more RNA inverse folding baselines such as RiboDiffusion or RhoDesign? These models are available on Github (e.g., https://github.com/ml4bio/RhoDesign)\n2. Can you design an experiment to validate the advantage of BFN? I can see the benefit of geometric algebra enhanced network but it's unclear BFN is the right choice. Wouldn't diffusion, GFlowNet, and other methods work as well?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. The idea of incorporating geometric algebra for RNA inverse folding is interesting and novel. It can generate representation not only at node-level, but also for edges, planes, which gives a richer representation of the geometric structure.\n2. The results show that RBFN achieves better performance than gRNAde, which is a well-known RNA inverse folding baseline.\n3. The paper is written clearly, with intuitive figures\n\nWeakness\n1. This paper presented ablation studies to study different modeling choices, but it didn't show why BFN is beneficial. Could the model perform better with a standard diffusion model enhanced with geometric algebra? This part is unclear.\n2. Comparison with additional RNA inverse folding baselines are needed, such as RiboDiffusion.", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer qKvq,\n\nWe express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript.\n\n> **W2 & Q1: Comparison with additional RNA inverse folding baselines are needed, such as RiboDiffusion. & Can you incorporate more RNA inverse folding baselines such as RiboDiffusion or RhoDesign? These models are available on Github (e.g., https://github.com/ml4bio/RhoDesign)** \n\nThank you for the valuable suggestion to incorporate additional RNA inverse folding baselines (RiboDiffusion and RhoDesign). We have extended our experiments and added results for these models in the rebuttal (see table below). While we currently cannot modify the manuscript due to submission system restrictions, **we will update the main text with these comparisons in the revised version upon acceptance**.\n\n\n| PDB ID | Description                                | ViennaRNA | FARNA | RDesign | Rosetta | gRNAde     | RBFN       | RhoDesign  | RiboDiffusion |\n| :----- | :----------------------------------------- | :-------- | :---- | :------ | :------ | :--------- | :--------- | :--------- | :------------ |\n| 1CSL   | RRE high affinity site                     | 0.25      | 0.20  | 0.4455  | 0.44    | **0.5719** | 0.4832     | 0.4643     | 0.4286        |\n| 1ET4   | Vitamin B12 binding RNA aptamer            | 0.25      | 0.34  | 0.3929  | 0.44    | **0.6250** | 0.4482     | 0.5428     | 0.5371        |\n| 1F27   | Biotin-binding RNA pseudoknot              | 0.30      | 0.36  | 0.3013  | 0.37    | 0.3437     | **0.5208** | 0.4211     | 0.3667        |\n| 1L2X   | Viral RNA pseudoknot                       | 0.24      | 0.45  | 0.3727  | 0.48    | 0.4721     | **0.5833** | 0.2593     | 0.5185        |\n| 1LNT   | RNA internal loop of SRP                   | 0.33      | 0.27  | 0.5556  | 0.53    | 0.5843     | **0.6375** | 0.3636     | 0.7273        |\n| 1Q9A   | Sarcin/ricin domain from E.coli 23S rRNA   | 0.27      | 0.40  | 0.4417  | 0.41    | 0.5044     | **0.8588** | 0.8148     | $\\underline{1.0000}$ |\n| 4FE5   | Guanine riboswitch aptamer                 | 0.29      | 0.28  | 0.4112  | 0.36    | 0.5300     | 0.5326     | **0.8209** | 0.7164        |\n| 1X9C   | All-RNA hairpin ribozyme                   | 0.26      | 0.31  | 0.3967  | 0.50    | 0.5000     | **0.5187** | 0.3833     | 0.6875        |\n| 1XPE   | HIV-1 B RNA dimerization initiation site   | 0.27      | 0.24  | 0.3834  | 0.40    | **0.7037** | 0.6522     | 0.6957     | $\\underline{1.0000}$ |\n| 2GCS   | Pre-cleavage state of glmS ribozyme        | 0.25      | 0.26  | 0.4518  | 0.44    | **0.5078** | 0.4990     | 0.2049     | 0.8310        |\n| 2GDI   | Thiamine pyrophosphate-specific riboswitch | 0.25      | 0.38  | 0.3523  | 0.48    | **0.6500** | 0.6042     | 0.2436     | 0.6410        |\n| 2OEU   | Junctionless hairpin ribozyme              | 0.23      | 0.30  | 0.5000  | 0.37    | **0.9519** | 0.5580     | 0.1905     | $\\underline{0.9535}$ |\n| 2R8S   | Tetrahymena ribozyme P4-P6 domain          | 0.27      | 0.36  | 0.5641  | 0.53    | 0.5689     | **0.7172** | 0.6415     | $\\underline{0.9937}$ |\n| 354D   | Loop E from E. coli 5S rRNA                | 0.28      | 0.35  | 0.4458  | 0.55    | 0.4410     | 0.8031     | **0.8261** | 0.6522        |\n|        | Overall recovery:                          | 0.27      | 0.32  | 0.4296  | 0.45    | 0.5682     | **0.6012**     | 0.4909     | $\\underline{0.7181}$ |\n\n**Key Observations and Justifications:**  \n1. **RiboDiffusion Performance:** While RiboDiffusion achieves high scores (e.g., 1.0000 on 1Q9A and 1XPE), we verified that these instances likely correspond to structures present in its training data (as noted in their repository documentation). Since the training code for RiboDiffusion is not publicly available, we cannot retrain or validate this behavior independently. We will add this clarification to the revised manuscript.  \n\n2. **RhoDesign Comparison:** For RhoDesign (RBFN variant), our model achieves competitive results across most benchmarks. Notably, RBFN outperforms RhoDesign on 10 out of 14 cases (e.g., 0.8588 vs. 0.8148 on 1Q9A, 0.7172 vs. 0.6415 on 2R8S). This suggests that our proposed architecture and training strategy yield robust generalization beyond the training distribution.  \n\n\n> **W1 & Q2: This paper presented ablation studies to study different modeling choices, but it didn't show why BFN is beneficial. Could the model perform better with a standard diffusion model enhanced with geometric algebra? This part is unclear. & Can you design an experiment to validate the advantage of BFN? I can see the benefit of geometric algebra enhanced network but it's unclear BFN is the right choice. Wouldn't diffusion, GFlowNet, and other methods work as well?** \n\nThank you for your insightful comment regarding the choice of Bayesian Flow Networks (BFN) over other modeling approaches such as standard diffusion models. We appreciate the need for a clearer justification and experimental validation of why BFN was selected for our RNA sequence generation task.\n\n**Justification Based on Existing Literature**\n\nOur decision to use BFN is grounded in existing research that highlights its advantages, particularly for discrete data tasks [1][2]. Specifically:\n\n- **Parameter Space Modeling**: As illustrated in Figure 2 of [2], BFN operates in parameter space rather than observation space, which inherently offers lower variance during sampling compared to standard diffusion models. This characteristic is crucial for generating high-quality discrete sequences like RNA.\n  \n- **Sampling Process Differences**: The sampling mechanism in BFN is fundamentally different from diffusion models. While diffusion models gradually denoise data through iterative steps, BFN directly models the flow of probability distributions in parameter space, leading to more efficient and effective sampling processes for discrete tasks.\n\nGiven these advantages, we adopted BFN based on its demonstrated success in handling discrete tasks such as molecular structure generation [2], drug design in continuous parameter space [3], and protein family design [4].\n\n**Experimental Validation Plan**\n\nTo further validate the benefits of using BFN over other methods, including standard diffusion models, we are currently implementing a comparative experiment:\n\n1. **Baseline Models**: We will extend a standard diffusion model or a GFlowNet variant, enhanced with geometric algebra, similar to our BFN approach. This ensures a fair comparison by maintaining consistent enhancements across all models.\n\n2. **Results Comparison**: By comparing the performance of BFN against these baselines, we aim to demonstrate the specific advantages of BFN in terms of efficiency, quality of generated sequences, and overall robustness.\n\n\nIn summary, while we believe BFN is the right choice for our application due to its unique properties and proven effectiveness based on existing literature and preliminary results, we are committed to providing empirical evidence through controlled experiments to support this claim.\n\n**References:**\n\n[1] Graves, Alex, et al. \"Bayesian flow networks.\" arXiv preprint arXiv:2308.07037 (2023).\n\n[2] Song, Yuxuan, et al. \"Unified generative modeling of 3d molecules with bayesian flow networks.\" The Twelfth International Conference on Learning Representations. 2024.\n\n[3] Qu, Yanru, et al. \"MolCRAFT: Structure-Based Drug Design in Continuous Parameter Space.\" Forty-first International Conference on Machine Learning. 2024.\n\n[4] Gong, Jingjing, et al. \"Steering Protein Family Design through Profile Bayesian Flow.\" ICLR. 2025.\n\nWe hope this clarifies our rationale and addresses your concerns. Thank you for your valuable feedback."}, {"review_id": "npr2o8Eu7K", "reviewer": "Reviewer_jqC4", "summary": "The paper introduces a novel model for RNA inverse design, i.e. the generation of RNA sequences given one (or several) 3D backbone structures. The model is based on the generative framework of Bayesian Flow networks, which allow to learn the distribution of discrete RNA sequences. The network architecture uses layers from recent graph neural networks based on Clifford algebra to enhance the geometric expressiveness of the model.  The paper further employs a new distribution for the sampling of time-steps which emphasizes times near t=0. The model achieves competitive result on various metrics in single- and multi-state RNA design benchmarks.", "questions": "- I think it would be good to include a plot of the beta distribution used for the time-step distribution for better intuition on the shape of the distribution\n- On page 4 it is indicated that for the neighborhood of a node the 32 nearest neighbors are used. I wonder why a fixed number of neighbors was chosen instead of defining a cutoff radius?\n- How long does inference take for models with different numbers of steps and compared to the baseline models (in particular gRNAde)", "limitations": "Yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "- Strengths:\n    - The paper has a clear target in improving RNA inverse design and overcoming common problems in this task like the existence of several conformers and target sequences for a given 3D structures. The proposed methods are sensible and are proven to be helpful in the ablation studies.\n    - The paper does provide errors obtained from runs with three different random seed for most conducted experiments\n- Weaknesses:\n    - It is very hard to follow the formal presentation of methods in the paper. The main problem here is that a large number of symbols and expressions are not properly defined anywhere which makes the framework only understandable with prior knowledge in both Geometric algebra and Bayesian flow networks. Below is an incomplete list of examples:\n        - There is no clear notation for scalars, vectors and higher grade elements of Clifford algebra. Sometimes vectors are written in bold sometimes not. Sometimes they are written with an arrow on top sometimes they are not.\n        - On page 3 , G is introduced as the 3D backbone structure, but the components of the tuple are not defined until section 3.1 and they also do not reference the backbone coordinates in the previous sentence.