We introduce a new framework that employs Malliavin calculus to derive explicit expressions for the score function -- i.e., the gradient of the log-density -- associated with solutions to stochastic differential equations (SDEs). Our approach integrates classical integration-by-parts techniques with modern tools, such as Bismut's formula and Malliavin calculus, to address linear and nonlinear SDEs. In doing so, we establish a rigorous connection between the Malliavin derivative, its adjoint (the Malliavin divergence or the Skorokhod integral), Bismut's formula, and diffusion generative models, thus providing a systematic method for computing $\nabla \log p_t(x)$. For the linear case, we present a detailed study proving that our formula is equivalent to the actual score function derived from the solution of the Fokker--Planck equation for linear SDEs. Additionally, we derive a closed-form expression for $\nabla \log p_t(x)$ for nonlinear SDEs with state-independent diffusion coefficients. These advancements provide fresh theoretical insights into the smoothness and structure of probability densities and practical implications for score-based generative modelling, including the design and analysis of new diffusion models. Moreover, our findings promote the adoption of the robust Malliavin calculus framework in machine learning research. These results directly apply to various pure and applied mathematics fields, such as generative modelling, the study of SDEs driven by fractional Brownian motion, and the Fokker--Planck equations associated with nonlinear SDEs.

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 will be made available online.