scDFM: Distributional Flow Matching Model for Robust Single-Cell Perturbation Prediction
Authors: Chenglei Yu
Chuanrui Wang
Bangyan Liao
Tailin Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Machine Learning
Single Cell
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: A central goal in systems biology and drug discovery is to predict the transcriptional response of cells to perturbations. This task is challenging due to the noisy, sparse nature of single-cell measurements and the fact that perturbations often induce population-level shifts rather than changes in individual cells. Existing deep learning methods typically assume cell-level correspondences, limiting their ability to capture such global effects.
We present **scDFM**, a generative framework based on conditional flow matching that models the full distribution of perturbed cells conditioned on control states.
By incorporating an MMD objective, our method aligns perturbed and control populations beyond cell-level correspondences. 
To further improve robustness to sparsity and noise, we propose the Perturbation-Aware Differential Transformer architecture (PAD-Transformer), a backbone that leverages gene interaction graphs and differential attention to capture context-specific expression changes.
**scDFM** outperforms prior methods across multiple genetic and drug perturbation benchmarks, excelling in both unseen and combinatorial settings. In the combinatorial setting, it reduces MSE by 19.6\% over the strongest baseline.
These results highlight the importance of distribution-level generative modeling for robust $\textit{in silico}$ perturbation prediction. The code is available at https://github.com/AI4Science-WestlakeU/scDFM.

Published as a conference paper at ICLR 2026
SCDFM: DISTRIBUTIONAL FLOW MATCHING FOR Ro- BUST SINGLE-CELL PERTURBATION PREDICTION
Chenglei Yu?.2; Chuanrui Wang"; Bangyan Li & Tailin Wu
1Zhejiang University
Department of Artificial Intelligence, School of Engineering, Westlake University (yuchenglei, wangchuanrui, wutailin lake. edu.
ABSTRACT
A central goal in systems biology and drug discovery is to predict the transcrip- tional response of cells to perturbations. This task is challenging due to the noisy, sparse nature of single-cell measurements and the fact that perturbations often induce population-level shifts rather than changes in individual cells. Ex- isting deep learning methods typically assume cell-level correspondences, limit- ing their ability capture such global effects. We present ScDFM, a generative framework based on conditional flow matching that models the full distribution of perturbed cells conditioned on control states. By incorporating an MMD ob- jective, our method aligns perturbed and control populations beyond cell-level correspondences. To further improve robustness to sparsity and noise, we propose the Perturbation-Aware Differential Transformer architecture (PAD-Transfommer), a backbone that leverages gene interaction graphs and differential attention to capture context-specific expression changes. SCDFM outperforms prior methods across multiple genetic and drug perturbation benchmarks, excelling in both un- seen and combinatorial settings. In the combinatorial setting, it reduces MSE by 19.6% over the strongest baseline. These results highlight the importance distribution-level generative modeling for robust in silico perturbation prediction. The code is available at http:://ithbb..oomAAASeeeeeeeeee
INTRODUCTION
Accurate prediction of the transcriptomic response of cells to genetic drug perturbations at single- cell resolution is a central challenge in functional genomics and drug discovery (Bunne et al. 2023] Lotfollahi al. 2023). Understanding these responses not only reveals complex gene regulatory networks, but also accelerates the design of novel therapeutic strategies and allows personalized medicine (Qi al. 2024 Viñas Torné al. 2025). However, given the exponential growth in the
number of potential gene or drug combinations, systematically screening all possible perturbation combinations by experimental means is practically infeasible (Roohani al. 2024). As result, the development of in silico models capable of accurately predicting cellular perturbation effects has become critical, and progress in this area remains primary barrier to advancements in the field.
A fundamental challenge in modeling single-cell perturbation responses lies in the unpaired nature of the data. Due to the destructive nature of RNA sequencing, it is impossible to observe the same cell both before and after perturbation. This makes cell-level pairing and supervision impossible, and standard pointwise losses ill-suited. Most existing models thus focus narrowly on recovering the mean expression profile, ignoring higher-order statistics such variance, skewness, or shifts in subpopulation proportions (Mejia et al. 20255 Yu et al. 22025) Chi et al. 2025). This turns out to be serious limitation; for example. |aaakkishhan al. (2025) shows that perturbations induce com- plex distributional changes beyond the mean that many current methods fail to capture. Moreover, benchmarks such as Systema (Viñas Torné et al. 2025) show that simply predicting the population
mean can perform even better than many sophisticated models under standard metrics (Ahlmann-
Eltze et al. 2025 Csendes al. 2025
"Equal contribution.
&Corresponding author.

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Published as a conference paper at ICLR 2026
Moreover, single-cell transcriptomic data present severe modeling difficulties due to their sparse, zero-inflated and noisy nature. Technical dropout often yields missing values that do not reflect true biological absence (Dai et al. (024). while batch effects and uneven sequencing depth distort
gene-gene correlations (Zhou et al. 2025 Moreover, perturbation effects are nonlinear and context-
dependent, making the modeling task substantially harder (Xing & Yau 2025
Song al. 2025).
As a result, models that treat genes as independent inputs or rely on shallow architectures struggle generalize to new cell types or out-of-distribution perturbations. When the model backbone treats genes independently without explicit gene co-expression relationships, it tends to overfit noise rather than extract meaningful biological signal. Empirical work reports that Geneformer (Theodoris al. 2023) and scGPT (Cui et al., (024) underperform simpler baselines with standard batch correction tools in zero-shot settings (Kedzierska al. 2025). This underscores the need for more expressive, perturbation--oouss. and noise-resistant model backbones.
We propose SCDFM, generative framework based on conditional flow matching (Lipman et al.) 2022). to accurately reconstruct the distribution of single-cell gene expression after perturbations. Our model tackles the two major limitations mentioned above: the neglect population- dis- tributional fidelity and the failure to account for noisy, interdependent gene regulation.
First, to address distribution-level fidelity, we incorporate the Maximum Mean Discrepancy (MMD) loss (Gretton et al.] 2006) into training objective. Unlike traditional loss functions between paired prediction and ground truth, the MMD loss directly quantifies the distance between the cell distribution generated by the model and the true post-perturbation cell distribution. By combin- ing the MMD loss with the flow matching objective, the model is guided to reproduce the overall population shift induced by perturbations rather than merely improving per-cell predictions, thereby better capturing distribution-level effects.
Second, to mitigate noise and capture complex regulatory dependencies, introduce the BerturbationAAaar Differential Transformer (PAD-Transformer). It incorporates gene-gene masked attention, where co-expression graph guides the model focus on biologically related genes, and employs differential attention to separate control and perturbation signals and highlight their interactions. By integrating these components into unified backbone, PAD-Transformer filters noise, preserves regulatory structure, and scales effectively to complex perturbation scenarios.
We evaluate our method on two challenging benchmarks: (1) the Norman combinatorial gene- perturbation dataset (Norman et al. 2019). and (2) the Combosiplex combinatorial drug- frnggperturbbtion dataset (Mathur et al.] 2022). On the Norman dataset, we assess generalization under two settings: (i an additive setting, where all singles and a subset of duals are used for training, and (ūi a hold- out setting, where specific dual combinations are completely excluded from training. Together, these evaluations demonstrate that the synergy between biologically structured attention and distribution- level training in scDFM significantly improves both accuracy and robustness of in silico single-cell perturbation prediction.
2 RELATED WORK
Foundation Models for Single-Cell Biology. Recent advances in large-scale pretraining have led to the development of foundation- style models for single-cell data, such as Geneformer (Theodoris) al.) 2023), scBERT (Yang et al. 2022), scFoundation (Khan et al., (202)), scGPT (Cui et al.p 2024.. and UCE (Rosen et al. 2023). These models are typically trained on large collections of unperturbed expression profiles to learn general-purpose embeddings of cells and genes, which can then be transferred to downstream tasks with minimal supervision. While such approaches enable broad applicability and data-efficiem, learning, several studies (Ahlmann-Elle) et al. 2025| Csendes let al. 2025 Kedzierska et al.) 2025) have shown that they may struggle capture perturbation- specific effects, especially when distributional changes go beyond the population mean.
Models Tailored to Perturbation Effects. A parallel line of work has proposed perturbation- specific models that more directly capture gene regulatory dynamics. CPA (Lotfollahi et al.) (023) models the compositional structure of perturbations by embedding genes and conditions in shared latent space, enabling extrapolation to novel combinations. GEARS (Roohani et al., 2024) incorpo- rates biological priors such as gene-gene co-expression and Gene Ontology into the model archi-
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Experimental data Input Data Target Distribution [e]
Differential Transformer
Liner
(c))
Gane Emb
Exprrtt
[sottesseg..k ( eestrannguuili
Mosk
Mull-
Q11
oz
1v
Pape
trejectory (Flow Matching Process)
Conditional velocity Prediction via PAD-Transtormer
(ca
[*]
Cell Encoder
X N
refecity
Perturbatonn Embedding
Fye O1 04
Emio
Transformer
Differential Transformer
AG
MMD
Game
Cell Encoder
Embh
Expr
t=0
Frrse
adaLN
adaLN
Emibeddig
Figure I: Overview of scDFM, which models cntturrbation-ppecifit cell state transitions as flow matching process from noise to perturbed expression. The TAD-Transformer predicts time-dependent velocities conditioned control cell context and perturbation embedding, while gene-ges gene masked attention and differential Transformer layers capture biological dependencies. Final distributional alignment is enforced via MMD regularization.
tecture, improving generalization to unseen perturbations. PerturbNet (Yu al. 2025 and GPer- turb (Xing & Yau 2025) further explore probabilistic formulations for unseen condition prediction.
