scDFM: Single-Cell Flow Matching

• scDFM is a single-cell Distributional Flow Matching framework that models the full distribution of perturbed cells, capturing population-level shifts in RNA-seq data.
• It integrates Conditional Flow Matching with an MMD objective and a PAD-Transformer backbone to simultaneously learn dynamic transport and end-state calibration.
• The method addresses challenges like dropout, noise, and non-additive effects, offering improved predictive insights in experiments where cell-paired measurements are infeasible.

Searching arXiv for papers on scDFM and closely related single-cell perturbation modeling. scDFM, short for single-cell Distributional Flow Matching, is a conditional generative framework for single-cell perturbation prediction that models the full distribution of perturbed cells conditioned on control states and perturbation identities, rather than treating perturbation response as a deterministic cell-level mapping (Yu et al., 6 Feb 2026). It is designed for settings in which RNA-seq is destructive, so the same cell is not observed before and after perturbation, and in which perturbations induce population-level shifts under sparse, noisy, and high-dimensional measurements. The framework combines Conditional Flow Matching (CFM) with a Maximum Mean Discrepancy (MMD) objective and uses a Perturbation-Aware Differential Transformer (PAD-Transformer) backbone to encode gene-level context, perturbation identity, and control-versus-perturbed differences (Yu et al., 6 Feb 2026).

1. Problem formulation and conceptual scope

scDFM addresses perturbation prediction in experiments where one observes control cells with gene expression vector $c_x \in \mathbb{R}_+^G$ and post-perturbation cells with expression $x \in \mathbb{R}_+^G$ under perturbation condition $c_p$, where $c_p$ is represented as a multi-hot vector over perturbations (Yu et al., 6 Feb 2026). The central modeling difficulty is that there is no one-to-one pairing between $c_x$ and $x$, because single-cell RNA-seq measurements are destructive. In this regime, perturbation effects are population-level rather than cellwise, and conventional per-cell regression objectives can overemphasize mean expression while missing higher-order structure such as variance, skewness, or multimodal subpopulations.

The framework therefore targets the conditional distribution

$p_\theta(x \mid c_x, c_p),$

with the explicit goal of reproducing the population-level distribution of perturbed cells and generalizing to unseen perturbations and combinatorial perturbations (Yu et al., 6 Feb 2026). The data description emphasizes several practical challenges that motivate this design: dropout and zero inflation, technical noise and batch effects, and non-linear, context-dependent perturbation effects. A plausible implication is that scDFM is intended not merely as a point predictor but as a distribution-aware simulator of perturbation response under realistic single-cell measurement conditions.

2. Flow-matching and distribution-level objectives

The core dynamical model is a conditional ODE

$$\frac{d X_t}{dt} = v_\theta(X_t \mid t, c_x, c_p), \qquad t \in [0,1],$$

where $v_\theta$ is a time-dependent velocity field conditioned on control profile and perturbation condition (Yu et al., 6 Feb 2026). Training uses Conditional Flow Matching. scDFM specifies a linear interpolation in expression space,

$x_t = (1-t)x_0 + tx_1,$

between a source sample $x \in \mathbb{R}_+^G$0 and a target sample $x \in \mathbb{R}_+^G$1, and minimizes an $x \in \mathbb{R}_+^G$2 objective between the learned velocity and the reference velocity along this path (Yu et al., 6 Feb 2026). The source distribution is described as a noisy gene expression distribution, while the target is the empirical perturbed expression distribution.

A distinctive element is the addition of MMD-based endpoint alignment. The paper argues that CFM enforces local trajectory consistency but does not guarantee that generated terminal states match the real perturbed population. To correct this, scDFM forms a one-step endpoint approximation,

$x \in \mathbb{R}_+^G$3

and penalizes the discrepancy between generated endpoints and real perturbed samples using an MMD term (Yu et al., 6 Feb 2026). The kernel is a mixture of Gaussian RBFs with bandwidths selected by a dynamic median heuristic per batch using factors $x \in \mathbb{R}_+^G$4.

The combined objective is

$x \in \mathbb{R}_+^G$5

with $x \in \mathbb{R}_+^G$6 in the reported experiments (Yu et al., 6 Feb 2026). In the paper’s interpretation, $x \in \mathbb{R}_+^G$7 teaches how cells move, while $x \in \mathbb{R}_+^G$8 teaches where the population should end up. This suggests that scDFM is explicitly structured to separate local transport dynamics from global distributional calibration.

