arXiv subjects

Ehsan Mirafzali

Publications and source records attributed to Ehsan Mirafzali.

7 recordsLinked to original sources

arXiv2026

Holographic generative flows with AdS/CFT

Holography, in the form of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, offers a natural setting for generative modelling. Data on a boundary manifold lifts into a higher-dimensional bulk through a propagator, and this extra dimension plays the role of a flow parameter. We exploit this structure to build GenAdS, an approach to generative flow matching in which the dynamics are represented by the evolution of fields in AdS, together with a residual correction learned by a neural network. Boundary samples are encoded as scalar sources, transported into the bulk along the flow, and decoded after numerical integration. Our paradigm combines a Fourier-space encoding scheme for the data as AdS sources, a normalised radial phase space in which to stage the flow-matching dynamics, and a Klein--Gordon backbone to guide the flow. On a two-dimensional checkerboard benchmark, our experiments show that most of the benefit of GenAdS comes from the Fourier representation and convolutional architecture. However, when we remove momentum-channel regularisation, our most physics-informed GenAdS variant rivals the strongest physics-free control on boundary violation. On MNIST, GenAdS models remain close to a convolutional baseline on fidelity while achieving significantly higher recall at comparable precision, suggesting a fidelity-coverage trade-off. Our findings establish GenAdS as a physically interpretable and experimentally controllable framework for generative modelling, with many avenues for future extension.

cs.LG↗

arXiv2026

Logarithmic derivatives of variational and singular stochastic partial differential equations

For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of Röckner, Shang and Zhang, we compute the logarithmic derivative of the law of the solution at a fixed time along a prescribed direction of the state space. The formula is intrinsic, being expressed through the Hilbert-Schmidt Malliavin derivative $Φ_r = \mathcal{D}_r X(t)$ and the covariance $γ_t = \int_0^t Φ_r Φ_r^{*} \,\mathrm{d}r$ alone, so that neither the inversion of the first variation used in finite dimensions nor uniform Malliavin-Sobolev bounds on the Tikhonov family are called upon. It is obtained from an integration-by-parts identity on an abstract Hilbert space, a Moore-Penrose construction of a covering field on Wiener space, and a trace formula for the Tikhonov limit, specialised to the equation through the representation $Φ_r = Y(t,r)\mathcal{B}(r,X(r))$ of the Malliavin derivative by the first variation; the stochastic $p$-Laplacian and the two-dimensional Navier-Stokes equation are treated in detail. Beyond the variational class, a scalar reduction gives an integration-by-parts identity for the law of a pairing $\langle u(t),φ\rangle$ which passes to the renormalised limit for the singular equations of Bruned, Chandra, Chevyrev and Hairer, and which is represented by a logarithmic derivative under second-order Malliavin smoothness and negative-moment hypotheses.

math.PR↗

arXiv2026

Score-Based Diffusion Models in Infinite Dimensions: A Malliavin Calculus Perspective

We study score-based diffusion modelling in infinite-dimensional separable Hilbert spaces through Malliavin calculus, extending the analysis of generative models beyond the finite-dimensional setting. The forward diffusion process is formulated as a linear stochastic partial differential equation (SPDE) driven by space--time coloured noise with a trace-class covariance operator, ensuring well-posedness in arbitrary spatial dimensions. Building on Malliavin calculus and an infinite-dimensional extension of the Bismut--Elworthy--Li formula, we derive a closed-form expression for the logarithmic derivative of the transition measure along Cameron--Martin directions, which serves as the natural infinite-dimensional analogue of the score function. Our operator-theoretic approach preserves the intrinsic geometry of Hilbert spaces and accommodates general trace-class operators, thereby incorporating spatially correlated noise without assuming semigroup invertibility. We validate the derived score formula numerically for several classes of linear SPDEs in both one and two spatial dimensions using spectral methods.

math.PR↗

arXiv2026

Spinverse: Differentiable Physics for Permeability-Aware Microstructure Reconstruction from Diffusion MRI