More recently, generative approaches such scDiffusion (Bunne et al. 2023 CellFlow (Klein (ct al. 2025), and UNLASTING (Chi et al.) 2025) have applied diffusion- based models to learn continuous trajectories of cellular transitions under perturbation. These models often framed in latent space and provide a principled way to interpolate between control and perturbed states. However, their performance is sensitive the choice of embedding space. Our proposed method draws connections to both lines of prior work. Like foundation models, we leverage attention-baase encoders to extract rich representations from unperturbed expression profiles. At the same time, our method explicitly models the perturbation-inducee transition dynamics in the expression space with flow matching (Lipman al. 2022).
3 METHOD
{91, 12.., J denote the set of G profiled genes. The pre- perturbation state of a cell is represented as c₂ the expression level of gene 9; in the unperturbed cell. Analogously, the post pestturrturbtionn state is x = (a. x(gG)) E RRC, where x(g)) denotes expression level of gene Ii after pertur- bation. The perturbation condition is encoded as a multi-hot vector Cp E {0, 1]d, with cplil = if the j-th perturbation (e.g., drug CRISPR guide) is applied. Formally, each training instance consists of (Ca, Cp. I), and the goal is to learn the conditional generative model po Cz cp) that captures the population-leved distribution of perturbed cell states and generalizes novel perturba- tion combinations not observed during training.
Problem Setup. Let G
RÇ, where (g))) is
Model Overview. Figure provides an overview of ScDFM, which is built on a flow matching architecture (Section[..1 The model learns continuous trajectories that transform noisy initial states into target perturbed expressions through iterative conditional refinement, conditioned the control expression C₂ and the perturbation signal cp. At each step, gene expression features are encoded with gene gene-gene gene correlation mask (Section 3.3), while perturbation and time embeddings injected into stacked PAD- Transformer blocks to capture perturbation-an -waee dynamics (Section (.4). To ensure fidelity at both local and global levels, training combines the conditional flow matching loss

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with multi-kemnel MMD regularizer (Section 3.2), while velocity head estimates instantaneous changes along the trajectory.
3.1 PERTURBATION PREDICTION WITH FLOW MATCIINNG
Flow Matching (FM) (Lipman et al. 2022 is continuous- generative modeling framework that learns a time-dependent velocity field to morph a source distribution into a target distribution along continuous trajectory. In this work, we make the first attempt apply the FM framework directly in the high-dimensions gene expression space. Specifically, the source distribution øo is defined as the noisy gene expression distribution, and the target distribution 41 is defined as the perturbed gene expression distribution. The transformation evolves over a denoising time interval i. E [0, 1], conditioned on both the control state er (the pre-perturbation gene expression) and the perturbation condition Cp (which may correspond single combinatorial perturbations).
Formally, our objective is to learn a conditional velocity field Ug (ar, 1, (,Cx;c)) izes the instantaneous rate of change of the cell state at denoising time l. The state evolution follows conditional ODE:
Gp). This field character-
dxx; dt where X, denotes the generated gene expression state at time t.
(o t, Cz, Cp).
During training, Conditional Flow Matching (CFM) minimizes the discrepancy between the pre- dicted velocity and the reference velocity v induced by predefined interpolation path Tr (20. 11), for which we adopt the linear form nz (xo, xi) = (1-1)zo+ Given a control state e, which tells us cell line identity and perturbation condition cp. we define 2o to noisy source expression, and 21 ~ 91 (- Cz, Cp) as the corresponding post-perturbation expression drawn from the target distribution. The training objective is given by:
21, (CF Ee,,, Exo--on. 1 [lvo(ara C, cp) v(rr No. 2 6, Cz, 11
This formulation enables the model to directly learn the conditional transformation from noisy in- termediate states to the true pestturturbation states, while explicitly incorporating both the initial control state and the perturbation condition.
3.2 FLOW MATCHING WITH MULTI- -ERRELL MMD REGULARIZATION
Our framework learns conditional flow matching (FM) process by optimizing a velocity field Uo that aligns the generated trajectories with reference perturbation dynamics. This encourages biologically plausible and coherent evolution over continuous time. However, FM alone enforces local dynamical consistency and does not guarantee that the terminal distribution of generated cells X statistically aligns with the ground-truth perturbed distribution X.
To address this limitation and ensure population-level fidelity, introduce distribution-leww regu- larization term based on the Maximum Mean Discrepancy (MMD), which directly compares the pre- dicted terminal distribution the empirical distribution of ground-truth perturbed cells. We choose MMD over KI. divergence or Wasserstein distance because it is directly sample-based, computa- tionally efficient, and robust under support mismatch, making it well-suited for nigh-dimensional single-cell settings.
One-step prediction and target endpoint distribution. each training step, for sampled inter- mediate state ~ Te (Zo, I // Cp), we apply the learned velocity field uo (z,21, cr cp) to compute a one-step prediction of the perturbed state:
è 2, + (1 - t) 1o(2, cas op), with Cu ~ DControl
where Deonnrll denotes the empirical distribution of pre- pee-peruurbation (control) cells. This formulation approximates the endpoint of the flow. This yields a batch of model-predicted terminal samples (2(") X,,ē [0, B). which we compare against empirical samples {x(²) ~ X, drawn from the ground-truth post-perturbation cell population.
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Multi-kernel MMD regularizer. To measure the discrepancy between X1 and X1, we the squared MMD with a mixture of Gaussian RBF kernels:
I.
kmix (x,, x') = L
exp
1e==='1" 2
(3)
2=1
where {at (ot)=1 are bandwidths selected via median heuristic. In practice, we estimate refer- ence scale from pairwise squared distances and generate small set of bandwidths by multiplying this scale with fixed factors, which stabilizes training across heterogeneous cell populations. While Conditional Flow Matching (CFM) provides a principled way to learn velocity fields that interpo- late between control and perturbed states, it primarily enforces local dynamical consistency along trajectories. This means that the model is trained to match instantaneous velocity fields but does not directly constrain the global distributional outcome. As a result, terminal distribution of gener- ated cells X may deviate from the ground-trute perturbed distribution X, leading to mismatches in population-level statistics biologically relevant gene expression patterns.
The final training objective combines pointwise loss and distribution-level signals:
[ = LCFM + X LMMD,
(4)
LMDD (0) Eegno 21 (tlea,ep) 1) (I1, ( Ca. Cp)
(5)
where À O balances trajectory consistency against endpoint distributional fidelity. This combina- tion allows the model to learn both the fine-grained trajectory of individual cells and the global shift in the cell population distribution, addressing both local and global fidelity. The specific configura- tion and algorithm of the MMD regularization is provided in Appendix A.2
3.3 INITIAL EMBEDDING OF GENE EXPRESSIONS
Gene Encoding. Given a control cell expression profile 6, E RRC a perturbation condition Ops the current timestep f, and the corresponding perturbed expression 2t RRG, we construct cell representations over a selected subset of genes S _ G:
(6) where ez and (S) denote the expression values restricted to the selected gene subset S. Here E, maps each expression value to a d-dimensional embedding, while E, E S| is a sequence of contextualized gene embeddings obtained from cross-attention based gene encoder. The cross- attention mask derived from gene-gene relationships within the dataset, ensuring that each gene token interacts only with biologically relevant neighbors.
hg E.(( + ES),), hi Ex((')) + E,(),
Gene-gene co-expression graph attention mask. Relying solely the cross-attention net- work insufficient to capture intrinsic dependencies among genes. reality, genes are or- ganized within complex regulatory networks, which directly determine the transcriptomic changes under perturbations. Ignoring such dependencies may cause the model to treat each gene as an independent feature, thereby limiting the biological plausibility of the predictions.
To address this issue, we construct a gene-gene co-expression graph from the training data and incorporate it into the attention mechanism. For any two genes i and j, we define the edge weight as the absolute Pearson correlation:
Cov( 25) o (xx)), o
Wsj
where and 2j are the expression vectors of genes and j across all cells. Based on the weight matrix W = (wyy), we apply a KNN strategy to select the most strongly correlated (positive or negative) neighbors for each gene, yielding a sparse adjacency matrix A.