3. PAD-Transformer backbone

The velocity field $x \in \mathbb{R}_+^G$9 is parameterized by the PAD-Transformer, a backbone designed to integrate perturbation conditioning, control-state context, and gene-level structure (Yu et al., 6 Feb 2026). Because of the large number of genes, training operates on subsets $c_p$0 with restricted control and perturbed representations. Initial token representations are formed by adding a value embedding $c_p$1 for expression values and a gene identity embedding $c_p$2:

$c_p$3

A central architectural prior is a gene–gene co-expression graph used as a sparse attention mask. Edge weights are defined by absolute Pearson correlation,

$c_p$4

computed over training cells, after which a kNN graph is constructed with, for example, $c_p$5 neighbors retained (Yu et al., 6 Feb 2026). The resulting sparse adjacency matrix constrains the gene encoder so that tokens attend only to correlated neighbors. The paper frames this as a mechanism for enforcing biologically meaningful locality and suppressing spurious interactions under noise and sparsity.

PAD-Transformer also adopts differential attention, in which the effective attention weights are the difference between two softmax distributions,

$c_p$6

with $c_p$7 a learnable scalar (Yu et al., 6 Feb 2026). This mechanism is used in both self-differential attention over the perturbed latent and cross-differential attention between perturbed latent and control representation. The stated purpose is to suppress nuisance patterns and emphasize perturbation-relevant differences.

Perturbation identity is embedded as $c_p$8 and injected at every layer by concatenation and MLP transformation, while time is encoded by a sinusoidal embedding followed by an MLP:

$c_p$9

Time conditioning modulates differential attention layers via adaLN-Zero style conditioning (Yu et al., 6 Feb 2026). After $c_p$0 layers, the final hidden state is decoded into the predicted velocity field. In the reported setup for Norman, the backbone uses hidden size $c_p$1, $c_p$2 layers, $c_p$3 heads, and dropout $c_p$4 (Yu et al., 6 Feb 2026).

4. Training procedure, preprocessing, and inference

The preprocessing pipeline uses library size normalization to 10,000 counts per cell, followed by log1p transformation (Yu et al., 6 Feb 2026). For the Norman dataset, the workflow selects the top 5,000 highly variable genes, then includes perturbation target genes, yielding 5029 total genes. For ComboSciPlex, it uses the top 5,000 highly expressed genes. Evaluation is reported on the top 1000 highly expressed genes, which the paper describes as focusing metrics on robust signals.

The training loop samples time $c_p$5, samples a gene subset $c_p$6, constructs a subgraph from the co-expression matrix, builds control and perturbed tokens, computes perturbation and time embeddings, runs PAD-Transformer, accumulates the CFM loss, computes batch endpoints $c_p$7, and then evaluates the batch MMD objective before a gradient update (Yu et al., 6 Feb 2026). The method re-samples gene subsets each step, which is presented as a scalability measure and as a way for the gene encoder to observe different subgraphs over training.

Reported training hyperparameters for Norman are: Adam optimizer, initial learning rate $c_p$8 with cosine decay to $c_p$9, batch size 96, training for 100,000 steps, and MMD weight $c_x$0 (Yu et al., 6 Feb 2026). Inference uses Euler integration with $c_x$1 steps, starting from $c_x$2 and iteratively updating

$c_x$3

For perturbation conditioning, gene perturbations in Norman share parameters between perturbation embeddings and gene identity embeddings, while drug perturbations in ComboSciPlex use a separate embedding table (Yu et al., 6 Feb 2026).

The practical usage description emphasizes population-level prediction: for a new condition, one provides control cell profiles $c_x$4, encodes the perturbation condition $c_x$5, runs the ODE solver, and aggregates predictions across many control cells to obtain the generated perturbed population (Yu et al., 6 Feb 2026). The same mechanism is used for in silico combinatorial perturbations by setting multiple entries of the multi-hot perturbation vector to one.

5. Benchmarks, metrics, and empirical results

The reported experiments cover two datasets: the Norman dataset, described as a Perturb-seq CRISPRa dataset in K562 cells with approximately 100 single-gene activations and 124 double perturbations, and ComboSciPlex, a drug perturbation dataset in A549 cells with 63,378 cells and 32 treatment conditions (Yu et al., 6 Feb 2026). On Norman, results are reported for both an additive setting, where single perturbations used in test combinations are seen during training, and a holdout setting, where some single perturbations and all combinations involving them are removed from training.

