Fractional Diffusion Bridge Models

AI summary

Overview

Research area: generative machine learning, diffusion bridge models, stochastic processes, and non-Markovian noise. Technical level: Advanced. This paper introduces Fractional Diffusion Bridge Models (FDBM), a framework that replaces standard Brownian motion with an approximation of fractional Brownian motion to model memory, roughness, and long-range dependencies in generative bridges.

What This Paper Is About

Most diffusion and bridge models use Brownian motion, which assumes independent increments and no memory. Many real systems, such as proteins and images, contain temporal correlations, anomalous diffusion, and long-range dependencies that Brownian motion cannot capture well. The paper builds a tractable generative diffusion bridge driven by fractional noise, with a Hurst index that controls path roughness and correlation, and tests it on paired protein conformation prediction and unpaired image translation.

Key Contributions

1. Proposes FDBM, a generative diffusion bridge framework driven by a Markov approximation of fractional Brownian motion (MA-fBM), enabling non-Markovian trajectories with controllable correlation and long-range dependence via the Hurst index.
2. Proves the existence of a coupling-preserving stochastic process for these generalized fractional dynamics, enabling paired data translation without losing the training data alignment.
3. Formulates the Schrödinger bridge problem with an MA-fBM reference process and derives a principled loss for learning unpaired stochastic transport trajectories.
4. Demonstrates FDBM on protein conformational change prediction and unpaired image translation, achieving lower Cα RMSD and lower FID than Brownian baselines, and releases public implementations.

Main Findings

• Coupling preservation in paired settings: FDBM preserves the intended data coupling, unlike SBALIGN in the toy experiment. ABM also preserves coupling, but FDBM offers a wider range of trajectories controlled by the Hurst index.
• Protein conformational prediction: FDBM improves over ABM and other baselines on the D3PM dataset. Best FDBM with H=0.2 reaches median RMSD 2.12 Å, mean 3.34 Å, and 48% of predictions under 2 Å, compared with ABM at median 2.40 Å, mean 3.49 Å, and 43% under 2 Å.
• Unpaired image translation: FDBM achieves lower Fréchet Inception Distance (FID) than Brownian-motion-based bridge baselines, indicating better generative quality in unpaired domain translation.
• Synthetic distribution transport: FDBM improves Wasserstein distance on Moons and T-Shape datasets. The best Hurst index is dataset-dependent: smoother paths (H=0.6–0.7) help Moons, while rougher paths (H=0.2) help T-Shape.
• Efficiency and flexibility: Replacing Brownian motion with MA-fBM incurs minimal additional computational cost relative to Augmented Bridge Matching. The same neural network architectures can be used with modified inputs and outputs.
• Limitations in unpaired training: The authors do not claim convergence to the Schrödinger bridge solution. Finetuning with MA-fBM performs reliably mainly near H=0.5 due to asymmetries between forward and reverse bridge transformations.

Methodology in Plain English

Standard diffusion bridges use a Brownian bridge, which is mathematically convenient but memoryless. The authors instead want to use fractional Brownian motion, which can produce rougher or smoother paths and long-range correlations. Directly using fractional Brownian motion makes the bridge drift intractable, so they approximate it by a weighted sum of Ornstein–Uhlenbeck processes driven by the same Brownian motion. This augmented process is Markovian, so they can derive closed-form bridge dynamics and simulate it efficiently.

For paired data, they prove that a stochastic differential equation exists whose solution preserves the coupling in the training pairs. They then train a neural network to approximate the intractable drift and derive a loss function from the Kullback–Leibler divergence. For unpaired data, they define a Schrödinger bridge problem relative to the approximated fractional process, generalize reciprocal-class and Markovian-projection ideas, and train with a pretraining plus finetuning procedure. They evaluate the method on synthetic data, protein conformational changes, and unpaired image translation.

Why This Matters

This work expands generative diffusion models beyond Brownian noise, which is a simplifying assumption rather than a faithful model of many real processes. It connects fractional stochastic processes, diffusion bridges, and Schrödinger bridges within a practical machine learning framework, opening a route to generative models that explicitly represent memory, roughness, and long-range dependence.

Real-world applications include:

• Drug discovery and structural biology: predicting how proteins change shape when a ligand binds, which can support virtual screening and mechanism studies.
• Protein structure modeling: generating ligand-bound holo structures from ligand-free apo conformations and improving conformational ensemble prediction.
• Unpaired image translation: domain adaptation, medical image conversion, style transfer, and restoration where paired images are unavailable.
• Scientific time-series and biological dynamics: modeling anomalous diffusion, cell differentiation trajectories, and other systems with temporal correlations.

Industry relevance spans pharmaceutical and biotechnology companies using AI for drug discovery, medical imaging and diagnostics companies, generative AI tool developers, and computational science teams that need better stochastic models for correlated data.

Future Directions

• Improve unpaired FDBM so finetuning works reliably across a wider range of Hurst indices, not only near H=0.5.
• Prove convergence of the proposed iterative Markovian fitting procedure with an MA-fBM reference process to the Schrödinger bridge solution.
• Learn the Hurst index or make it data-adaptive, and optimize the number and coefficients of Ornstein–Uhlenbeck approximation terms.
• Scale the method to larger proteins, higher-resolution 3D images, and additional modalities, while benchmarking on tasks that specifically reward long-range memory modeling.
• Combine FDBM with other non-standard noise processes, flow matching, equivariant architectures, or optimal transport solvers.

Target Audience

This paper benefits machine learning researchers working on diffusion models, bridge models, and generative stochastic processes; computational biologists and chemists interested in protein conformational prediction; applied mathematicians studying Schrödinger bridges, optimal transport, and fractional processes; and practitioners in unpaired image translation or scientific time-series modeling. It is most accessible to readers with prior exposure to diffusion models or stochastic differential equations.

Authors’ abstract

We present Fractional Diffusion Bridge Models (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_α$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.