Technology Category
Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Learning with ManifoldsSearch and Optimization: Mixed Discrete/Continuous Search
Stochastic Interpolants in Hilbert Spaces
🤖 AI Summary
This work addresses the limitations of existing stochastic interpolation methods, which are confined to finite-dimensional spaces and thus struggle to model generative tasks between arbitrary distributions in function spaces. For the first time, the authors extend stochastic interpolation theory to infinite-dimensional Hilbert spaces, establishing a rigorous mathematical framework grounded in functional analysis and stochastic differential equations. They provide well-posedness guarantees and explicit error bounds, thereby overcoming dimensional constraints and enabling controllable generation under complex conditions. The proposed method achieves state-of-the-art performance on PDE-driven function space benchmark tasks, offering a general and efficient tool for generating high-dimensional continuous distributions in scientific computing.
Technology Category
Application Category
📝 Abstract
Although diffusion models have successfully extended to function-valued data, stochastic interpolants -- which offer a flexible way to bridge arbitrary distributions -- remain limited to finite-dimensional settings. This work bridges this gap by establishing a rigorous framework for stochastic interpolants in infinite-dimensional Hilbert spaces. We provide comprehensive theoretical foundations, including proofs of well-posedness and explicit error bounds. We demonstrate the effectiveness of the proposed framework for conditional generation, focusing particularly on complex PDE-based benchmarks. By enabling generative bridges between arbitrary functional distributions, our approach achieves state-of-the-art results, offering a powerful, general-purpose tool for scientific discovery.
Problem
Research questions and friction points this paper is trying to address.
stochastic interpolants
Hilbert spaces
infinite-dimensional
generative modeling
function-valued data
Innovation
Methods, ideas, or system contributions that make the work stand out.
stochastic interpolants
Hilbert spaces
infinite-dimensional generative modeling
conditional generation
function-valued data
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