Looped flows train recurrent reasoning with local denoising objectives. Progressively decreasing noise levels and shared noise temporally align these objectives, encouraging recurrent states that remain useful across updates. At inference, we integrate probability flow coupled with recurrent states, enabling inference-time scaling and diverse valid solutions.
Humans and machines often solve harder problems by spending more time on computation. In deep learning, looped models implement this idea during inference by recurrently updating a hidden state. In practice, however, their training backpropagates through only one or a few updates, making it hard to train early updates to support future ones.
We propose looped flows, an approach that sidesteps this issue by training the recurrence with local denoising objectives. By imposing temporal association across denoising objectives through progressively decreasing noise levels and shared noise, the model is incentivized to learn recurrent states that transfer useful computation over time, even when gradients cover only a few updates.
We then formulate inference as integrating the velocity of a probability flow parameterized by the learned denoiser, coupled with recurrent states. This allows solving harder problems by spending more computation through a finer temporal grid and enables multiple valid predictions from different initial noise samples. Across six reasoning benchmarks including two multi-solution benchmarks, looped flows outperform prior state-of-the-art looped models overall, achieving 58.8% test accuracy on ARC-AGI-1 and 12.2% on ARC-AGI-2.
We train the recurrence on a sequence of progressively easier denoising tasks. Decreasing noise levels and shared noise temporally align the objectives, encouraging globally useful recurrent states even when gradients cover only one or a few updates.
Inference integrates probability flow coupled with recurrent states. A finer temporal grid increases inference-time computation, improving Sudoku-Extreme accuracy from 74.5% at 8 steps to 97.9% at 128 steps.
On N-Queens and Graph Coloring, looped flows recover different valid solutions from different initial noise samples through probability transport. Diversity is retained even under deterministic ODE integration.
Looped models increase inference-time computation by recurrently updating a hidden state with shared parameters. In practice, training backpropagates through only one or a few updates, making it hard to train early updates to support future computation. Looped flows address this challenge with temporally aligned local denoising objectives, using progressively decreasing noise levels and shared noise.
For a problem c and target solution x1, draw a noise sample x0. Share the noise-target pair across increasing timesteps t to construct interpolants with decreasing noise levels:
The stateful denoiser predicts the solution and updates the recurrent state at each step. A local cross-entropy loss supervises the prediction, with gradients stopped between denoising steps. The shared noise-target pair imposes an additional association across timesteps.
Starting from noise, the stateful denoiser reads the problem, flow state, and recurrent state, predicts a solution, and updates the recurrent state. For deterministic ODE integration, a forward Euler step then updates the flow state:
A finer temporal grid increases inference-time computation. The noise-backtracking scheme in Algorithm 2 approximates stochastic integration and generally improves accuracy and solution coverage. Different initial noise samples yield diverse predictions even under deterministic ODE integration.
We build on TRM's architecture, adding noisy-input and time embeddings: about 5M parameters on Sudoku and 7M on the other tasks. Under the paper's evaluation protocol, looped flows improve over prior state-of-the-art looped models on five of six benchmarks and remain competitive on Maze-Hard.
Compared with TRM, ARC-AGI-1 improves from 44.6% to 58.8% and ARC-AGI-2 from 7.8% to 12.2% in single-trajectory evaluation (pass@2).
| Method | Parameters | Sudoku-Extreme | Maze-Hard | ARC-AGI-1 | ARC-AGI-2 |
|---|---|---|---|---|---|
| Direct prediction | 27M | 0.0 | 0.0 | 21.0 | 0.0 |
| Looped transformer | 7M | 61.3 | — | — | — |
| HRM | 27M | 55.0 | 74.5 | 40.3 | 5.0 |
| TRM | 5–7M | 87.4 | 85.3 | 44.6 | 7.8 |
| FPRM | 7M | 94.2 | 87.0 | 47.5 | 6.2 |
| GRAM | 10M | — | — | 52.0 | 11.1 |
| Looped flows | 5–7M | 97.9 ± 0.4 | 86.7 ± 1.1 | 58.8 ± 1.8 | 12.2 ± 1.9 |
Solution accuracy (%) for Sudoku and Maze; pass@2 (%) for ARC. Looped flows: mean ± standard deviation across 3 seeds. Bold indicates the best result in each column. Paper, Table 1.
| Method | Parameters | Sudoku-Extreme | Maze-Hard | ARC-AGI-1 | ARC-AGI-2 |
|---|---|---|---|---|---|
| PTRM | 5–7M | 98.8 (100) | 86.7 (100) | — | 9.7 (25) |
| EqR | 5M | 99.8 (128) | — | — | — |
| GRAM | 10M | 97.0 (20) | — | — | — |
| Looped flows | 5–7M | 99.3 ± 0.2 (5) | 86.9 ± 1.5 (5) | 59.5 ± 1.9 (5) | 12.2 ± 0.9 (5) |
Parentheses give the number of inference trajectories. Looped flows use best-Q selection over 5 trajectories; other methods use their reported ensemble settings. Accuracy (%) for Sudoku/Maze and pass@2 (%) for ARC. Looped flows: mean ± standard deviation across 3 seeds. Paper, Table 1.
On Sudoku-Extreme, increasing inference steps from 8 to 128 with a finer temporal grid improves accuracy from 74.5% to 97.9%. This is complementary to inference-time ensembling over independent trajectories.
Training rollouts use up to 16 denoising steps. At inference, the temporal grid can be made finer without retraining. Details in Section 5.1.