Diffusion MRI (dMRI) is sensitive to microstructural barriers, yet most existing methods either assume impermeable boundaries or estimate voxel-level parameters without recovering explicit interfaces. We present Spinverse, a permeability-aware reconstruction method that inverts dMRI measurements through a fully differentiable Bloch-Torrey simulator. Spinverse represents tissue on a fixed tetrahedral grid and treats each interior face permeability as a learnable parameter; low-permeability faces act as diffusion barriers, so microstructural boundaries whose topology is not fixed a priori (up to the resolution of the ambient mesh) emerge without changing mesh connectivity or vertex positions. Given a target signal, we optimize face permeabilities by backpropagating a signal-matching loss through the PDE forward model, and recover an interface by thresholding the learned permeability field. To mitigate the ill-posedness of permeability inversion, we use mesh-based geometric priors; to avoid local minima, we use a staged multi-sequence optimization curriculum. Across a collection of synthetic voxel meshes, Spinverse reconstructs diverse geometries and demonstrates that sequence scheduling and regularization are critical to avoid outline-only solutions while improving both boundary accuracy and structural validity.

cs.CV↗

arXiv2025

Malliavin Calculus for Score-based Diffusion Models

We introduce a new framework based on Malliavin calculus to derive exact analytical expressions for the score function $\nabla \log p_t(x)$, i.e., the gradient of the log-density associated with the solution to stochastic differential equations (SDEs). Our approach combines classical integration-by-parts techniques with modern stochastic analysis tools, such as Bismut's formula and Malliavin calculus, and it works for both linear and nonlinear SDEs. In doing so, we establish a rigorous connection between the Malliavin derivative, its adjoint, the Malliavin divergence (Skorokhod integral), and diffusion generative models, thereby providing a systematic method for computing $\nabla \log p_t(x)$. In the linear case, we present a detailed analysis showing that our formula coincides with the analytical score function derived from the solution of the Fokker--Planck equation. For nonlinear SDEs with state-independent diffusion coefficients, we derive a closed-form expression for $\nabla \log p_t(x)$. We evaluate the proposed framework across multiple generative tasks and find that its performance is comparable to state-of-the-art methods. These results can be generalised to broader classes of SDEs, paving the way for new score-based diffusion generative models.

cs.LG↗

arXiv2025

A Malliavin calculus approach to score functions in diffusion generative models

Score-based diffusion generative models have recently emerged as a powerful tool for modelling complex data distributions. These models aim at learning the score function, which defines a map from a known probability distribution to the target data distribution via deterministic or stochastic differential equations (SDEs). The score function is typically estimated from data using a variety of approximation techniques, such as denoising or sliced score matching, Hyvärien's method, or Schrödinger bridges. In this paper, we derive an exact, closed-form, expression for the score function for a broad class of nonlinear diffusion generative models. Our approach combines modern stochastic analysis tools such as Malliavin derivatives and their adjoint operators (Skorokhod integrals or Malliavin Divergence) with a new Bismut-type formula. The resulting expression for the score function can be written entirely in terms of the first and second variation processes, with all Malliavin derivatives systematically eliminated, thereby enhancing its practical applicability. The theoretical framework presented in this work offers a principled foundation for advancing score estimation methods in generative modelling, enabling the design of new sampling algorithms for complex probability distributions. Our results can be extended to broader classes of stochastic differential equations, opening new directions for the development of score-based diffusion generative models.

stat.ML↗

arXiv2025

ReMiDi: Reconstruction of Microstructure Using a Differentiable Diffusion MRI Simulator

We propose ReMiDi, a novel method for inferring neuronal microstructure as arbitrary 3D meshes using a differentiable diffusion Magnetic Resonance Imaging (dMRI) simulator. We first implemented in PyTorch a differentiable dMRI simulator that simulates the forward diffusion process using a finite-element method on an input 3D microstructure mesh. To achieve significantly faster simulations, we solve the differential equation semi-analytically using a matrix formalism approach. Given a reference dMRI signal $S_{ref}$, we use the differentiable simulator to iteratively update the input mesh such that it matches $S_{ref}$ using gradient-based learning. Since directly optimizing the 3D coordinates of the vertices is challenging, particularly due to ill-posedness of the inverse problem, we instead optimize a lower-dimensional latent space representation of the mesh. The mesh is first encoded into spectral coefficients, which are further encoded into a latent $\textbf{z}$ using an auto-encoder, and are then decoded back into the true mesh. We present an end-to-end differentiable pipeline that simulates signals that can be tuned to match a reference signal by iteratively updating the latent representation $\textbf{z}$. We demonstrate the ability to reconstruct microstructures of arbitrary shapes represented by finite-element meshes, with a focus on axonal geometries found in the brain white matter, including bending, fanning and beading fibers. Our source code is available online.

eess.IV↗