This static graph serves biologically grounded prior and is used to construct sparse attention mask in the gene encoder Eg. thereby constraining the attention mechanism to focus on biologically meaningful interactions.
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3.4 BACKBONE DESIGN
A core challenge in modeling single-cell perturbation responses lies in the noisy, sparse, and high- dimensional nature of the data. To address this, introduce the Perturbation-Aware Differential Transformer (PAD-Tranffrmmrr), a backhone that injects perturbation signals at every layer while employing differential attention suppress spurious correlations from noisy genes. By jointly en- coding control and perturbed states, the model captures explicit dependencies between them. Its architecture combines differential attention module with a perturbation-awars latent refine- ment block, enabling robust modeling of perturbation-specee dynamics under challenging single- cell conditions.
Differential Attention Module. Standard Transformers often prone to over-attending to irrel- evant tokens, especially in noisy biological data. This is particularly problematic in perturbation modeling, where only subset of genes may respond while others should remain suppressed. To address this, we incorporate a differential attention mechanism (Ye et al. 2025). which computes attention as the difference between softmax distributions:
Q1K] K Vil
92k] vihk
11 = softmax
A2 = softmax
(8)
where À is learnable scaling factor, Q: Wo, X, K; WK, Yr and WvY. Ordiff' = A AA2,
DiffAttn Y) = Σ
Oai V
Latent Refinement. At each layer, PAD- Transforfee refines the latent representation hy via three operations:
1. Perturbation injection. The perturbation condition cp is embedded as cp, broadcast, con- catenated with hy, and passed through an MLP adapter to obtain the injected representation:
T MLP (
|l 1r & epl)
(10)
2. Self-differential attention. Applied to Te to suppress noisy activations and refine infor- mative variations within the latent representation:
i T DilTAun(XX
-, =
Ti lemb)-
(11)
3. Cross-differential attention. Incorporates the control representation he as reference to guide refinement of the perturbed latent:
h +++ h DifEAttn(X Te, Y The femb).
(12)
The timestep encoded as temb MSi(SSinss())), where SinCos(t) denotes sinusoidal embedding of at multiple frequencies. This embedding provides adaLN-Zero modulation (Peebles) Xie 2023) for every self-differential attention and cross-differential attention layer.
Output. state:
After L layers, Cp is concatenated again and the decoder produces the predicted perturbed
D( 17 ep.).
(13)
PAD-Transfi TrD-Transfrrmrr leverages perturbation-awwe differential attention to refine latent trajectories, en- suring robust modeling both cell-level dynamics and population-level transcriptional shifts. The complete algorithmic workflow and training procedure provided in Appendix A.3
4 EXPERIMENT
4.1 EXPERIMENTAL SETUP
Baselines. We benchmark our method against a broad set of baselines spanning simple statisti- cal models to deep learning and foundation-model approaches. These include autoencoder-based

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model CPA (ootfolahi et al. 2023) graph-based model GEARs (Roohani al., 2024). two single- cell foundation models Geneformer (Thoodoris et al.) 2023) and scGPT (Cui et al. 2024), as well naive linear regression baseline [Ahlmann-Eltze
al. 2025 and statistical mean expression over control cells. In addition, we include CellFlow (Klein et al. 2025), a flow-matching model trained in reduced 50-dimensional PCA space and conditioned perturbation identity, as well as State (Adduri et al. 2025), a one-step perturbation response model that conditions on basal and per- turbation embeddings. All models are trained using the log of expression values for all genes, but at evaluation time the prediction target is restricted to the top 1,000 highly variable genes. Geneformer evaluated with pretrained weights, whereas scGPT and STATE are trained from scratch without pretraining (Appendix A.5
Metrics. We evaluate model performance from three complementary perspectives. (1) At the thanc-transcriptome level, MAE, MSE, Pearson A and mean L2 errors quantify pointwise recon- struction accuracy across all genes. We additionally include Pearson á and Σ20. computed using the pertuuraapttt mean the reference. These variants mitigate control-perturbation baseline bias and follow the updated evaluation practice introduced by Systema (Viñas Torné et al. 2025).
(2) At the distribution level, the Discrimination Score (DS) (Roohani al. | (202) assesses whether predicted populations under different perturbations remain well separated, penalizing collapsed or averaged responses. (3) At the differential expression (DE) level, Spearman p quantifies the rank correlation between predicted and real log fold changes, computed only on the set of statis- tically significant DE genes. This metric evaluates whether the model correctly captures both the directionality and the relative ordering of differential expression signals (details in AoppendixA..4.4
Data. We evaluate method two single- cell perturbation datasets: Norman and Com- boSciPlex. The Norman dataset (Norman et al. 2019) consists of genetic perturbations (CRISPR- overexpression) in the K562 cell line, including both single and double per- turbations (Appendix A.4.I). The ComboSei- Plex dataset involves drug perturbations, which differ in modality from the genetic perturba- tions in Norman, thus allowing assessment of cross-domain gencralization (Srivatsan et al. 2020 Lotfollahi et al. (023) A.4... To ensure robust evaluation, we perform mul- tiple random re-splits and report averaged re- sults for Norman dataset. Finally, we include ablation analysis to showcase the effect of each component of our model.
Single Courn))
Siyle Rook)
Doubde (usA)
Additive
Holdout
Figure 2: The additive setting (Section 4.2 tests generalization unseen doubles when all sin- gles are observed, while the holdout setting (Sec- tion 4.3) evaluates prediction of entirely unob- served singles and their combinations
We apply two evaluation splits: an additive setting, where all single perturbations used in combina- tions for testing have been observed individually in training; and a holdout setting, where certain sin- gle perturbations and all combinations involving them together are entirely withheld during training- These two settings together allow us to assess performance both in additive and more challenging generalization (holdout) regimes.
4.2 EVALUATING ON SEEN SINGLE PERTURBATION, (ADDITIVE SETTING)
In the first experiment, we use the additive split, where test combinations are composed of single perturbations already seen during training (Fig. 2 left). Table shows that our model achieves the best or near-best results across all categories: at the global reconstruction level it yields the lowest L2, MSE, and MAE; at the distribution level achieves the highest discrimination score with a more compact error distribution (Fig. 3) and at the DE level it obtains the strongest DE-Spearman P. Notably, the Additive baseline itself is competitive, consistent with recent findings (Ahlmann-
Eltze et al. 2025).
By contrast, existing methods show instability. scGPT drops sharply in DE-Spearman p. CPA suf- fers from large pointwise errors, and even Geneformer and GEARS trade off lower I.2 for weaker
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distributional alignment. In comparison, approach maintains balanced profile across metries, achieving robustness without sacrificing one dimension performance for another. We further include study the CEBPE+CEBPA perturbation prediction in Appendix AA.7
Quantitatively, our model reduces the MSE by 19.6% compared to CellFlow (0.00315 vs. 0,00392), the second best performing model, and obtains lower MAE (0.02155 vs. 0.02207), while also achieving the highest discrimination score (0.9737). Under the stricter Systema-style metrics, further attains the strongest Pearson 220 (0.9260), indicating that the improvements persist even when evaluation penalizes expression-overlap effects.
Table I: Comparison of different methods across evaluation metrics on the Norman additive split.
Model
L2
MSE
MAE
DE-Spearman P
Pearson
DS
Pearson å
Pearson Ago
Control
3,9937
0.018339
0,03953
N.A.
N.A.
0.5135
-0.1695
.0.1297
0.9024
scGPT
3.4122
0.01349
0,03796
1,07e.5 5
0.5304
0.5404
0.2165
0.2414
Geneformer
1.9122
0.8241
GEARS
3.5531
0.01387
0,06624
0.5624
0.7421
0.8601
-0.0089
0.2032
0.0713
0.2254
STATE
17,3330
0.30059
0,24705
0,5288
0,0108
0.51355
-0,0069
0,2515
CellHow
0.8678
SCDFM (OuIN)
1.7043
0.00315
0.02155
0.5705
0.8853
0.9737
0.8468
0.9260
10.00
Mean
Median
8.00
Prediction error (L,)
6.00
4.00
Additive
GEARS
(eurs)
Figure 3: Double perturbation prediction error (L2), Our method achieves the lowest enror distribu- tion, outperforming both additive and baseline models.
4.3 EVALUATING ON UNSEEN PERTURBATIONS (HOLDOUT SETTING)
In this experiment, we adopt a holdout split to test how well our model generalizes to perturbations not seen alone during training. Specifically, we remove subset of single perturbations along with all double perturbations involving them from the training set, and use these held-out conditions for testing (Fig. rigght). This allows to assess generalization both for unseen individual perturbations and for their combinatorial effects.
Since this task poses greater challenges, we observe larger performance gaps across models. As shown in Tab. 2 each baseline exhibits distinct limitations: scGPT fails to capture perturbation direction, resulting in negative DE-Spearman pt GEARS improves correlation metrics but suffers from large pointwise errors; Geneformer achieves low L2 and MAE yet falls short in preserving distributional structure. In contrast, our method combines structural priors and MMD regularization to achieve both low error and strong distributional fidelity, ensuring robust generalization.