Some reasoning problems have multiple valid solutions. On N-Queens and Graph Coloring, looped flows recover diverse valid solutions from different initial noise samples. We evaluate validity and coverage over 20 independent inferences per test instance.
| Method | Parameters | 8 × 8 board | 10 × 10 board | ||
|---|---|---|---|---|---|
| Accuracy ↑ | Coverage ↑ | Accuracy ↑ | Coverage ↑ | ||
| Direct prediction (8 layers) | 27M | 40.4 ± 1.1 | 13.7 ± 1.1 | 13.6 ± 0.5 | 1.6 ± 0.2 |
| Direct prediction (32 layers) | 100M | 40.2 ± 1.3 | 13.6 ± 1.1 | 13.1 ± 0.4 | 1.6 ± 0.2 |
| Looped transformer | 7M | 68.4 ± 3.7 | 23.6 ± 1.9 | 50.0 ± 7.6 | 6.2 ± 3.2 |
| HRM | 27M | 78.7 ± 2.9 | 26.7 ± 1.3 | 37.4 ± 0.3 | 4.7 ± 0.1 |
| TRM | 7M | 66.8 ± 5.7 | 36.1 ± 22.5 | 17.5 ± 11.2 | 2.0 ± 1.3 |
| AR transformer | 10.6M | 96.3 ± 1.0 | 84.8 ± 0.8 | 90.0 ± 2.2 | 53.2 ± 0.8 |
| MDLM | 12.6M | 96.1 ± 1.5 | 87.2 ± 0.6 | 74.3 ± 6.6 | 47.4 ± 2.2 |
| GRAM | 10M | 99.7 ± 0.3 | 90.3 ± 1.9 | 89.7 ± 2.7 | 57.5 ± 3.4 |
| Looped flows | 7M | 99.9 ± 0.1 | 91.4 ± 0.4 | 94.4 ± 0.7 | 61.5 ± 0.1 |
Accuracy (%) measures the validity of the first sample. Coverage (%) is the fraction of distinct valid solutions recovered from 20 samples. ↑ Higher is better. Looped flows: mean ± standard deviation over 3 repeated tests. Paper, Table 2.
| Method | Parameters | 8-vertex graphs | 10-vertex graphs | ||
|---|---|---|---|---|---|
| Conflicts ↓ | Coverage ↑ | Conflicts ↓ | Coverage ↑ | ||
| Direct prediction (8 layers) | 27M | 179.3 ± 4.0 | 19.9 ± 0.2 | 198.7 ± 5.0 | 6.7 ± 0.1 |
| Direct prediction (32 layers) | 100M | 174.0 ± 18.0 | 19.1 ± 1.7 | 227.7 ± 34.5 | 6.5 ± 1.9 |
| Looped transformer | 7M | 136.0 ± 16.1 | 20.5 ± 1.5 | 157.3 ± 9.0 | 7.2 ± 0.7 |
| HRM | 27M | 109.7 ± 1.5 | 21.8 ± 0.3 | 164.3 ± 21.6 | 8.9 ± 1.7 |
| TRM | 7M | 109.3 ± 3.1 | 22.3 ± 0.6 | 170.7 ± 17.9 | 6.8 ± 0.3 |
| AR transformer | 10.6M | 19.0 ± 11.3 | 83.0 ± 0.7 | 61.3 ± 8.3 | 40.0 ± 0.3 |
| MDLM | 12.6M | 2.7 ± 0.6 | 84.5 ± 4.0 | 12.0 ± 7.0 | 48.2 ± 1.4 |
| GRAM | 10M | 2.7 ± 2.1 | 85.8 ± 0.5 | 3.3 ± 1.5 | 51.3 ± 2.8 |
| Looped flows | 7M | 0.7 ± 0.6 | 89.4 ± 0.7 | 1.0 ± 1.0 | 55.2 ± 0.3 |
Conflicts are summed over test graphs using each graph's most frequent complete coloring from 20 samples (↓ lower is better). Coverage is the fraction of distinct valid solutions recovered (%, ↑ higher is better). Looped flows: 3 repeated tests. Paper, Table 2.
On around 65,000 Sudoku-Extreme test instances, TRM fails on 12.6% of cases. Among these failures, 88.3% fail to converge and 11.7% converge to spurious attractors. Looped flows recover 89.9% of the non-convergence cases and 98.0% of the spurious-attractor cases, resolving 90.9% of the TRM failures in this analysis.
We ablate time conditioning, training on interpolants, decreasing noise levels, and shared noise across timesteps. Each removal reduces ARC performance in this comparison; removing both time conditioning and interpolants produces the largest drop.
These ablations support the contribution of the flow formulation and temporally aligned denoising objectives to recurrent reasoning.
| Configuration | ARC-AGI-1 | ARC-AGI-2 |
|---|---|---|
| Looped flows | 58.8 ± 1.8 | 12.2 ± 1.9 |
| w/o time conditioning | 56.4 | 9.9 |
| w/o interpolant | 51.5 | 9.9 |
| w/o time & interpolant | 43.6 | 5.0 |
| w/o decreasing noise | 51.6 | 9.9 |
| w/o noise sharing | 56.4 | 10.8 |
pass@2 accuracy (%). Paper, Table 3.
In the Sudoku comparison, FLM without recurrence memorizes the training data. Self-conditioning on the denoised prediction or hidden features improves generalization, but learning recurrent states across decreasing noise levels performs best. Unlike looped flows, the self-conditioned baselines are trained with two forward passes at the same flow timestep.
@article{suleymanzade2026thinkingloopedflows,
title={Thinking with Looped Flows},
author={Suleymanzade, Ayhan and Lee, Chanhyuk and
Eijkelboom, Floor and Boffi, Nicholas M. and
Ceylan, İsmail İlkan and Kim, Jinwoo},
year={2026},
journal={arXiv preprint arXiv:2609.11801},
eprint={2609.11801},
archivePrefix={arXiv},
primaryClass={cs.LG},
url={https://arxiv.org/abs/2609.11801}
}