Geometric deep learning for local growth prediction on abdominal aortic aneurysm surfaces

Abdominal aortic aneurysms (AAAs) are progressive focal dilatations of the abdominal aorta. AAAs may rupture, with a survival rate of only 20\%. Current clinical guidelines recommend elective surgical repair when the maximum AAA diameter exceeds 55 mm in men or 50 mm in women. Patients that do not meet these criteria are periodically monitored, with surveillance intervals based on the maximum AAA diameter. However, this diameter does not take into account the complex relation between the 3D AAA shape and its growth, making standardized intervals potentially unfit. Personalized AAA growth predictions could improve monitoring strategies. We propose to use an SE(3)-symmetric transformer model to predict AAA growth directly on the vascular model surface enriched with local, multi-physical features. In contrast to other works which have parameterized the AAA shape, this representation preserves the vascular surface's anatomical structure and geometric fidelity. We train our model using a longitudinal dataset of 113 computed tomography angiography (CTA) scans of 24 AAA patients at irregularly sampled intervals. After training, our model predicts AAA growth to the next scan moment with a median diameter error of 1.18 mm. We further demonstrate our model's utility to identify whether a patient will become eligible for elective repair within two years (acc = 0.93). Finally, we evaluate our model's generalization on an external validation set consisting of 25 CTAs from 7 AAA patients from a different hospital. Our results show that local directional AAA growth prediction from the vascular surface is feasible and may contribute to personalized surveillance strategies.

Joint Manifold Learning and Optimal Transport for Dynamic Imaging

Dynamic imaging is critical for understanding and visualizing dynamic biological processes in medicine and cell biology. These applications often encounter the challenge of a limited amount of time series data and time points, which hinders learning meaningful patterns. Regularization methods provide valuable prior knowledge to address this challenge, enabling the extraction of relevant information despite the scarcity of time-series data and time points. In particular, low-dimensionality assumptions on the image manifold address sample scarcity, while time progression models, such as optimal transport (OT), provide priors on image development to mitigate the lack of time points. Existing approaches using low-dimensionality assumptions disregard a temporal prior but leverage information from multiple time series. OT-prior methods, however, incorporate the temporal prior but regularize only individual time series, ignoring information from other time series of the same image modality. In this work, we investigate the effect of integrating a low-dimensionality assumption of the underlying image manifold with an OT regularizer for time-evolving images. In particular, we propose a latent model representation of the underlying image manifold and promote consistency between this representation, the time series data, and the OT prior on the time-evolving images. We discuss the advantages of enriching OT interpolations with latent models and integrating OT priors into latent models.

Data-Agnostic Augmentations for Unknown Variations: Out-of-Distribution Generalisation in MRI Segmentation

Medical image segmentation models are often trained on curated datasets, leading to performance degradation when deployed in real-world clinical settings due to mismatches between training and test distributions. While data augmentation techniques are widely used to address these challenges, traditional visually consistent augmentation strategies lack the robustness needed for diverse real-world scenarios. In this work, we systematically evaluate alternative augmentation strategies, focusing on MixUp and Auxiliary Fourier Augmentation. These methods mitigate the effects of multiple variations without explicitly targeting specific sources of distribution shifts. We demonstrate how these techniques significantly improve out-of-distribution generalization and robustness to imaging variations across a wide range of transformations in cardiac cine MRI and prostate MRI segmentation. We quantitatively find that these augmentation methods enhance learned feature representations by promoting separability and compactness. Additionally, we highlight how their integration into nnU-Net training pipelines provides an easy-to-implement, effective solution for enhancing the reliability of medical segmentation models in real-world applications.

Active Learning for Deep Learning-Based Hemodynamic Parameter Estimation

Hemodynamic parameters such as pressure and wall shear stress play an important role in diagnosis, prognosis, and treatment planning in cardiovascular diseases. These parameters can be accurately computed using computational fluid dynamics (CFD), but CFD is computationally intensive. Hence, deep learning methods have been adopted as a surrogate to rapidly estimate CFD outcomes. A drawback of such data-driven models is the need for time-consuming reference CFD simulations for training. In this work, we introduce an active learning framework to reduce the number of CFD simulations required for the training of surrogate models, lowering the barriers to their deployment in new applications. We propose three distinct querying strategies to determine for which unlabeled samples CFD simulations should be obtained. These querying strategies are based on geometrical variance, ensemble uncertainty, and adherence to the physics governing fluid dynamics. We benchmark these methods on velocity field estimation in synthetic coronary artery bifurcations and find that they allow for substantial reductions in annotation cost. Notably, we find that our strategies reduce the number of samples required by up to 50% and make the trained models more robust to difficult cases. Our results show that active learning is a feasible strategy to increase the potential of deep learning-based CFD surrogates.

