arXiv:2605.31106v1 Announce Type: new Abstract: Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, de
This research addresses a fundamental limitation in Riemannian diffusion models, enabling their application to a broader range of data structures with a novel computational approach.
It advances the mathematical and computational foundations of generative AI, potentially expanding its capabilities to complex, non-Euclidean data relevant across scientific and engineering domains.
The ability to approximate the heat kernel on general manifolds via PINNs removes a significant bottleneck for diffusion models, making them more versatile for manifold-supported data.
- · AI researchers
- · Generative AI developers
- · Machine learning on complex data
- · Physics-informed neural network advancements
- · Legacy manifold approximation methods
More sophisticated generative models can be developed for data residing on non-flat spaces, such as molecular structures or climate data.
This could lead to breakthroughs in areas requiring generative modeling of scientific data where traditional Euclidean assumptions do not hold.
The broader application of generative AI to complex scientific data could accelerate discovery and design in fields like materials science and drug discovery.
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Read at arXiv cs.LG