4.4 EVALUATING DRUG PERTURBATIONS
Table 3 reports performance on the ComboSciPlex dataset (Srivatsan et al. 2020). which measures drug rather than genetic perturbations. At the global reconstruction level. model achieves the lowest L2, MSE, and MAE, indicating the most accurate transcriptome recovery. At the distribu- tion level, it maintains competitive discrimination scores, slightly below CPA but more stable than
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Table 2: Comparison of different methods across evaluation metrics on the Norman holdout split.
Setting
Model
L2 L
MSE 1
MAE
DE-Spearman p
Pearson A
DS
Pearson Á
Pearson 2gg 1
Control
0.0955
N.A.
N.A.
0.5217
0.1618
0.1982
seGPT
2.5007
0.0080
0,0259
-0.1139
0.4503
0.5680
0.0747
0.0798
GEARS
2.5641
0.0075
0.3569
0.6646
0.8271
0.6356
0.7914
Single
Geneformer
1.6962
0.0036
0.0191
0.3669
0.6985
0,8070
0.56220
0.6513
CPA
5,8060
0.0356
0.0853
0.1168
0.2837
0.5796
-0,028
0.0801
STATE
18.2543
0.3333
0,2693
0.6116
0.0004
0.5236
0.0154
0.2386
CollFlow
1,6758
0.0035
0.0191
0,2860
0.7109
0,8072
0.6138
0.6753
SCDFM (ours)
1.6186
0.0030
0.0190
0.6957
0.7127
0.8914
0.6689
0.8116
Control
4,1882
0.0207
0.0423
N.A.
N.A.
0.5322
-0.1303
+0,0265
ScGPT
3,5171
0.0153
0,0362
-0.0655
0.5693
0.5578
0.2814
0,2652
GEARS
3.7458
0.0156
0.0708
0,2543
0.7552
0.8766
0.6407
0.8413
Double
Geneformer
2,0819
0.0050
0,0237
0.3468
0.7361
0.8067
0.6245
0.7261
CPA
3,7891
0.0357
0.0796
0,3652
0.4176
0.6$11
0,2432
0.2870
CellFFow
2.1042
0.0049
0.0236
0,5074
0.8095
0.8622
0.6780
0.7155
(ours)
0.0047
0.5676
0.8357
scGPT. At the DE level, it achieves the highest Pearson A and DE-Spearman p. demonstrating su- perior recovery of differential expression patterns. Overall, these results show that approach generalizes effectively to drug perturbations, combining low pointwise error with consistent capture of perturbation-speeiff signals.
Table 3: Comparison of different methods across evaluation metrics on Combosciplex.
Model
L2
MSE
MAE
DE-Spearman p
Pearson A
DS
Control
5,3716
0.0324
0.0698
N.A.
N.A.
0.5714 14
scGPT
1.6934
0.0031
0.0251
-0.1261
0.8322
0.8571
CPA
1.6592
0.0029
0.0240
0.7906
0,8150
0,8980
SCDFM (ours)
1.6567
0.0028
0.0220
0.8289
0.8933
0.8776
4.5 ABLATION STUDY
To assess the contribution of each component, we perform ablation experiments on the Norman dataset under the holdout split. Figure quantitative results. Removing the gene-gene mask or the Differential Transformer backbone reduces correlation with ground truth, highlighting the importance structural priors and noise suppression. Dropping MMD regularization causes the sharpest decline, underscoring its critical role in distribution-level fidelity.
In addition to quantitative metrics, we also visualize representative perturbations in Fig. Without MMD, generated cells deviate substantially from the ground truth distribution. In contrast, our full model preserves the global geometry and population structure, demonstrating that MMD essential for stable and biologically consistent predictions.
CNNS+UBASHSA
CEBPE+cOntrol
PTPN12+Z8TB25
00007
00007
D000
Figure 4: UMAP visualizations of perturbed cell states. Removing MMD leads to clear distribu- tional mismatches, where generated cells deviate from the ground truth manifold.

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Published as a conference paper at ICLR 2026
L2
DE Spearman
Pearson Delta
2.0
0.65
0.80
0.620
-2.1%
1.9
0.60
0.78
0.773
8.6%
1.8
+10.2%
0.55
0.76
2.44
-2.4%
-1.9%
-20.9%
1.6
0.45
0.72
1.5
0.40
0.70
wio Gere Graph
w/o MND
wio Cross
wio Diff
ScDFM tours)
wio Gene Graph
wio MMD
wio Crass
wio DIfT
ScDFM tours)
w/o Gene Graph
w/o MMD
wi Grass
wio Diff
SCDFM tours)
Figure Ablation study on the Norman holdout setting.
5 CONCLUSION
In this paper, we have presented SCDFM, distribution-aware flow matching framework for bust single-cell perturbation prediction. By integrating conditional flow matching with MMD-based alignment and a perturbation-awar differential Transformer, method captures both local dy- namics and global population shifts. Extensive evaluations across genetic and drug perturbations demonstrate that ScDFM achieves strong generalization unseen combinations while maintaining low error and high biological fidelity.
Overall, our work paves the way for distribution-aware generative models that twins of cellular responses, providing a foundation for in silico screening and systems-level biology. Limitations and directions for future work discussed in Appendix A.6
digital
ACKNOWLEDGEMENT
We thank Minsi Ren and Tian Xia for helpful discussions and valuable feedback the manuscript. We also gratefully acknowledge the support from the Westlake University Research Center for In- dustries of the Future and the Westlake University Center for High-Perforee Perffrrerrrm Computing. The content of this work is solely the responsibility of the authors and does not necessarily represent the official views of the funding entities.
ETHICS STATEMENT
This study exclusively uses publicly available single-cell gene expression datasets, accordance with their respective data use agreements. No personally identifiable or patient-specific informa- tion was used this work. The methods developed in this study are intended for in silico model- ing and hypothesis generation in biological research. While our approach could potentially assist drug discovery or therapeutic design in the future, it is not intended for direct clinical use. We encourage responsible and transparent of generative models in biology, and caution against over-interpretatton model predictions without rigorous experimental validation.
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CONTENTS
Appendix
14
A.1
The Use of Large Language Models (LLMs)
14
A.2
MMD details
14
A.3
Training and inference details
14
A.4
Experimental Sctup
15
A.5
Baselines
20
A.6
Discussion about Limitations and Future Works
A.7
Case study
22
A.8
Consistency Under Randomized Evaluation
22
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A APPENDIX
A. THE USE OF LARGE LANGUAGE MODELS (LLMS)
In this paper, we employ LLMs as general-purpose assist tools for refinement and language polishing. All core research ideas, datasets, and scientific conclusions presented in this paper are our own original contributions.
A.2 MMD DETAILS
Dynamic kernel selection for MMD. We implement MMD with multi-kernel Gaussian RBF mixture:
L Σ 1=1
kiii (xc, x') = exp
|| ||
20;
Rather than fixing the kernel bandwidths {ar), we dynamically adjust them at each training step. Specifically, we compute pairwise squared distances within the batch:
Du = ||, x
take the median of all off--daggnna entries as a reference scale m, and generate multiple bandwidths as
/se m. E (0.5, 1.0, 4.0).
This procedure ensures that kernel bandwidths adapt to the distributional scale the current mini- batch, while the use of multiple scales improves robustness to heterogeneous gene expression ranges.
The unbiased squared MMD estimate is then computed as
MMD*(X,Y) mmm 1)
Σ $
1 kmix (Ki, x.)) + n 1)
kimix (yi
Σ i,
kimix (1ir
mE
Algorithm 1: Dynamic multi-kemnel RBF selection and unbiased MMD
Input: Real samples = [m generated samples {15)5=1- factors (0.5, 1.0, 2.0, 4.0}
Output: MMD (X, Y)
Pairwise distances: (t + [||,,,, =1, Dav [.5]+ lly: 1---9: 33 Dzuli. J 11;--;1111 y 1:11:
Batch-adaptive bandwidths (median heuristic): m max (median D [i. j}, e):
Σ <- (a /sml S).
for E do
8+ zo'=zi:
Kar xe c DDon, Ky te DDyy Kzy te PDyy
Σ Krn Σ diag( Kez)
m(m 1) +
Σ Kyy diag(Ky))
Uyy
n(n 1) +e
Σ KKay
Mayy
10
u(o) Wer + Tyy 2ury.:
11 return (X,) Σ
v(o).:
A.3 TRAINING AND INFERENCE DETAILS
Setup. Let the dataset
{(cz, 21, ep)) measured over G genes with a gene-gent graph W E G At training time we do not consume all G genes at once. Instead, for each item we sample an index set 7 C (1....,G) G) of size [ & G (policy in Sec. A.2 or Algorithm 2}. We then restrict expression vectors to this set, C2 = (c)(gr))ier and x; = (xx(g:)) (ieg)) and extract the masked gene subgraph M, W[T, T].