\n        - The notation for a Clifford algebra G_{p,q,r} and the parameters p,q,r are mentioned but never properly defined\n        - In the first paragraph of page 5 the index k is reused multiple times and sometimes wrongly instead of the correct index n (for the number of graphs). Other indices like f and f’ are never defined\n        - In line 165 the initialization of geometric algebra features is unclear, the symbol q was also never defined before\n        - …\n    - It is never explicitly stated which Clifford algebra is used within the model. I assume its the Euclidean geometric algebra since this is what the reference architecture uses. In this case the interpretation of elements as points, lines, planes and solid structures is wrong. Points, lines and planes would be appropriate interpretations for projective geometric algebra, whereas the geometric interpretation of Elements of the Euclidean geometric algebra corresponds to scalars, vectors, directed area elements (pseudovectors) and directed volume elements (pseudoscalars). A discussion for why this algebra was chosen over other geometric algebras which have also found use in recent deep learning architectures is missing\n    - Overall the results of RBFN are only marginally better than the baselines. Given that inference time is probably higher due to the iterative procedure, the use cases for the model are limited.\n    - There is the related work [1] which is not cited, but seems to achieve similar results on the single-state benchmark dataset\n\n[1] Manzourolajdad, A., & Mohebbi, M. (2025). Secondary-Structure-Informed RNA Inverse Design via Relational Graph Neural Networks. Non-coding RNA, 11(2), 18.", "was_revised": true, "final_justification": "I had many concerns about the presentation of the original submission, which had a lot of missing information and undefined expressions which made it very hard to follow the logic of the paper. During rebuttal the authors clarified many of these points. I remain hesitant to vote for acceptance of the work, since in my opinion a resubmussion would allow for a more accurate judgement of the presentation quality of the revised paper. Due to the improvements made during rebuttal I have raised my score to 3.", "rebuttal": "Dear reviewer jqC4,\n\nWe express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript. We denote geometric algebra by GA.\n\n> **W1: Question about formal presentation**\n\nThank you to the reviewer for pointing out this issue. We have revised the paper and will update it in the updated manuscript. Due to the word limit, we provide a brief summary of the revisions:\n\n- For **section 2.1 Problem Definition**, to avoid content confusion, we removed the introduction of G and revised the description to re-explain nucleotides so that readers can understand the structure of nucleotides and the content of backbone atoms. The specific symbol content has been moved to section 3.1.\n\n- For **section 2.2 GA**, we have rephrased the language and added explanations for the symbols, for example:\n\n'''\n\n $\\mathbb{G}_{p,q,r}$ denotes a GA over a real vector space with $p$ basis vectors of positive signature ($e_i^2 = +1$), $q$ of negative signature ($e_j^2 = -1$), and $r$ of zero signature ($e_k^2 = 0$), satisfying $p + q + r = n$ for an $n$-dimensional space.\n\n'''\n\n- **For section 2.3  Bayesian Flow Networks**, we have reorganized the relevant formulas using the symbols and expressions from diffusion models to make it easier for readers to understand. For example:\n\n'''\n\nBFNs introduce a continuous-time parameter $t\\in[0,1]$ and define two key distributions: (1) the \\textit{sender distribution} $q_t(\\mathbf{x}_t)$ representing the data distribution at time $t$, and (2) the \\textit{receiver distribution} $q_t(\\mathbf{x}_t|\\mathbf{x}_0)$ describing the forward process from the original data $\\mathbf{x}_0$ to the corrupted data $\\mathbf{x}_t$. The core of BFNs is the Bayesian inversion mechanism, where the reverse process is given by:\n\n$q_t(\\mathbf{x}_0|\\mathbf{x}_t) = \\frac{q_t(\\mathbf{x}_t|\\mathbf{x}_0)q_0(\\mathbf{x}_0)}{q_t(\\mathbf{x}_t)}$\n\n'''\n\n- **For section 3.1 GA GNN for RNA Three-Dimensional Structure Modeling**, we have provided a more detailed explanation of the symbols and have re-unified some individual symbols. Additionally, we have added symbolic identification for coordinates related to atoms, for example:\n\n'''\n\nFor an RNA structure with $m$ conformational states and $n$ nucleotides, we represent it as $m$ graphs $\\{ \\mathcal{G}^{(1)}, \\dots, \\mathcal{G}^{(m)}\\}$. \n\n...\n\n$\\mathfrak{q}(\\mathbf{v})$ is obtained based on the three-dimensional coordinate relationships. For a multivector $\\mathbf{v} = s + \\vec{v} + \\mathbf{b} + t\\mathbf{I}$ where $s$ is scalar, $\\vec{v}$ vector, $\\mathbf{b}$ bivector, and $t\\mathbf{I}$ pseudoscalar components, the quadratic form is $\\mathfrak{q}(\\mathbf{v}) = s^2 + \\|\\vec{v}\\|^2 - \\|\\mathbf{b}\\|^2 - t^2$ in $\\mathcal{G}_{3,0,0}$. \n\n'''\n\n- **For section 3.2 on Bayesian Flow Networks for RNA Generation**, we have concretized the relevant symbols and updated the descriptions to make them more focused on the task of RNA design, thereby making it easier for readers to understand. For example:\n\n'''\n\nBefore training, we define a \\textbf{uniform prior} over the 4 RNA nucleotides: $p_0(\\mathbf{y}) = \\text{Categorical}(\\mathbf{y}|\\pi_0)$ with $\\pi_0 = [1/4, 1/4, 1/4, 1/4]^\\top$. \n\n...\n\n'''\n\n**We hope that these detailed revisions and enhancements will make the methods section easier for readers to understand, thereby contributing to scientific research.