We hope this framework sparks new ideas for noisy-data scenarios in science and beyond. Check out our paper for technical details, proofs, and full experiments! Thanks for reading—let us know what you think.

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Results show that Inverse Flow can often match or outperform even supervised baselines that have access to the ground truth —and it excels on data when standard assumptions (like independent or Gaussian noise) are violated.

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We validated these methods on:
1. Synthetic tasks (Navier-Stokes, 8-Gaussians).
2. Semi-synthetic image denoising with varied noise (Gaussian, correlated, SDE-based).
3. Real fluorescence microscopy.
4. Single-cell genomics, where we improved clarity of cell types and developmental trajectories.

Why is this cool?
1. It handles any continuous noise, including correlated or non-Gaussian noise.
2. It generalizes to cases where the noise distribution only has a simulation procedure (like certain SDEs).
3. No ground-truth are needed—just the noisy measurements and knowledge of the noise model.

1. IFM extends conditional flow matching by replacing unknown clean data with a generated guess in each training iteration.
2. ICM eliminates the need to simulate ODEs during training; it applies our generalized consistency training to directly get a one-step mapping from noise to clean data.

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Key insight: Instead of using real clean data (which we don’t have), we estimate it on the fly by mapping noisy data backward through a learned ODE or direct consistency function. This effectively “inverts” the noise process—hence Inverse Flow.

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We propose two approaches:
1. Inverse Flow Matching (IFM)
2. Inverse Consistency Model (ICM)
Both learn how to “reverse” a given (potentially complex) noise process to recover the underlying clean signal distribution from only the noisy observations.

(3/9)

Traditional methods (e.g., diffusion models, flow matching, consistency models) need clean training data to learn generative models. But what if your data is always noisy and ground truth remains unseen? That’s where our Inverse Flow framework steps in.

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