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Gene/context encoding. The gene encoder Eg consumes the ordered identity sequence G1 (gt)er together with mask M, and produces contextualized gene tokens Z₁ Eg(G,; My) E Rlixxd In parallel, the value embedder E, maps each scalar expression to T2 element-wise. We form aligned token sequences
he (c + Zr, n. Fr (x. + Z,1,
so that identity and value are summed for the same genes (index-wis) alignment).
PAD- TAD-Transformrr block. Given perturbation embedding Ep Emb(cy) and time embedding femb = MLP(SinCos(t)), PAD- Transformer applies (i) perturbation injection via an adapter, (ii) self-differential attention on the perturbed latent, and (iii) cross-differential attention against hes with adal NN-Zero modulation by femb at every layer. After L. layers we decode the velocity vo and form the one-step endpoint approximation
(T) #1 Ft + () vo-
Timestep Embedding. The timestep & E [0, 1] is encoded as femb = MLP(SinCos(t),, where SinCos(t) denotes a sinusoidal embedding of at multiple frequencies.
SinCos(t) (sin(w))), with {wj} } a set of predefined frequencies.
sin(w,),, cos(wtt))
(14)
Objective. The training loss is LFM + LMMD. Conditional Flow Matching LFM supervises the instantaneous velocity field along the path xi ~ T. Ca., Cp). To align terminal distribu- tions, we compute LMMD MMD² X1) on mini-batches of endpoints restricted to /. We use Gaussian RBF multi-kernel mixture with dynamic bandwidths: at each step reference scale is esti- mated from the median of off-diagonal pairwise squared distances within the batch, and small of bandwidths {or is generated by multiplying this scale with fixed factors (e.g.. {0.5, 1.0, 2.0, 4.0)). The unbiased estimator drops self-similarities. Full details appear in Appendix A.2 and Algorithm
Training procedure. Algorithm 2 summarizes batching over items, per-item gene-subset sam- pling, masked gene/context encoding, PAD-Transformer passes, endpoint construction, dynamic kernel selection, and parameter updates. Subset sampling is re-drawn every step unless otherwise noted; fixing the RNG seed and the ordering of G1 ensures reproducibility.
Inference. At test time we only predict selected subset of genes. We choose / using the same policy as in training (e.g., a fixed target subset, or a deterministic sampler), build Z₁ = (G;; M,) once, and evolve (1)) from 1=0 to using TAD--Tranformer and an ODE stepper (Euler/Heun). The final output is 2 genes I (Algorithm 3 If full-vocabuld volabulcyyy outputs desired, post-hoc imputation head can be added, but not used in our experiments.
Complexity. Both masked attention in F., and differential attention in PADD-Trassfrmmer scale as (( per layer, dynamic MMD adds O(B)) pairwise evaluations within the batch. Choosing S=[[] balances accuracy and compute.
A.4 EXPERIMENTAL SETUP
A.4.1 NORMAN DATASET
The Norman dataset a foundational benchmark for modeling single-cell responses to combinato- rial genetic perturbations (Norman et al. 22019). Originating from the work oorman et al.) (2019), the experiment utilized CRISPR activation (CRISPRa) in the K562 human cell line to systematically upregulate target genes. The resulting Perturb-seq data profiles cellular responses to approximately 100 single-gene and 124 dual-gene activations, making it rich and challenging dataset for evaluat- ing model's 's ability to predict complex, combinatorial effects (Norman et al.] (2019).
To ensure consistency with recent benchmarks, our study used the publicly available, scFouddation- reprocessed version of the Norman dataset (Ahlmann-Elie al. 2025). The specific data file can
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Algorithm 2: Training scDFM (PAD-Transformer) with gene-subset encoding
Input: Dataset D = {(cx, F1, c,) with G genes; gene graph W: subset sampler SampleSubset(G, s): FM schedule æy; MMD scales S: weight X; layers L
Output: Trained parameters 0
for mini-batch 8 C do
Sample (1 1) (per item or per batch);
// Build batch tensors on the same gene index set per item foreach 21. E do
I < SampleSubset(G,nt
G1 (g))ee MI+WW[
sample from nz Cr,C
Z1 - [g Mr):
// contextual gene identities
he f (c(ck)) + Zri + Ex(2)) Zι
Cp + Emb(c,); temb + +MLP(Si
4 < PAD-Transfone Transformer(he, hes eps lambi L); Ug Decode Velootyeel (h"r, pp);;
10
LFM +- CFM(U8; " (1) S...pp .c,,):: 1);
11
a t 26 (1-()) vs:
12
collect =/44) and xz(" into batch sets X1, X1; + dynamic bandwidths via batch median heuristic on X, (Appendix A.2
14
15
LADD < (X,,
16
+ LPM + A AMMD: Update A + 0 VoL:
Algorithm 3: Inference with PAD- Transformer (gene-subset input)
Input: Trained 0; control profile ca; condition Cy. subset index 7 (same policy training); gene graph W: steps K
Output: Predicted perturbed expression 2 on genes 1 G (gi)ies; M [[[W J]; Z₁ + FG My);
Initialize 2n ~ Qo (c.g., noise; or use standard source); for 0 to K do
i + k/K; Lemb <- MMP(SinCos(()): tp Emb(e));
he E.(c Zr:
1 + Ex(()) Zs;
h <- PAD-Tranee PAD-Transformer((b, he, Ep, tembi L);
Uo + DecodeVelocityylee ep];
(I) (1) + At
2h+1
// Buler; Heun/ODE solver optional
return (11)
ØK
(T)
prediction is on the selected genes only
be downloaded directly from this Figshare link We processed the data using the following standard pipeline in Scanpy:
Normalization: Library size was normalized to a target sum of 10,000 counts per cell using sc. pp. normalize total.
Log Transformation:
Expression values were log-transformed with sc.pp loglp to stabilize variance.
Gene Selection: We selected the top 5,000 highly variable
genes
using
sc.pp.hight. variable genes.
This procedure, combined with the inclusion of the perturbed target genes, resulted in final feature set of 5,029 genes for model training. For evaluation, focused the top 1,000 most highly expressed genes, a common practice that ensures the assessment is performed on robust biological signals Ahlmann-Eltze et al. (2025).
To guarantee the robustness of our findings, all experiments were conducted using four independent random main/validation/testt splits, and report the mean performance across these runs. The exact
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Alvespimycia pimicm
Panobinostat
Tanespimycin
Carmofur
Cediranib
Crizotinib
Curcumin
Dacinostat
Danusertib
Dasatinib
Givinostat
PCI-34051
Pirarubicin
SRT1720
SRT2104
SRT302A
Alvespimycin
Carmofur
Cediranib
Crizotinib
Train Combo
Curcumin
Test Combo
Dacinostat
Danusertib
Dasatinib
Givinostat
PCI-34051
Panobinostat
Pirarubicin
SRT1720
SRT2104
SRT3025
Sorafenib
Tanespimycin
Figure 6: ComboSciPlex combination data split. Rows and columns denote small-molecule com- pounds. Filled deep blue cells indicate drug pairs used for training; red outlined cells denote held- out test pairs.
preprocessing configurations and data split indices will be made fully available in our public code repository ensure complete reproducibility.
A.4.2 COMBOSCIPLEX
ComboSciPlex dataset. We use the ComboSciPlex drag-combination dataset (Lotfollahi al. 2023), follow-up extension of the SciPlex chemical barcoding platform (Srivatsan et al.) 2020). designed measure single-cell transcriptional responses pairwise drug perturbations. In the preprocessed release used by prior work (Klein et al. 2025 Lotfollahi et al.] 2023). the dataset contains 63,378 single-cell transcriptomes and total of 32 treatment conditions (single agents and pairwise combinations) in A549 (see Fig. (6). Programmatic access is available via pertpy (pertpy data. combosciplex ()) we additionally mirror the exact files used our runs (Figshare DOI below). For our experiments, we follow common practice and restrict the expression space to the top 5,000 highly expressed genes after QC/nomaalzaaion which has been shown to work well this dataset family. The train/test split is illustrated in Fig. 6 all available single-agent profiles are included training, and evaluation is conducted on held-out drug combinations (red outlines) to probe combinatorial generalization; the precise list of train/test pairs and their counts
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released with code. Downloads: pertpy loader (documentation) and Figshare (preprocessed subset)
A.4.3
TRAINING SETTING FOR s SCDFM
All experiments were performed cluster equipped with NVIDIA H800 GPUs.
Norman training setup.
We train scDFM on the Norman dataset with the following configuration:
Optimizer LR schedule: Adam with an initial learning rate 10-5, decayed by cosine schedule to Mmin 10
Batch size: 96.
Training length: 100,000 optimization steps.
Distribution regularization: MMD loss with weight À = 0.5 (dynamic multi-kernel RBF; details Appendix).