**\n\n> **W2: Question about Clifford algebra**\n\nThank you for your professional feedback on the use of Clifford algebra in our paper. We will make the necessary revisions in the updated manuscript.  \n\nYou accurately identified a key conceptual confusion in our paper: in Figure 2, we incorrectly used the interpretative framework of projective GA, misinterpreting 0-vectors, 1-vectors, 2-vectors, and 3-vectors as \"points, lines, planes, and solid structures,\" respectively.  \n\nAs you correctly pointed out, in standard 3D Euclidean GA ($\\mathcal{G_{3,0,0}}$):\n\n- scalars represent quantities without direction\n\n- vectors represent directed line segments\n\n- bivectors represent directed planes/area elements\n\n- trivectors/pseudoscalars represent directed volumes  \n\nWe acknowledge that this was an oversight in our presentation, stemming from a confusion between different variants of GA. While we used Euclidean GA for calculations in our model design, we inadvertently adopted the common interpretative framework of projective geometry from computer graphics when explaining its biological significance. The intuitive interpretation of nucleotide positions in RNA molecular structures led us to erroneously map Euclidean GA elements onto the projective geometric interpretation. \n\nWe will make the following significant revisions in the updated manuscript:\n\n1. Fully correct the interpretation in Figure 2, clearly distinguishing the proper interpretative framework of Euclidean GA. \n\n2. Add a dedicated subsection (Section D.3) to discuss why we chose $\\mathcal{G_{3,0,0}}$ over other GAs (such as projective GA $\\mathcal{G_{3,0,1}}$ or conformal GA  $\\mathcal{G_{4,1,0}}$):\n\n- **Spatial Representation**: RNA structures reside in standard 3D Euclidean space, where $\\mathcal{G}_{3,0,0,}$ directly models spatial relationships without extra dimensions. RNA design focuses on nucleotide relative positions, orientations, and local geometry naturally captured by vectors and bivectors in Euclidean GA.  \n- **Task Alignment**: While projective GA (PGA) excels at representing points/planes, RNA design prioritizes vector relationships and local geometry over absolute positions. $\\mathcal{G}_{3,0,0}$ balances mathematical simplicity with sufficient expressiveness for key RNA structural features.  \n- **Computational Efficiency**: $\\mathcal{G}_{3,0,0}$ (8 dimensions) offers lower complexity than PGA (16D) or conformal GA (32D). This efficiency is critical for RNA inverse design involving long sequences (thousands of nucleotides).  \n\n\n3. Re-examine all sections of the paper that involve geometric interpretations to ensure conceptual consistency.  \n\nWe appreciate your identifying this critical issue. Your feedback helped us correct conceptual errors and deepened our understanding of the theoretical foundations of GA in RNA structure modeling. Your professional insights significantly enhanced the rigor and scientific value of our work.\n\n> **W3&Q3: Performance & inference time of RBFN**\n\nThank you for your attention to performance and the computational cost of RBFN.\n\nWhile RBFN shows modest but meaningful improvements over gRNAde, the gains align with RNA’s structural complexity:\n\n- 5.8% higher recovery (0.6012 vs. 0.5682),\n- 17.0% lower scRMSD error (10.83 vs. 13.04).\n\nGiven RNA’s extreme flexibility and shallow energy landscapes—where sub-angstrom precision dictates function—even incremental gains in structural accuracy (e.g., 2.21 Å scRMSD reduction) directly impact biological interpretability.\n\n\n\n**For the computational cost**, we re-ran the benchmarks on a GeForce RTX 3090. The experimental results show that RBFN requires only 3.05 seconds for long RNA (4269nt), **21.7 times faster than** gRNAde(AR), which takes 66.28 seconds. These results demonstrate that RBFN can efficiently handle ultra-long RNA on consumer-grade GPUs, making it suitable for laboratory-level deployment.\n\n\nHere are the test results:\n\n| **Length (PDB ID)** | **gRNAde (time)** | **RBFN (time)** | \n| ------------------- | ----------------- | --------------- | \n| 25 (2A43)           | **0.3222s**           | 0.7063s     | \n| 50 (6TF0)           | **0.6245s**          | 0.7026s     | \n| 75 (2HOM)           | 0.9811s           | **0.6911s**     |\n| 120 (3B4A)          | 1.5539s           | **0.7124s**     | \n| 159 (6D8M)          | 2.0698s           | **0.7207s**     | \n| 249 (9MEE)          | 3.2962s           | **0.7526s**     | \n| **4269 (8TOC)**     | **66.2848s**      | **3.0469s**     | \n\n> **W4&Q1: related work & a plot of the beta distribution**\n\nWe will supplement the **revision with a citation and discussion** of this literature to more comprehively present the current progress and challenges in the research, and we will also **add a plot of the beta distribution**. Using secondary structure is a great idea, which has also given us some inspiration. In our future work, we will further explore how to utilize this information.\n\n> **Q2: Using 32 nearest neighbors instead of a fixed truncation radius**\n\n\n\nThank you for this insightful question. We use fixed neighbor counts (32) instead of distance cutoffs to address fundamental challenges in RNA modeling:\n\n1.  **Solving Density Heterogeneity:**  \n    RNA regions show extreme density variations:  \n    - **Stems:** Tight packing (P-P ~5.9 Å → 6-10 neighbors at 5Å cutoff)  \n    - **Loops:** Loose packing (P-P 7-8 Å → 4-7 neighbors)  \n    - **Pseudoknots:** Ultra-dense (→15-20+ neighbors)  \n    Fixed cutoffs cause **unstable neighbor counts**, risking redundancy in dense areas and missed interactions in sparse regions.