Probability-Flow ODE in Infinite-Dimensional Function Spaces

Diffusion model (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021b; Kingma et al., 2021) is a class of generative model that adds noise to real data to train the score network and sequentially approximate the time-reversed process (Föllmer and Wakolbinger, 1986; Anderson, 1982) to generate samples from the true data distribution. This model has shown remarkable empirical success in numerous domains such as image generation (Song et al., 2021b,a), video generation (Luo et al., 2023), medical data processing (Song et al., 2022; Chung and Ye, 2022; Akrout et al., 2023), and audio generation (Kong et al., 2020). However, "classical" diffusion models formulated on finite-dimensional Euclidean spaces limit their applicability to function generation problems as they can only generate function values realized on a fixed discretization of the function's domain (Li et al., 2020) and cannot capture functional properties of a data such as integrability or smoothness (Kerrigan et al., 2023). Motivated by such a limitation of finite-dimensional models, there has been a line of works extending the finite-dimensional diffusion model to infinite-dimensional Hilbert spaces; for instance, Hagemann et al. (2023); Kerrigan et al. (2023); Lim et al. (2023a,b); Pidstrigach et al. (2023); Phillips et al. (2022); Baldassari et al. (2023). Kerrigan et al. (2023) proposes a discrete-time model that serves as an analog of Ho et al. (2020) in infinite-dimensional space, and Hagemann et al. (2023) introduces a finite-dimensional approximation of an infinite-dimensional SDEs and utilizes the time-reversal formula in finite-dimensional spaces. Lim et al. (2023a); Franzese et al. (2023); Pidstrigach et al. (2023) propose continuous-time models by extending the SDE framework of Song et al. (2021b) to infinite dimensions based on semigroup theory (ref. Da Prato and Zabczyk (2014)); however, their consideration is limited to a relatively simple class of SDEs, such as Langevin type SDE or SDEs with constant-time diffusion coefficients. Later, Lim et al. (2023b) proved a general form of time-reversal formula which encompasses various choices of SDEs such as VPSDE, VESDE, sub-VPSDE (Song et al., 2021b) and variance scheduling (Nichol and