Gene-gene mask: kNN graph with k = 30 built from signed (both positive and negative) gene-gene correlations; the resulting mask is used in the gene encoder's self-attention.
Perturbation embedding: for CRISPR activations, the perturbation embedding for gene g shares parameters with the gene identity embedding Eg(g) (no separate perturbation- embedding table).
Backbone width/depth: hidden size d = 512, L = layers, H 8 attention heads, dropout 0.1 (applied attention and MLP))
Inference: Euler ODE rollout with K 100 uniform steps over E [0, 1] (i.e., At 0.01).
Unless otherwise noted, results are averaged over four independent random train/val/test splits using the same preprocessing and masking pipeline.
ComboSciPlex setting (difference from Norman). Unlike Norman, where the perturbation em- bedding for gene shares parameters with the gene identity embedding (g), in ComboSciPlex dedicated perturbation embedding table Ep for small molecules. The drug embeddings
in E, are learned end-to-end and are not tied to gene tokens; multi-drug conditions are embedded via E.p and injected every layer through the same adapter mechanism in Norman. All other hyperparameters (optimizer, LR schedule, backbone width/depth, kNN=30 signed gene-gene mask, - for MMD) follow the Norman setup.
A.4.4 METRIC DEFINITIONS
To comprehensively assess both pointwise prediction accuracy and distributional fidelity, we report multiple evaluation metrics adapted from prior work single-cell perturbation modeling. Most implementations follow the standard routines provided in the celleval library
L2 (Mean-level Perturbation Distance). To quantify expression deviation at the perturbation level, we compute the average L2 distance between predicted and ground-truth **mean gene ex- pression **ctors** for each perturbation. Formally, for each non-control perturbation p P. define:
1.2 1.2
Σ Il Hpll2 PEP
(15)
P
where H, = No EN E under perturbation p. from ground-truth and predicted expression respectively. This metric evaluates global shifts between predicted and real perturbation responses in gene expression space.
and A , Σー x(P)
are the empirical mean vectors over Np cells
https:://githuu..com//ttt httm://arreettittuutt//eeleeee
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MSE, MAE. These metries quantify absolute cell-level error between predicted and ground-truth gene expression. Let X, X E RNN be the predicted and real expression matrices. Then:
MSE = NG 1= j=
1 A Σ
Xij
X35
2
. MAE L NG
Li=1
G
Xij
815
Perturbation Discrimination Score (PDS). This metric is a rank-based retrieval score based on pseudobulk similarity; it evaluates whether predicted perturbation effects resemble their true terparts more than other unrelated perturbations.
Let X* E IG and X E IR denote the predicted and true log-normalized expression ma- trices under perturbation (excluding control), with ny and my cells respectively. We first compute pseudobulk vectors by averaging across cells:
mE
Uk
yk
X
(16)
TE
TTR
For each perturbation p. we calculate the L, distance between its predicted pseudobulk Y, and all ground truth pseudobulks (y where N denotes the total number of perturbation categories:
dnr Σ Upg Vt.gl,
(17)
D
where Gp denotes the set of genes directly perturbed by p (which are excluded from the comparison).
We sort {dp 1.W in ascending order, and record the rank of the true target p:
rankp arg sort, dpc-
(18)
Finally, the discrimination score for perturbation p defined as:
PDS, 1 rank, -
(19)
N 1
A perfect match yields PDSp 1. The overall score is averaged across all predicted perturbations:
1 PDS
Σ PDSp
(20)
N
To measure biological relevance, compute the Spearman correlation be- tween predicted and true log fold-changes for genes that are significantly differentially expressed in ground truth (adjusted p-value < 0.05). This focuses evaluation on meaningful changes and filters out low-signal noise.
DE-Spearman-Sig.
Pearson A. This measures the difference in Pearson correlation matrices between predicted and true cell-wise gene expression. For each gene, the Pearson correlation vector across cells is com- puted, and the average LI difference between these vectors defines A.
Pearson Å. This metric evaluates whether the model captures sample-specific heterogeneity be- yond the average perturbation effect, following the principles of the Systema framework. Unlike standard evaluations that use the control group as baseline, we employ the mean expression of the specific perturbation from the training set the reference isolate non-systematic variations. Let x; E C and x; E R denote the predicted and true log-normalized expression vectors for cell i Irain E G represent the centroid (mean expression profile) of perturbation computed from the training data. We define the residual vectors as:
under perturbation p. respectively. Let x (p)
8 .prd
x;
x (P) Irain"
S X; (p)
(21)
19

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Published as a conference paper at ICLR 2026
The Pearson å score is computed as the average Pearson correlation coefficient (p) between these residual vectors across all cells in the test set (N):
Pearson Å
1 N
Σ pl predd di,
(22)
higher score indicates that the model successfully predicts the specific cellular response variations that deviate from the population mean.
Pearson 220- To focus the evaluation the most biologically relevant variations and mitigate the impact of noise from invariant genes, this metric restricts the Pearson Á calculation to the top- 20 genes with the highest variance. Let G20 denote the subset of 20 genes exhibiting the highest variance in the true residuals (8e across the perturbation population. The metrie is defined as:
N
Pearson Â20
E P (re.p 6 uuue (920l)
(23)
where [G20] denotes indexing the vectors to include only the genes in the subset. This serves as stricter test of the model's precision capturing high-varianes gene programs.
A.5 BASELINES
Control (no-change). A naive identity baseline that predicts no perturbation effect:
Cz
ie., the post-perturbation profile is taken to the pre- pre-perturbation control profile.
Additive (linear superposition). For a combination of two single perturbations and b, we ap- proximate the combined effect as the sum of single-agent deltas measured (or predicted) relative to control:
Lats = c₂ + (x(a) cz) + (z - -x).
where (a)) and (z) are the single-perturbed profiles (when available, from the same split) aligned to the same gene index as This baseline encodes strictly additive interaction model without
higher-order non-linear effects.
scGPT (Transformer without pretraining in our setting). A gene-token Transformer that rep- resents cell as sequence of gene tokens with value and identity embeddings and predicts the perturbed expression conditioned on and Cp. In our experiments, we train scGPT from scratch on our splits (no external pretraining), keeping architecture capacity comparable to ours and injecting Cp at the input/adapter layers for conditioning.
Geneformer (pretrained foundation model). large pretrained Transformer for single-cell tran- scriptomics (pretrained on large corpora of scRNA-seg profiles). We fine-tune the released check- point on our task/splits to map (CasGp) Gp) to post-perturbation expression. Concretely, perturbation information is injected via InSilicoPerturbbr, and then we map the CLS token's representation to perturbed expression with linear probe.
GEARS (graph-based perturbation predictor). A graph-aware baseline that encodes intergene structure (e.g., coexpression/regulatory graphs) via message passing and predicts gene-level re- sponses under single and combinatorial perturbations. Combinations are modeled by jointly con- ditioning on multiple targets in the graph encoder/decoder. We use the official implementation and default training recipe adapted to our preprocessing and splits.
CPA (Compositional Perturbation Autoencoder). compositional latent-space model which an encoder maps C. to latent representation, learned embeddings encode perturbation (and op- tional covariates/dose), and decoder reconstructs the perturbed state. Combination treatments are composed by adding condition embeddings in latent space (plus optional dose scalings), enabling zero-shot composition. We follow the public implementation with our data normalization and split protocol.
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Published as a conference paper at ICLR 2026
State (State Transition Model). We followed the official State implementation and trained the model from scratch using the publicly released code (Adduri et al. 2025). We did not use the pretrained State embedding model the pretrained State transition model for two reasons. First, the released pretrained State transition weights are designed specifically for the "drug type + dosage" formulation and do not support drug combinations gene perturbations, which is the evaluation setting of our work. Second, while pretrained embeddings are provided, no fine-tuning pipeline is included in the original release, and adapting the pretrained embedding model to the combination- prediction setting would require reimplementation of components that were not publicly available. Training from scratch therefore ensured a fair and reproducible experimental setup aligned with the setting of our benchmarking experiments.
CellFlow. CellFlow (Klein et al. 2025 models perturbation response as a continuous transfor- mation between control and perturbed cell states in a low-dimensional latent space, where a flow- matching objective is used to learn the velocity field conditioned on perturbation identity. During inference, the model integrates the learned flow starting from control cell to obtaín its predicted perturbed state. In our implementation, following the official open-source reference, we first re- duce gene expression to 50-dimensional PCA space. which serves as the latent representation for both control and perturbed profiles. The predicted latent state is subsequently mapped back to full gene space using the PCA decoder for evaluation. All hyperparameters and data preprocessing steps follow the public implementation unless otherwise specified to ensure strict comparability with reported results.
Implementation note. All baselines trained/evalluate under the same pre- pre-processing. gene index alignment, and four random train/val/test splits as our method (means reported).
A.6 DISCUSSION ABOUT LIMITATIONS AND FUTURE WORKS
Distributional Flow Matching. Our results suggest that enforcing distribution-level alignment core requirement for modeling realistic perturbation effects. Beyond biology, the idea of distribution-awar flow matching may also benefit tasks in vision and generative modeling. Ex- ploring this direction remains exciting topic for future work.