\n\n2.  **Mathematical Necessity for GA-GNN:**  \n    - **32 radial basis functions (RBFs)** encode distances.  \n    - **Matching neighbor count (32)** ensures each RBF interval is sufficiently sampled.  \n    - Critical for stable **geometric product calculations** and **multivector normalization**.\n\n3.  **Biophysical Justification:**  \n    - Core functional interactions occur within **~15-20 sequence neighbors** (local motif size).  \n    - **32 neighbors** intentionally **captures essential long-range tertiary contacts** while aligning with RBF dimensions.\n\n\nIn summary, the strategy of using a fixed number of neighbors allows our model to **robustly handle the density variability of RNA structures** while maintaining the mathematical rigor of GA representations."}, {"review_id": "tQmorzEPHj", "reviewer": "Reviewer_DZoU", "summary": "This paper introduces RBFN, a novel generative model for the RNA inverse design problem, which aims to generate valid nucleic acid sequences for a given 3D RNA structure. It combines a geometric-algebra-enhanced graph neural network with a Bayesian flow network (BFN) and addresses two key challenges: the \"one-to-many\" nature of the problem and the conformational flexibility of RNA. The work provides clear theories and promising experimental results.", "questions": "1. What is the metric used in Table 1?\n\n2. In Table 2, does AR refer to autoregression? The RBFN is only compared with the baseline gRNAde with AR. Are there additional comparisons with other models for this task, such as diffusion models or VAEs? Why not compare RBFN with the methods from Table 1?\n\n3. Is there any demonstration that the geometric algebra module provides geometric inductive bias? Why do the multiple vectors 0/1/2/3-... improve expressivity? Or the performance gain may come from the increasing parameters.", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper introduces geometric algebra to encode RNA 3D backbone geometry, yielding richer structural representations than traditional coordinate or hand-crafted features.\n\n2.  The work has effective ablation studies, removing either the geometric algebra module or the custom sampling schedule degrades performance markedly. The reports with mean and standard deviations add confidence that gains aren’t due to random chance.\n\n3.  The results show gains in diversity metrics (UCR/USR). They support the major hypothesis that a generative flow-based model is better suited for exploring the vast sequence space of RNA.\n\nWeaknesses: \n\nThe paper should explain the motivation for using geometric algebra and a Bayesian Flow Network. It currently does not describe in detail the limitations of the existing RNA modeling methods used as baselines, which components those approaches lack, and why geometric algebra together with a Bayesian Flow Network can address these gaps.", "was_revised": true, "final_justification": "I have reviewed the authors' response. Overall, it addresses the concerns reasonably well, but I find the motivation still unconvincing, largely due to its presentation. While the bullet points in the rebuttal help clarify some points, the authors did not clearly revise specific lines or paragraphs to improve readability, aside from a few high-level changes. As a result, it is difficult to assess the effectiveness of the revision, so I am keeping my score as borderline.", "rebuttal": "Dear reviewer DZoU,\n\nWe express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript.\n\n> **W: Motivation for using geometric algebra and a Bayesian Flow Network**\n\nWe sincerely thank the reviewers for their in-depth questions regarding the motivation behind our method and the limitations of existing RNA modeling approaches. Below, we will systematically elaborate on four aspects: **the shortcomings of current methods**, **the rationale for introducing GA**, **the design logic of BFN**, and the synergistic mechanism by which both **address key issues**. We will also illustrate the innovativeness of RBFN with experimental results. The relevant content will be updated in the subsequent version of the paper.\n\n**1. Core Limitations of Existing RNA Modeling Methods**\n\nThe current mainstream RNA design methods (such as Rosetta) face the following critical bottlenecks:\n\n**(a) Shallow 3D Structure Modeling**\n\nExisting tools over-rely on secondary structure predictions (e.g., ViennaRNA), neglecting tertiary interactions. This necessitates error-prone manual feature engineering (e.g., defining distance/torsion constraints in Rosetta), leading to poor generalization to complex conformations and noise sensitivity.\n\n**(b) Inefficiency in Discrete Sequence Generation**\n\nAutoregressive decoding (e.g., gRNAde) suffers from cumulative errors in long sequences (>500nt), where early prediction errors cascade through attention mechanisms. Additionally, insufficient distribution modeling treats sequence generation as discrete optimization, lacking probabilistic sampling of diverse, stable RNAs, resulting in low diversity/compatibility.\n\n**(c) Absence of Multi-State Modeling**\n\nExisting tools (e.g., RDesign) mostly focus on designing single states with fixed backbones, whereas biologically active RNAs (e.g., riboswitches) often need to dynamically switch between multiple conformations. \n\n**2. Motivation and Advantages of Introducing GA**\n\nGeometric algebra (GA) intrinsically encodes geometric inductive biases through its mathematical structure, directly addressing the 3D modeling limitations in existing methods. (See Q3 response for details.)