We propose a novel class of graph neural networks based on the discretised Beltrami flow, a non-Euclidean diffusion PDE.

This paper argues that dataset structure is important in image recognition tasks (among other tasks). Specifically, we focus on the nature and genesis of correlational structure in the actual datasets upon which DNNs are trained. We argue that DNNs are implementing a widespread methodology in condensed matter physics and materials science that focuses on mesoscale correlation structures that live between fundamental atomic/molecular scales and continuum scales. Specifically, we argue that DNNs that are successful in image classification must be discovering high order correlation functions. It is well-known that DNNs successfully generalize in apparent contravention of standard statistical learning theory. We consider the implications of our discussion for this puzzle.

Numerical simulations provide key insights into many physical, real-world problems. However, while these simulations are solved on a full 3D domain, most analysis only require a reduced set of metrics (e.g. plane-level concentrations). This work presents a hybrid physics-neural model that predicts scalar transport in a complex domain orders of magnitude faster than the 3D simulation (from hours to less than 1 min). This end-to-end differentiable framework jointly learns the physical model parameterization (i.e. orthotropic diffusivity) and a non-Markovian neural closure model to capture unresolved, 'coarse-grained' effects, thereby enabling stable, long time horizon rollouts. This proposed model is data-efficient (learning with 26 training data), and can be flexibly extended to an out-of-distribution scenario (with a moving source), achieving a Spearman correlation coefficient of 0.96 at the final simulation time. Overall results show that this differentiable physics-neural framework enables fast, accurate, and generalizable coarse-grained surrogates for physical phenomena.

We propose Microscopic Propagator Imaging (MPI) as a novel method to retrieve the indices of the microscopic propagator which is the probability density function of water displacements due to diffusion within the nervous tissue microstructures. Unlike the Ensemble Average Propagator indices or the Diffusion Tensor Imaging metrics, MPI indices are independent from the mesoscopic organization of the tissue such as the presence of multiple axonal bundle directions and orientation dispersion. As a consequence, MPI indices are more specific to the volumes, sizes, and types of microstructures, like axons and cells, that are present in the tissue. Thus, changes in MPI indices can be more directly linked to alterations in the presence and integrity of microstructures themselves. The methodology behind MPI is rooted on zonal modeling of spherical harmonics, signal simulation, and machine learning regression, and is demonstrated on both synthetic and Human Diffusion MRI data.

We introduce and demonstrate a new paradigm for quantitative parameter mapping in MRI. Parameter mapping techniques, such as diffusion MRI and quantitative MRI, have the potential to robustly and repeatably measure biologically-relevant tissue maps that strongly relate to underlying microstructure. Quantitative maps are calculated by fitting a model to multiple images, e.g. with least-squares or machine learning. However, the overwhelming majority of model fitting techniques assume that each voxel is independent, ignoring any co-dependencies in the data. This makes model fitting sensitive to voxelwise measurement noise, hampering reliability and repeatability. We propose a self-supervised deep variational approach that breaks the assumption of independent pixels, leveraging redundancies in the data to effectively perform data-driven regularisation of quantitative maps. We demonstrate that our approach outperforms current model fitting techniques in dMRI simulations and real data. Especially with a Gaussian mixture prior, our model enables sharper quantitative maps, revealing finer anatomical details that are not presented in the baselines. Our approach can hence support the clinical adoption of parameter mapping methods such as dMRI and qMRI.

Transformer Neural Networks are driving an explosion of activity and discovery in the field of Large Language Models (LLMs). In contrast, there have been only a few attempts to apply Transformers in engineering physics. Aiming to offer an easy entry point to physics-centric Transformers, we introduce a physics-informed Transformer model for solving the heat conduction problem in a 2D plate with Dirichlet boundary conditions. The model is implemented in the machine learning framework MLX and leverages the unified memory of Apple M-series processors. The use of MLX means that the models can be trained and perform predictions efficiently on personal machines with only modest memory requirements. To train, validate and test the Transformer model we solve the 2D heat conduction problem using central finite differences. Each finite difference solution in these sets is initialized with four random Dirichlet boundary conditions, a uniform but random internal temperature distribution and a randomly selected thermal diffusivity. Validation is performed in-line during training to monitor against over-fitting. The excellent performance of the trained model is demonstrated by predicting the evolution of the temperature field to steady state for the unseen test set of conditions.

Diffusion-weighted imaging (DWI) is a type of Magnetic Resonance Imaging (MRI) technique sensitised to the diffusivity of water molecules, offering the capability to inspect tissue microstructures and is the only in-vivo method to reconstruct white matter fiber tracts non-invasively. The DWI signal can be analysed with the diffusion tensor imaging (DTI) model to estimate the directionality of water diffusion within voxels. Several scalar metrics, including axial diffusivity (AD), mean diffusivity (MD), radial diffusivity (RD), and fractional anisotropy (FA), can be further derived from DTI to quantitatively summarise the microstructural integrity of brain tissue. These scalar metrics have played an important role in understanding the organisation and health of brain tissue at a microscopic level in clinical studies. However, reliable DTI metrics rely on DWI acquisitions with high gradient directions, which often go beyond the commonly used clinical protocols. To enhance the utility of clinically acquired DWI and save scanning time for robust DTI analysis, this work proposes DirGeo-DTI, a deep learning-based method to estimate reliable DTI metrics even from a set of DWIs acquired with the minimum theoretical number (6) of gradient directions. DirGeo-DTI leverages directional encoding and geometric constraints to facilitate the training process. Two public DWI datasets were used for evaluation, demonstrating the effectiveness of the proposed method. Extensive experimental results show that the proposed method achieves the best performance compared to existing DTI enhancement methods and potentially reveals further clinical insights with routine clinical DWI scans.

Learning representations for structured data with certain geometries (observed or unobserved) is a fundamental challenge, wherein message passing neural networks (MPNNs) have become a de facto class of model solutions. In this paper, we propose an energy-constrained diffusion model as a principled interpretable framework for understanding the mechanism of MPNNs and navigating novel architectural designs. The model, inspired by physical systems, combines the inductive bias of diffusion on manifolds with layer-wise constraints of energy minimization. As shown by our analysis, the diffusion operators have a one-to-one correspondence with the energy functions implicitly descended by the diffusion process, and the finite-difference iteration for solving the energy-constrained diffusion system induces the propagation layers of various types of MPNNs operated on observed or latent structures. On top of these findings, we devise a new class of neural message passing models, dubbed as diffusion-inspired Transformers, whose global attention layers are induced by the principled energy-constrained diffusion. Across diverse datasets ranging from real-world networks to images and physical particles, we show that the new model can yield promising performance for cases where the data structures are observed (as a graph), partially observed or completely unobserved.