Representation and Path Design. In this work, we adopt simple linear interpolant in the log- normalized gene expression space as the default reference path for flow matching. While this choice yields stable training and interpretable trajectories, it is arguably suboptimal: biological processes may follow nonlinear, branching, or manifold-constrained transitions that are poorly approximated by linear paths in log space. We believe future research should explore both the choice of repre- sentation space and the design interpolation trajectories. A more principled understanding of which interpolants best capture perturbation dynamics could significantly enhance the realism and generalization of generative perturbation models. However, we leave these directions for future exploration.
Scalability to Multi-Context Datasets. Our study focuses on two representative datasets, Norman and ComboSciPlex. but does not include more recent large-scale resources such as ARC-state (Ad- duri et al. 2025), which became available during our work. These datasets span multiple cell lines and perturbation types, offering a more comprehensive benchmark for generalization. Extending our method to such diverse contexts will be important for future work.
Structural Priors and Graph Topology. We utilized a Pearson correlation-based graph to con- struct the attention mask, serving as a computational proxy for gene regulatory networks. While this approach effectively introduces biological structure, it primarily captures linear co-expression patterns and may miss complex non-linear or causal dependencies. We view ScDFM as a flexible framework where the graph topology is a modular component. Future iterations could significantly benefit from incorporating more sophisticated priors, such as those derived from Mutual Information (MI) or causal discovery algorithms, to better capture the non-linear dynamics of gene regulation.
Scalability to for genetic and drug perturbations. A critical future direction is applying scDFM broader, large-
Datasets. Our study currently focuses on representative benchmarks
21

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Published as a conference paper at ICLR 2026
Table 4: Additive setting variance across four random folds.
Model
1.2
MSE
MAE
DE-Spearman P
Pearson A
DS
Control
0.13003
0.00100
0.00100
N.A.
N.A.
0.00297
Additive
0.03077
0.00015
0.00015
0.04000
0.01090
0.00899
scGPT
0.10928
0.00096
0.00096
0,20351
0.02374
0.00811
Geneformer
0.07275
0.00045
0.00133
0.13790
0,02357
0.03896
GEARS
0.16239
0.00133
0.00133
0.07530
0.02577
0.01582
CPA
0.26981
0.00301
0.00301
0.08628
0,00357
0.04691
Ours
0.04685
0.00023
0.00023
0.12040
0.01278
0,01153
scale initiatives such the Virtual Cell Challenge (Roohani et al. 2025). Validating the model
on such massive resources, spanning multiple cell lines and diverse perturbation mechanisms like CRISPRi/a, will be crucial for demonstrating robustness across heterogeneous biological contexts and ad advancing foundation-scall perturbation modeling.
A.7 CASE STUDY
Case Study: CEBPE+CEBPA. To better illustrate model behavior, we analyze the joint pertur- bation of CEBPE and CEBPA. For this case study, first selected the top 20 genes showing the largest absolute expression changes relative to the control state. For each gene, we plotted the dis- tribution of expression changes across single cells boxplots (after subtracting the control mean), and overlaid the model predictions as blue dots representing the mean predicted shift. This setup enables direct comparison between empirical distributions and different predictive models.
As shown in Fig. 7 and Fig. [ the seven panels cover all baselines considered: Additive, Con- trol, CPA, GEARS, Geneformer, ScGPT, and SCDFM. The control and additive baselines capture only coarse global trends and fail to reproduce non-linear responses. CPA and GEARS improve aligmment but still deviate on synergistic largets. Geneformer and scGPT provide partial improve- ments, yet often misestimate magnitudes or underperform on non-additive regulation. By contrast, our framework consistently places predictions within the observed variance and faithfully recovers synergistic upregulation in genes such CEACAM20 and LSTI. Together, these results confirm the necessity of explicitly modeling gene-gene interactions and highlight the advantage our approach in capturing complex combinatorial perturbation effects.
A.8 CONSISTENCY UNDER RANDOMIZED EVALUATION
Variance across random folds. Tables 4 and and report the variance each metric across four ran- dom splits for the additive and holdout (single/double) settings, respectively (values rounded five decimals). Lower variance indicates greater run-to-run stability. In the additive setting, methods that rely on simple composition (e.g., the Additive baseline) unsurprisingly exhibit small variance, while learned baselines such as CPA and GEARS show substantially larger variability. Our SCDFM maintains consistently low and competitive variance across metries (e.g., small L.2/MSE/MAF and modest discrimination score), indicating stable training despite modeling the full conditional dis- tribution. the holdout regime, variances increase for all methods-moot so for the double set- ting-reflecting the higher difficulty of extrapolating to fully unseen combinations. SCDFM remains competitive (e.g., second-lowest L2 variance under the double setting) and markedly more stable than CPA, which exhibits the largest variability. Entries marked "N.A." correspond to metries that are undefined for a given baseline (e.g., Pearson for Control). Overall, these results comple- ment the main-text means by showing that ScDFM delivers only strong accuracy but also robust behavior under random data splits.
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Published as a conference paper at ICLR 2026
Change gene expression control
Two-gene perturbation (Additive)
change in gene expression after perturbing CEBPE+CEBPA
(log-normallin mormallize counts)
Control mellnel
hediciion
Truth (cianibiion)
2
-2
SAT2
NCF2
COL18A1
BTNLB
ASAHI
SIGLEC14
MEP1A
MROHB
LMO4
MSRB1
Ak8
PLPP1
TMSB10
UPP1
WFDC12
CSFIR
CFH
CKMTZ
CEACAMZO
LSTI
Change expression control llognoomalizzed counts)
Two-gene perturbation (Control)
change in gene expression after perturbing CEBPE
Cuntral meline
hedicions-
Truth (obbribution)
SAT2
NCF2
COL1BA1
8TNL8
ASAH1
SIGLEC14
MEP1A
MROHB
LMO4
MSRB1
Ak8
PLPP1
TMSB10
UPP1
WFDC12
CSFIR
CFH
CKMTZ
CEACAMZO
LST1
Change expression control (log (log-normal nognoomaalzedd counts)
Two-gene perturbation (CPA)
change in gene expression after perturbing CEBPE+CEBPA
meline
hediciinn
Truth (clstribttion
2
SAT2
NCF2
COL18A1
8TNLB
ASAHI
SIGLEC14
MEP1A
MROH8
LMO4
MSRBI
Ak8
PLPP1
TMSB10
UPP1
WFDC12
CSFIR
CFH
CKMTZ
CEACAMZO
LSTZ
Change expression over control (log-normm rng-normalized counts)
Two-gene perturbation (GEARS)
change in gene expression after perturbing CEBPE+CEBPA
Cuntrol aellne)
Pudictions
Truth (bbtnbbtion)
2
-2-
SAT2
NCF2
COL18A1
BTNLB
ASAH1
SIGLEC14
MEPIA
MROHB
LMO4
MSRB1
AK8
PLPP1
TMS810
UPP1
WFDC12
CSFIR
CFH
CKMT2
CEACAM2O
LSTI
Figure 7: Comparison across additive, control, CPA and GEARS
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Published as a conference paper at ICLR 2026
Change gene expression control
Two-gene perturbation (Geneformer)
change in gene expression after perturbing CEBPE+CEBPA
dog-normalized counts)
Contnol ( seline
Irdictions
(etttibution
2
-2
SATI
NCF2
COL18A1
BTNL8
ASAHI
SIGLEC14
MEP1A
MROHB
LM04
MSRB1
AKB
PLPP1
TMSB10
UPP1
WFDC12
CSFIR
CFH
CKMT2
CEACAMZO
LSTI
Change gene expression control
Two-gene perturbation (sCGPT)
change in gene expression after perturbing CEBPE+CEBPA
log-normalized counts)
Control ( aselinal
Reedcctions
(clstnbtttio)
2
-2
SATI
NCF2
COL18A1
8TWL8
ASAHI
SIGLEC14
MEP1A
MROHB
LMO4
MSRB1
Ak8
PLPP1
TMSB10
UPP1
WFDC12
CSFIR
CFH
CKMT2
CEACAMZO
LSTI
Change gene expression control llognnormaizzdd counts)
Two-gene perturbation (sCDFM (Ours))
change in gene expression after perturbing CEBPE+CEBPA
Control & meline)
Predictions.
(cistnbutioo)
2
SA72
NCF2
COL18A1
BTNL8
ASAH1
SIGLEC14
MEPIA
MROHB
LMO4
MSRB1
Aka
PLPP1
TMSB10
UPP1
WFDC12
CSFIR
CFH
CKMT2
CEACAMZ0
LST1
Figure 8: Comparison of Geneformer, scGPT, and SCDFM (Ours) on the CEBPE*CEBPA case.
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Published as a conference paper at ICLR 2026
Table 5: Holdout setting variance across four random folds.