\n\n**3. Design Logic and Advantages of BFN**\n\nWe adopted Bayesian Flow Networks (BFN) based on its demonstrated success in handling discrete tasks such as molecular generation [1], drug design in continuous parameter space [2], and protein sequence design [3]. BFN addresses the efficiency and diversity issues in discrete sequence generation by evolving distributions in a continuous parameter space:\n\n**(a) Mechanism of Continuous Time Distribution Evolution**\n\nBFN models sequence generation as the evolution of a probability distribution over time ( $t \\in [0,1]$ ), directly sampling complete sequences via continuous gradient optimization, thus eliminating cumulative errors in autoregressive decoding. \n\n**(b) Low-Variance Sampling and Small Alphabet Adaptation**\n\nBFN directly models the categorical distribution of the four bases, minimizing the KL divergence between the model distribution and the true data distribution. By leveraging continuous gradients, it enhances sampling efficiency, resulting in a higher sequence recovery rate (60.12%) for RBFN compared to gRNAde (56.82%).\n\n[1] Song, Yuxuan, et al. \"Unified generative modeling of 3d molecules with bayesian flow networks.\" The Twelfth International Conference on Learning Representations. 2024.\n\n[2] Qu, Yanru, et al. \"MolCRAFT: Structure-Based Drug Design in Continuous Parameter Space.\" Forty-first International Conference on Machine Learning. 2024.\n\n[3] Gong, Jingjing, et al. \"Steering Protein Family Design through Profile Bayesian Flow.\" ICLR. 2025.\n\n**4. Synergistic Mechanism of Geometric Algebra and BFN**\n\nThe 3D structural features encoded by the GA module (e.g., bivector plane relationships) **serve as the prior distribution** for BFN, guiding sequence generation to ensure spatial compatibility. During BFN’s evolution process, the GA module continuously validates structural feasibility and corrects deviations via gradient backpropagation, enabling dynamic feedback.\n\nIn the revised manuscript, we will strengthen the explanation of the method's motivation at Introduction, clearly outlining the limitations of existing methods and the targeted improvements of RBFN in the introduction and related work sections. We believe that RBFN, through the deep integration of Geometric Algebra and Bayesian Flow Networks, provides a new methodological framework for the field of RNA design.\n\n> **Q1: Metric in Table1**\n\nNative Sequence Recover, we will update the version and place it in the table caption. Thank you for pointing out this issue.\n\n> **Q2: Table2's explain and compare methods**\n\n\nYes, AR (Autoregressive) in Table 2 refers to Autoregressive Modeling.\n\nIn the original submission, RBFN was primarily compared with gRNAde (the current SOTA model) because its publicly available code and evaluation metrics facilitated direct comparison. However, during the Rebuttal phase, we further supplemented the following experiments:\n\n- RhoDesign: An RNA design method based on 3D structure.\n\n- RiboDiffusion: An RNA design method based on a diffusion model, which generates sequences by progressively reversing noise.\n\nDue to space limitations, the results of the additional comparative experiments can be found in the Rebuttal for qKvq.\n\nThe methods in Table 1 are mainly based on secondary structure for generating RNA sequences and perform poorly in 3D structure. Therefore, we primarily compared with the SOTA method, gRNAde. Additionally, to demonstrate the improvement of RBFN in 3D structure metrics (such as scTM-score), we retained the comparison with ViennaRNA (a classic 2D tool).\n\n> **Q3: geometric inductive bias and improve expressivity**\n\nWe are grateful for the reviewer's in-depth attention to the GA. We provide a targeted response to this issue:\n\n**1. Does the Geometric Algebra module provide geometric inductive bias?**\n\n\nYes, GA intrinsically embeds powerful geometric inductive bias into the model, significantly enhancing its capability to represent complex RNA three-dimensional structures. This advantage stems directly from GA's mathematical structure, which encodes physical-world geometric relationships through:  \n**1) Hierarchical multivectors** (scalars, vectors, bivectors, trivectors/pseudoscalars) and  \n**2) Fundamental geometric operations** (outer product, inner product, rotors).  \nCritically, each multivector level captures distinct geometric semantics:  \n- **Vectors (1-vectors)** represent directional displacement (e.g., distances between atoms like C4' and P);  \n- **Bivectors (2-vectors)** characterize planar orientation (e.g., base pair relationships);  \n- **Trivectors/pseudoscalars (3-vectors)** model volumetric properties and global conformational changes.  \n\nThis hierarchical representation enables the model to **directly learn** essential spatial patterns—distances, angles, and torsion angles—**without relying on error-prone manual feature engineering**. The critical role of GA's inductive bias is unequivocally demonstrated in ablation studies: removing GA features (\"w/o GA\") caused a significant performance drop, reducing sequence recovery rate from 60.1% to 57.0%. These results confirm that GA's geometric priors are fundamental to the model's enhanced ability to capture and exploit RNA structural information.\n\n\n**2. How do multivectors (0/1/2/3...) enhance expressive power?**\n\n\n **The hierarchical geometric semantics and compositional capabilities of multivectors fundamentally enhance the expressive power of RNA structure models.