The order in which tokens are predicted in language modeling has a profound impact on the quality of generated sequences. Autoregressive and masked diffusion models iteratively fill in tokens and can exploit such orderings across generation steps, but how to leverage an ordering for one-step parallel language modeling is unclear. We introduce asynchronous flow map language models (aFMLMs), which incorporate prediction order into one-step modeling by learning the flow maps of asynchronous flows. These flows specify an ordering through a tokenwise time schedule that controls the relative denoising progress of tokens. We formulate a general class of asynchronous flows that admit both prescribed and learned, state-adaptive schedules, show that they define valid flow maps, and develop new methods for their flow map distillation. The resulting models can generate all tokens in a single forward pass by internalizing the ordering encoded in the flow into their hidden computations. On Sudoku solving and code generation for GSM8K math word problems, aFMLM establishes new state-of-the-art one-step performance.
Figures & tables
Figure 1: Overview. (Left) We formulate asynchronous flows that denoise tokens at different rates. To rhyme with “mitten,” “kitten” emerges early and guides the preceding words. (Right) We distill these flows into an asynchronous flow map that internalizes this ordering and generates the full sequence in one forward pass.
Figure 2: Accuracy of a FLM under different inference-time schedulers on Sudoku-Hard (left) and GSM8K (right). A well-chosen scheduler improves performance over the synchronous one (Sync.). Alg. order † denotes the a FLM whose denoiser is trained only along the algorithmic order.
Sudoku
GSM8K
Model
Easy
Medium
Hard
Accuracy
MDLM ( Sahoo et al., 2024 )
63.6
16.2
1.7
0.0
Duo ( Sahoo et al., 2025 )
69.8
24.6
2.4
0.0
CANDI ( Pynadath et al., 2025 )
69.0
36.1
4.0
0.0
FLM ( Lee et al., 2026 )
69.1
21.5
2.8
0.0
S-FLM ( Deschenaux and Gulcehre, 2026 )
77.2
36.0
5.2
0.0
Table 1: One-step exact solving accuracy (%) on Sudoku and zero-shot accuracy (%) on GSM8K (see Section C.2 for baseline details). For a FLM, the schedule does not matter in one-step generation.
Figure 4
Schedule
Teacher
Distillation
Total
Learned
1.7h
3.3h
5.0h
Prescribed
0h
3.2h
3.2h
Table 2: Training cost (GPU hours) on Sudoku-Hard.
Figure 5: A visualization of code solutions to a GSM8K problem generated by one-step a FMLM (EB) (middle) and a FMLM (Sync.) (bottom) in their layer-wise order, compared with the generation order of a FLM (EB) (top). Columns show when 10%, 50%, 85%, and 100% of the tokens are decided.
Schedule
Sudoku-Hard
GSM8K
Synchronous
0.500
0.091
Asynchronous
0.516
0.271
Table 3: Spearman correlation with the schedule-induced ordering.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Power schedules τj(t)=tαj with β=log3 (left) and their speeds ktj (right). Darker curves denote higher-priority positions, with exponents ranging from αj=1/3 (darkest) to αj=3 (lightest), and the dashed line denotes the synchronous schedule.
TinyGSM / GSM8K
Sudoku
Architecture (DiT)
Hidden size
768
512
adaLN cond. dim
128
128
Layers
12
8
Attention heads
12
8
FFN multiplier
4
4
Appendix
Table 4: Per-task hyperparameters. All models on the same task share the architecture and optimizer, and differ only in the training and inference schedules.
Figure 7: Algorithmic ordering for a hard Sudoku puzzle. Plain digits are the given clues, and labels r1 – r3 indicate the round in which each empty cell is resolved. Cells resolved in the current round are shaded in darker blue, while previously assigned rounds remain visible. Cells still blank after round 3 are resolved in later rounds (not shown).
Model
10%
50%
85%
100%
a FMLM (EB)
6
8
11
12
a FMLM (Sync.)
2
7
10
12
Appendix
Table 5: Transformer block (out of 12) at which each column of Figure 5 is taken.