Setting
Model
L2
MSE
MAE
DE-Spearman
Pearson A
DS
Control
0.17045
0.00040
0.00103
N.A.
N.A.
0.00244
scGPT
0.08057
0.00035
0.00115
0.09429
0.03844
0.01939
Geneformer
0.15349
0,00056
0.00128
0,13532
0,05961
0.06346
Single
GEARS
0.14342
0.00067
0.00213
0.22257
0.01262
0.02299
CPA
0.66121
0.00678
0.01183
0,28781
0.02085
0.06246
scDFM (ours)
0.10947
0.00042
0.00036
0.16871
0.03498
0.03304
Control
0,26210
0.00242
0.00323
N.A.
N.A.
0.00192
scGPT
0.22748
0.00208
0.00330
0.23503
0.05280
0.00667
Geneformer
0.33132
0.00160
0.00320
0.28817
0.05467
0.07668
Double
GEARS
0.26108
0.00215
0.00474
0,23465
0.01816
0.01295
CPA
0,93954
0.01017
0.01746
0.14223
0,03166
0.04369
scDFM (ours)
0.24457
0.00128
0.00218
0.10788
0,01025
0.01895
25


Reviews:
[{"review_id": "KMuIohgHL6", "reviewer": "Reviewer_P4kP", "summary": "This paper presents scDFM, a novel generative framework for predicting the transcriptional response of single cells to perturbations. It models the perturbation as a continuous-time flow that transforms the control cell distribution into the perturbed one, and it incorporates an MMD loss to force the entire distribution of generated cells to match the ground-truth distribution. The authors demonstrate through experiments on genetic and drug perturbation datasets that scDFM outperforms existing baselines.", "questions": "1. The ablation study shows that removing the graph prior worsens performance. Does this suggest the model is critically dependent on this prior, limiting its ability to discover novel, non-obvious gene interactions that are not captured by simple correlation? How would the model perform if the biological prior was based on a more sophisticated measure than Pearson correlation?\n\n2. What is the computational trade-off of using MMD?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written and easy to follow. The idea is original.", "weaknesses": "The model does not learn the gene-gene interaction network but it is given it as a biologically grounded prior. This graph is constructed from simple absolute Pearson correlation on the training data, which prevents the model from discovering novel, non-obvious, or non-linear gene relationships that aren't captured by basic correlation.\n\nThe flow matching framework learns a path from control to perturbed. As the paper acknoledges, it uses a simple linear interpolant as the reference path, which is a significant oversimplification. Real biological processes follow complex, non-linear manifolds, and this assumed path may not be biologically realistic.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bviSnm5CS6", "reviewer": "Reviewer_MKtS", "summary": "This paper represents a generative framework called scDFM based on conditional flow matching that predicts response to cellular perturbation (genetic and molecular). There are three parts to scDFM framework: (1) the control flow matching where it models the transition from initial state of the cell lines to perturbed state. (2) multi-kernel MMD regularizer which ensures population-level fidelity. (3) backbone design which is a perturbation aware transformer which addresses the noisy, sparse, and high dimensional\nnature of the data. Finally, the model is evaluated on two tasks of genetic and molecular perturbation on two datasets: Norman and Sciplex3 and it marginally outperforms the baselines.", "questions": "- Are the results stated in the paper on the final prediction or the delta (prediction - control) ? Could you please report the eval on delta as well?\n- Why dataset Sciplex3 is called ComboSciPlex in the paper? \n- Is there an ablation study where we could see the benefit of flow matching against diffusion models?\n- Could you explain the experimental setting ? For example, how many genes are trained test on? do you only look at HVGs?\n- I'm still not convinced why just predicting the mean expression vector is not good enough. In fact, the experimental results shows many of the mean based models have very strong results. Is it possible to design an experiment that shows mean expression vector is not enough?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is well written and easy to understand.\n- This paper addresses two interesting and important questions in a single framework: genetic and molecular perturbation.\n- The framework uses a biological prior which strengthens the model.\n- The authors use flow matching instead of diffusion models and autoencoder (like prior methods) which is an interesting architectural choice and the reason behind it is sound .", "weaknesses": "The main weakness regarding this paper is Section 4. Experiments:\n- The experimental results are not strong. In most cases, scDFM just barely outperforms the baselines.\n- The baselines used in this paper are not the latest and best in the field.\n- The dataset is limited and since the model is not showing strong results, it is not clear how scDFM would perform on other datasets. \n\nThe framework for molecular perturbation has some limitations:\n- It cannot be generalized to unseen molecules.\n- The experiments are done only on a subset of drugs in sciplex3", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tE6sW99Lpf", "reviewer": "Reviewer_u6LS", "summary": "This paper proposes scDFM, a conditional flow-matching model for single-cell perturbation prediction. The method extends standard flow-matching by adding a Maximum Mean Discrepancy (MMD) loss to align generated and empirical cell-state distributions and introduces a Perturbation-Aware Differential Transformer (PAD-Transformer) with a gene–gene co-expression attention mask. The model is evaluated on the Norman and ComboSciPlex datasets in both additive and hold-out regimes. Results show modest improvements over previous methods such as GEARS, CPA, Geneformer, and scGPT.", "questions": "See my weaknesses", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "1. The paper is well-written and clear, with a strong visual presentation and reproducibility statement.\n2. Applying flow matching directly in the expression space is a reasonable and technically clean adaptation of continuous generative models to the single-cell domain.\n3. The idea of incorporating a population-level regularizer (MMD) is conceptually sound and aligns with the motivation to capture distributional, rather than per-cell, perturbation effects.\n4. The inclusion of differential attention and gene-graph masking demonstrates awareness of biological structure.", "weaknesses": "1. Limited novelty and incremental contribution\n\nFlow matching for biological state modeling has already appeared in multiple works. The MMD term is a straightforward sample-based regularizer with no new theoretical or algorithmic insight. The PAD-Transformer largely reuses existing building blocks (Differential Transformer + GEARS-style gene-masking). As a result, the paper reads as a combination of known components rather than a fundamentally new modeling principle.\n\n2. Outdated benchmarking and missing key baselines\n\nThe evaluation follows the GEARS-style additive/holdout split, which has since been shown to be insufficient due to expression-vector overlap between training and test sets. Recent frameworks such as PertEval and Systema explicitly address these issues; none of these are considered here. Furthermore, the paper cites but does not evaluate against the most relevant contemporary models (CellFlow, State), making it difficult to assess progress over the true state of the art.\n\n3. Lack of analysis of computational cost and scalability\n\nFlow matching and MMD introduce non-trivial overhead compared to autoencoder or diffusion models, yet there is no runtime or memory comparison. Without such analysis, it is unclear whether the method is practical for larger or multi-tissue datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LSGWeBfNO8", "reviewer": "Reviewer_JmMZ", "summary": "scDFM is a generative framework for single-cell perturbation prediction based on conditional flow matching. To model population-level effects, it combines a flow matching objective with a Maximum Mean Discrepancy (MMD) loss. The model's backbone, the Perturbation-Aware Differential Transformer (PAD-Transformer), incorporates a gene co-expression graph to guide its attention mechanism. The method is evaluated on combinatorial genetic and drug perturbation datasets.", "questions": "See weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The combination of flow matching for trajectory modeling and MMD loss for distributional alignment is a sound approach for this task.\n\n- The PAD-Transformer architecture, which incorporates biological priors via a co-expression graph, is an interesting design choice.\n\n- The evaluation incorporates several cell-eval metrics which is encouraged.", "weaknesses": "- MMD has already been considered in STATE [1], and flow-matching has already been covered in CellFlow [2]. Therefore the novelty needs to be carefully positioned. It is unclear why performing flow matching in the original space is a meaningful novelty and improvement over e.g. cell-flow.\n- It is unclear how the results compare to CellFlow, given its similarity. Also, the paper’s citation on CellFlow [2] seems wrong.\n- It is unclear how multi-step generation increases performance.\n- There seems to be a discrepancy from [3]. In particular, in [3], the additive baseline performs better than Geneformer, but Fig. 3 here seems to show an opposite trend. Furthermore, it is unclear how the results compare to well-established benchmarks in [3] on unseen perturbation and drug perturbations.\n- The intuition on the gene attention mask remains unclear. The exact same argument could be applied to image or language data, but bi-directional attention without masking is still widely adopted.\n\n[1] Adduri, Abhinav K., et al. \"Predicting cellular responses to perturbation across diverse contexts with State.\" bioRxiv(2025): 2025-06.\n[2] Klein, Dominik, et al. \"CellFlow enables generative single-cell phenotype modeling with flow matching.\" bioRxiv (2025): 2025-04.\n[3] Ahlmann-Eltze, Constantin, Wolfgang Huber, and Simon Anders. \"Deep-learning-based gene perturbation effect prediction does not yet outperform simple linear baselines.\" Nature Methods (2025): 1-5.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)