** Geometric Algebra achieves this through its graded multivector structure, where each level captures distinct physical properties: _scalars_ encode global attributes like energy states; _vectors_ represent directional features (e.g., atomic displacements like C3'→P distances); _bivectors_ define planar orientations critical for base-pairing geometries; and _trivectors/pseudoscalars_ characterize volumetric relationships governing RNA tertiary conformations. These geometric primitives combine compositionally through operations like the geometric product (uv=u⋅v+u∧v), enabling simultaneous modeling of symmetric (scalar) and antisymmetric (bivector) interactions – a capability absent in traditional RNA design frameworks. Further amplifying this expressivity are GA's **inherent operational advantages**:\n\n - The **outer product** (u∧v) constructs higher-dimensional subspaces (e.g., generating a bivector plane from two vectors), capturing nonlinear geometric relationships essential for complex motifs;\n - **Rotors** implement efficient spatial transformations (R=e−θB/2) without Euler/matrix decomposition, streamlining conformational modeling;\n - **Parameter efficiency** emerges from GA's compact representation – a single bivector in G3,0,0 space can encode rotational and planar data that would require multiple disconnected parameters in classical GNNs. This unified mathematical framework thus provides both richer structural representation and greater computational economy for RNA design tasks.\n\n\n\n**3. Is the performance improvement due to an increase in the number of parameters?**\n\nThe performance improvement is mainly attributed to the structural advantages of Geometric Algebra, rather than a mere increase in the number of parameters. gRNAde (with 1,104,311 parameters) achieved only a 56.82% sequence recovery rate in the single-state task, while RBFN (with 1,106,977 parameters) reached 60.12%, indicating that the structural design of Geometric Algebra is more efficient."}, {"review_id": "jBYjoQ0gpp", "reviewer": "Reviewer_LyRL", "summary": "This paper utilized geometric algebra to represent the 3D structure of RNA. Then they proposed a Geometric Algebra-Enhanced Bayesian Flow Network (RBFN) for RNA inverse design. RBFN improves RNA sequence generation by combining geometric algebra for better 3D structure modeling with a Bayesian Flow Network to sample sequences that correspond to a given RNA 3D backbone. The method outperforms existing RNA design tools in benchmarks, showing improvements in sequence recovery and structural consistency.", "questions": "- Does the method work for more dynamic RNA structures?\n- What are the computational costs for large-scale RNA design tasks?", "limitations": "- Computational Cost: The method could be computationally expensive for large RNA molecules.\n- Data Quality Dependence: The method relies on accurate 3D RNA structural data, which may not always be available.\n- Lack of Wet-Lab Testing: Without experimental validation, the real-world applicability is uncertain.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Originality: Combining geometric algebra with Bayesian Flow Networks for RNA design is novel.\n- Significance: It has potential applications in RNA-based therapies and synthetic biology.\n- Experimental Validation: The method shows superior performance in benchmark tests. The structure evaluation is very good, which may be ignored by previous works.\n- Clarity: The paper is well-structured and clear for a technical audience.\n\nWeaknesses:\n- Complexity: The technical concepts, particularly geometric algebra and Bayesian Flow Networks, may be challenging for some readers.\n- Data Scarcity: The method depends on the availability of high-quality RNA structural data.\n- Limited Wet-Lab Validation: There’s no real-world experimental validation of the method.", "was_revised": true, "final_justification": "This method seems solid and useful, so I will remain my score of 5.", "rebuttal": "Dear reviewer LyRL,\n\nWe express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript.\n\n> **W1: Complexity: The technical concepts, particularly geometric algebra and Bayesian Flow Networks, may be challenging for some readers.**\n\nWe understand the reviewer's concerns about the complexity of Geometric Algebra (GA) and Bayesian Flow Networks (BFN). To lower the barrier to understanding, we will take the following measures in the revised manuscript to ensure that the technical details are clearer and more comprehensible:\n\n**1. Strengthening and layered explanation of the background section**\n\nWe will introduce the core concepts of Geometric Algebra in a layered manner, including the basic framework, core operations, practical usage examples, and provide sufficient references for further learning. Additionally, we will further clarify the mechanisms of BFN, including the evolution of continuous-time distributions, differences from traditional diffusion models, handling of discrete data, and visual diagrams of time-step sampling to make it clearer for the readers.\n\n**2. Enhancing the writing of the paper**\n\nWe will clarify the descriptions of symbols and related elements, and unify the writing style. We have also written pseudocode in the appendix for reference, and we will add tables explaining the relevant symbols to make it easier for everyone to understand.\n\n> **W2 & L2: Data Scarcity: The method depends on the availability of high-quality RNA structural data. & Data Quality Dependence: The method relies on accurate 3D RNA structural data, which may not always be available.