Score-Based Diffusion in Infinite Dimensions

• Score-based diffusion is a generative modeling method that leverages the score function to reverse a forward diffusion process in infinite-dimensional spaces.
• It employs a rigorous mathematical framework using SPDEs and Malliavin calculus to achieve well-defined Gaussian distributions and spatially correlated noise.
• The approach facilitates direct inference and generative tasks for functional data, enabling advanced applications in fields like computational physics and scientific computing.

Score-based diffusion refers to a broad class of generative models that learn the score function—the gradient of the log-density—of a data distribution. By leveraging this score, they define a reverse-time stochastic process, often driven by stochastic partial differential equations (SPDEs) or stochastic differential equations (SDEs), to synthesize new samples and solve inverse problems. The theoretical foundation for score-based diffusion models has recently been extended from finite-dimensional spaces (such as images or vectors) to infinite-dimensional Hilbert spaces (such as function spaces or physical fields), using advanced tools from stochastic analysis, Malliavin calculus, and operator theory (Mirafzali et al., 27 Aug 2025). These advances enable generative modeling and inference directly in the functional regime, which is essential for applications in computational physics, scientific computing, and beyond.

1. Mathematical Framework for Infinite-Dimensional Score-Based Diffusion

Score-based diffusion in infinite dimensions is rigorously formulated by considering the forward process as a linear SPDE on a separable Hilbert space $H$. The canonical formulation is

$$du(t) = A u(t)\,dt + Q^{1/2} dW_t,\qquad u(0) = u_0\in H,$$

where:

• $A$ is a densely defined unbounded linear operator (e.g., Laplacian with boundary conditions), generating a strongly continuous semigroup $S(t) = e^{tA}$,
• $Q^{1/2}$ is a Hilbert–Schmidt operator representing the spatial covariance structure of the noise,
• $W_t$ is a cylindrical Wiener process.

Well-posedness is assured under the trace-class assumption on $Q$, enabling the mild solution: $u(t) = S(t) u_0 + \int_0^t S(t-s) Q^{1/2} dW_s,$ with the stochastic integral understood in the Hilbert-space Itô sense. The condition $$\int_0^t \| S(s) Q^{1/2} \|_{\mathrm{HS}}^2 ds < \infty$$ guarantees that $u(t)$ is an $H$-valued random variable, even in arbitrary spatial dimensions.

This framework allows spatially correlated (colored) noise, with the trace-class structure ensuring that noise injection is mathematically compatible with the geometry of $H$ and avoids the divergence issues endemic to white noise in infinite dimensions.

2. Forward Diffusion and Covariance Structure

In the forward diffusion, the process begins with a deterministic initial element $u_0$ undergoing successive diffusion and drift according to the above SPDE. The linear dynamics ensures that for each $t > 0$ the distribution of $u(t)$ is Gaussian in $H$. The mean is $S(t)u_0$, and the covariance operator is

$$\gamma_{u(t)} = \int_0^t S(s) Q^{1/2} [Q^{1/2}]^* S(s)^* ds,$$

which is trace-class and positive semi-definite. This operator encodes both the diffusion imposed by the semigroup and the spatial correlations from $Q$. The trace-class requirement is central to ensuring the process is well defined and that subsequent score computations are meaningful in $H$.

3. Malliavin Calculus and Infinite-Dimensional Score Function

A principal advance is the derivation of the score function (the Fréchet derivative of the log-density) directly in the infinite-dimensional setting using Malliavin calculus and an extension of the Bismut–Elworthy–Li formula. The Malliavin derivative of $u(t)$ at $r \in [0, t]$ is

$D_r u(t) = S(t - r) Q^{1/2},$

which leads to an explicit operator formula for the score via: $$\langle \nabla \log p_{u(t)}(u), h \rangle_H = -\langle \gamma_{u(t)}^\dagger (u - S(t)u_0), h \rangle_H,$$ for all $h$ in the range of $\gamma_{u(t)}^{1/2}$, where $\gamma_{u(t)}^\dagger$ is the Moore–Penrose pseudoinverse. This characterization allows computation of the score along any admissible Cameron–Martin direction without needing to approximate or discretize the Hilbert space.

This avoids finite-dimensional projections and preserves the geometric and statistical structure inherent to $H$.

4. Operator-Theoretic and Geometric Treatment

The operator-theoretic approach ensures that all analysis and computation remain within the intrinsic geometry of the Hilbert space. By explicitly defining the relevant semigroups, covariance operators, and noise structures, the model accommodates general (possibly degenerate) $Q$ and non-invertible semigroups, which are essential for, e.g., dissipative or ill-posed SPDEs. When invertibility is lacking, the formulas adapt via pseudoinverses, maintaining mathematical rigor for a broad class of infinite-dimensional processes.

Spatial correlations are naturally encoded through $Q$; this is crucial in practice for applications involving, for example, random fields in continuum mechanics or spatially varying material models.

5. Connection to Functional Data Analysis and Computational Approaches

By reinterpreting score estimation as an infinite-dimensional regression task, the framework makes explicit links to functional data analysis. This enables the use of computational strategies including:

• Kernel methods: Leveraging reproducing kernel Hilbert space (RKHS) techniques to estimate or regularize the score.
• Neural operator architectures: Employing neural networks tailored to function spaces, such as DeepONets or Fourier Neural Operators, which approximate mappings between elements of $H$, for direct learning of the score function in high-batch and high-resolution regimes.

Such strategies exploit the infinite-dimensional formulation to achieve discretization invariance and the ability to transfer models across resolutions.

6. Applications and Broader Implications

A rigorous foundation for infinite-dimensional score-based diffusion has implications for any domain where data is naturally functional or field-valued. Direct applications include:

• Scientific computing and PDE modeling: The framework enables generative modeling for solution fields, uncertainty quantification, and data-driven calibration, all without ad hoc discretization.
• Physical inverse problems: The ability to model and infer probability densities directly on functionals allows principled prior specification for ill-posed inverse problems.
• High-dimensional generative tasks: By bypassing finite-dimensional constraints, the method opens avenues for high-resolution image, audio, or field generation, facilitating super-resolution and transfer across computational grids.

The methodology accommodates general trace-class noise operators, making it suitable for complex statistical models with spatial correlation that arise in meteorology, hydrodynamics, or computational imaging.

The approach is general enough to be extended to nonlinear SPDEs with state-dependent diffusion, contingent on further analytic work to generalize the infinite-dimensional Malliavin calculus and Bismut formulae beyond the linear case.

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In sum, infinite-dimensional score-based diffusion, as formalized through operator-theoretic Malliavin calculus (Mirafzali et al., 27 Aug 2025), lays a mathematical and computational foundation for generative modeling and inference on function spaces. The exact closed-form expression for the score function enables dimension-independent modeling, rigorously incorporates spatially correlated noise, and directly links to established and emerging tools in statistical functional analysis and neural operator learning.