Baselines include Control, Additive, scGPT, Geneformer, GEARS, CPA, STATE, and CellFlow (Yu et al., 6 Feb 2026). Evaluation uses multiple metric families. Global reconstruction metrics are L2, MSE, and MAE. Distribution-level behavior is assessed by DS (Discrimination Score / PDS). Differential-expression fidelity is assessed by DE-Spearman $c_x$6. Heterogeneity and correlation structure are assessed by Pearson $c_x$7, Pearson $c_x$8, and Pearson $c_x$9 (Yu et al., 6 Feb 2026).

On Norman, additive split, scDFM reports L2 = 1.7043, MSE = 0.00315, MAE = 0.02155, DE-Spearman = 0.5705, DS = 0.9737, Pearson $x$0, and Pearson $x$1 (Yu et al., 6 Feb 2026). The paper highlights that this corresponds to a 19.6% MSE reduction relative to CellFlow, whose MSE is 0.00392. It also notes that the Additive baseline is surprisingly strong, but scDFM still improves both reconstruction and distributional metrics, which the authors interpret as evidence that it captures non-additive effects.

On Norman, holdout split, scDFM reports the best values across the tabulated methods for both unseen singles and unseen doubles on most metrics (Yu et al., 6 Feb 2026). For held-out single perturbations, the reported values are L2 = 1.6186, MSE = 0.0030, MAE = 0.0190, DE-Spearman = 0.6957, Pearson $x$2, DS = 0.8914, Pearson $x$3, and Pearson $x$4. For held-out double perturbations with unseen singles, scDFM reports L2 = 2.0309, MSE = 0.0047, MAE = 0.0235, DE-Spearman = 0.5676, Pearson $x$5, DS = 0.9189, Pearson $x$6, and Pearson $x$7.

On ComboSciPlex, scDFM reports L2 = 1.6567, MSE = 0.0028, MAE = 0.0220, DE-Spearman = 0.8289, Pearson $x$8, and DS = 0.8776 (Yu et al., 6 Feb 2026). The paper describes these results as showing generalization across perturbation modality, from CRISPRa to drugs.

Qualitative analyses reinforce the distribution-level claim. UMAP visualizations show that removing the MMD objective causes generated cells to deviate from the ground-truth manifold, while full scDFM better follows the real population geometry (Yu et al., 6 Feb 2026). A case study on CEBPE+CEBPA highlights genes such as CEACAM20 and LST1, where scDFM is reported to match directionality and remain within observed variance better than several baselines.

6. Interpretation, limitations, and terminological ambiguity

The design of scDFM encodes a specific methodological position: perturbation prediction should be treated as distribution transport in full expression space, not merely as latent reconstruction or mean-profile regression (Yu et al., 6 Feb 2026). The use of MMD indicates a preference for sample-based distribution matching under support mismatch, while the PAD-Transformer introduces biological structure through graph masking and differential control-versus-perturbed attention. A plausible implication is that the framework is especially suited to settings where the principal scientific object is a perturbed population manifold rather than a single canonical response vector.

The reported limitations are explicit. The interpolation path is linear in log-normalized expression space, whereas real biological trajectories may be non-linear, branching, and manifold-constrained (Yu et al., 6 Feb 2026). The co-expression graph is built from absolute Pearson correlation and kNN, so it captures only linear dependencies and does not encode causality. Experiments are limited to Norman and ComboSciPlex rather than larger multi-context datasets such as ARC-state or Virtual Cell Challenge. The method also carries computational cost: attention scales as $x$9 on gene subsets, and MMD scales as $p_\theta(x \mid c_x, c_p),$0 in batch size, though the gene-subset sampling strategy partially mitigates this.

Future directions named in the source include manifold-aware paths, learned or causal graphs, scaling to larger multi-cell-type datasets, multi-omics integration, incorporating dose and time-course covariates, and using scDFM as a digital twin for in silico screening and optimization in drug discovery (Yu et al., 6 Feb 2026). The implementation is available at the GitHub repository linked by the paper.

The term scDFM is also potentially ambiguous in adjacent literature. One source notes that the acronym was not explicitly defined there and could refer informally to SCDM, the “Score-Based Channel Denoising Model” for digital semantic communications (Mo et al., 18 Jan 2025). Another source states that the term could plausibly be used for a semantic-feature multiple-access concept that the paper itself names SFDMA, “Semantic Feature Division Multiple Access” (Ma et al., 2024). In the dedicated single-cell perturbation context, however, scDFM specifically denotes single-cell Distributional Flow Matching (Yu et al., 6 Feb 2026).