**\n\nThe definition of the RNA inverse design task dictates that **sequence generation must be based on existing three-dimensional structures** (such as the fixed backbone redesign task described in the paper). This limitation stems from the strong correlation between RNA function and its three-dimensional conformation. Current methods, including RBFN, require generating sequences conditioned on input structures, thus they cannot operate without the support of high-quality structural data. The current RNASolo dataset contains only about 18,955 RNA structures, as of 24-07-2025, which leads to a data scarcity issue in model training. Future improvements can be made in two directions:\n\nThe first direction is, as the performance of RNA structure prediction tools improves, **predicted three-dimensional structures could supplement real data**. For example, similar to how ESM-IF[1] uses AlphaFold2-predicted protein structures for inverse folding training, RNA design could alleviate data bottlenecks by constructing virtual training sets based on predicted structures, thereby expanding sample diversity and improving performance. \n\nThe second direction is to overcome data dependency, future work will explore **joint modeling of sequences and structures**, optimizing both simultaneously through multi-task learning, reducing reliance on single-sided data, much like existing works in protein design[2], aiming at the co-generation of RNA sequences and structures.\n\n[1] Hsu, Chloe, et al. \"Learning inverse folding from millions of predicted structures.\" International conference on machine learning. PMLR, 2022.\n\n[2] Campbell, Andrew, et al. \"Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design.\" International Conference on Machine Learning. PMLR, 2024.\n\n> **W3 & L3: Limited Wet-Lab Validation: There’s no real-world experimental validation of the method. & Lack of Wet-Lab Testing: Without experimental validation, the real-world applicability is uncertain.**\n\nWe understand the reviewer's concern regarding wet lab validation. **Due to the current limitations of our laboratory conditions**, equipment resources, and experimental timelines, we are unable to independently complete functional validation experiments for RNA sequence design (such as in vitro folding validation, structural characterization, or functional activity testing) at this time. We are also very interested in conducting wet lab validations, and we commit to open-sourcing our code upon acceptance of the paper, **encouraging researchers worldwide to conduct validations in their own laboratories**. For example, we will provide standardized RNA sequence design templates to facilitate rapid experimentation by collaborating labs.\n\n> **Q1: Does the method work for more dynamic RNA structures?**\n\nTheoretically, RBFNs are capable of handling more dynamic RNA structures. The core idea is to **treat the dynamic conformations as a collection of multiple static states**. However, incorporating more conformations will increase the memory usage due to the expansion of the feature tensor dimensions.\n\n> **Q2 & L1: What are the computational costs for large-scale RNA design tasks? & Computational Cost: The method could be computationally expensive for large RNA molecules.**\n\nThank you for your attention to the computational cost of RBFN. We re-ran the benchmarks on a GeForce RTX 3090 (24GB VRAM) and have made the test code publicly available. The experimental results show that **RBFN requires only 3.05 seconds and 1.84GB of VRAM for long RNA (4269nt)**, significantly outperforming gRNAde, which takes 66.28 seconds. These results demonstrate that RBFN can efficiently handle ultra-long RNA on consumer-grade GPUs, making it suitable for laboratory-level deployment.\n\nThe following is how we add the test code for calculating time and memory usage:\n\n```python\n\n    # Time & Memory Monitoring, Before Running the Model\n    torch.cuda.reset_peak_memory_stats(device)\n    time_start = time.time()\n    \n    # Inference (50-step sampling, batch=1)\n    with torch.no_grad():\n        sequence = model.design(pdb_path, steps=50)\n\n    #After the operation is completed, get the time and GPU memory usage.\n    time_end = time.time()\n    peak_mem = torch.cuda.max_memory_allocated(device) / 1024**2  # MB\n    return time_end - time_start, peak_mem\n```\n\nHere are the test results:\n\n| **Length (PDB ID)** | **gRNAde (time)** | **RBFN (time)** | **Memory (RBFN)** |\n| ------------------- | ----------------- | --------------- | ----------------- |\n| 25 (2A43)           | **0.3222s**           | 0.7063s    | 537.15 MB         |\n| 50 (6TF0)           | **0.6245s**           | 0.7026s    | 548.04 MB         |\n| 75 (2HOM)           | 0.9811s           | **0.6911s**     | 557.44 MB         |\n| 120 (3B4A)          | 1.5539s           | **0.7124s**     | 572.70 MB         |\n| 159 (6D8M)          | 2.0698s           | **0.7207s**     | 583.30 MB         |\n| 249 (9MEE)          | 3.2962s           | **0.7526s**     | 614.96 MB         |\n| **4269 (8TOC)**     | **66.2848s**      | **3.0469s**     | **1839.95 MB**    |"}]

Author rebuttal:

Meta-review: This paper addresses the important task of inverse RNA design and proposes a Bayesian flow network. Experimental results show that the method outperforms existing state-of-the-art models. The submission underwent a long discussion during the rebuttal process. Although the authors made substantial revisions, high-level concerns remain. Reviewers pointed to unclear motivation for using Bayesian flow networks (DZoU) and noted that parts of the method description were difficult to follow (jqC4). After reading the rebuttal, I feel these issues are not fully resolved. On the other hand, the work makes valuable contributions in capturing RNA conformations and in addressing challenges posed by limited data, which could inspire future research in tackling RNA modeling problems. Taken together, I view this as a promising though imperfect contribution, and I lean toward a weak accept.
Decision: Accept (poster)