PDE-DKL: PDE-constrained deep kernel learning in high dimensionality

Many physics-informed machine learning methods for PDE-based problems rely on Gaussian processes (GPs) or neural networks (NNs). However, both face limitations when data are scarce and the dimensionality is high. Although GPs are known for their robust uncertainty quantification in low-dimensional settings, their computational complexity becomes prohibitive as the dimensionality increases. In contrast, while conventional NNs can accommodate high-dimensional input, they often require extensive training data and do not offer uncertainty quantification. To address these challenges, we propose a PDE-constrained Deep Kernel Learning (PDE-DKL) framework that combines DL and GPs under explicit PDE constraints. Specifically, NNs learn a low-dimensional latent representation of the high-dimensional PDE problem, reducing the complexity of the problem. GPs then perform kernel regression subject to the governing PDEs, ensuring accurate solutions and principled uncertainty quantification, even when available data are limited. This synergy unifies the strengths of both NNs and GPs, yielding high accuracy, robust uncertainty estimates, and computational efficiency for high-dimensional PDEs. Numerical experiments demonstrate that PDE-DKL achieves high accuracy with reduced data requirements. They highlight its potential as a practical, reliable, and scalable solver for complex PDE-based applications in science and engineering.

Deep Networks are Reproducing Kernel Chains

Identifying an appropriate function space for deep neural networks remains a key open question. While shallow neural networks are naturally associated with Reproducing Kernel Banach Spaces (RKBS), deep networks present unique challenges. In this work, we extend RKBS to chain RKBS (cRKBS), a new framework that composes kernels rather than functions, preserving the desirable properties of RKBS. We prove that any deep neural network function is a neural cRKBS function, and conversely, any neural cRKBS function defined on a finite dataset corresponds to a deep neural network. This approach provides a sparse solution to the empirical risk minimization problem, requiring no more than $N$ neurons per layer, where $N$ is the number of data points.

Deep vectorised operators for pulsatile hemodynamics estimation in coronary arteries from a steady-state prior

Cardiovascular hemodynamic fields provide valuable medical decision markers for coronary artery disease. Computational fluid dynamics (CFD) is the gold standard for accurate, non-invasive evaluation of these quantities in vivo. In this work, we propose a time-efficient surrogate model, powered by machine learning, for the estimation of pulsatile hemodynamics based on steady-state priors. We introduce deep vectorised operators, a modelling framework for discretisation independent learning on infinite-dimensional function spaces. The underlying neural architecture is a neural field conditioned on hemodynamic boundary conditions. Importantly, we show how relaxing the requirement of point-wise action to permutation-equivariance leads to a family of models that can be parametrised by message passing and self-attention layers. We evaluate our approach on a dataset of 74 stenotic coronary arteries extracted from coronary computed tomography angiography (CCTA) with patient-specific pulsatile CFD simulations as ground truth. We show that our model produces accurate estimates of the pulsatile velocity and pressure while being agnostic to re-sampling of the source domain (discretisation independence). This shows that deep vectorised operators are a powerful modelling tool for cardiovascular hemodynamics estimation in coronary arteries and beyond.

Sparsifying dimensionality reduction of PDE solution data with Bregman learning

Classical model reduction techniques project the governing equations onto a linear subspace of the original state space. More recent data-driven techniques use neural networks to enable nonlinear projections. Whilst those often enable stronger compression, they may have redundant parameters and lead to suboptimal latent dimensionality. To overcome these, we propose a multistep algorithm that induces sparsity in the encoder-decoder networks for effective reduction in the number of parameters and additional compression of the latent space. This algorithm starts with sparsely initialized a network and training it using linearized Bregman iterations. These iterations have been very successful in computer vision and compressed sensing tasks, but have not yet been used for reduced-order modelling. After the training, we further compress the latent space dimensionality by using a form of proper orthogonal decomposition. Last, we use a bias propagation technique to change the induced sparsity into an effective reduction of parameters. We apply this algorithm to three representative PDE models: 1D diffusion, 1D advection, and 2D reaction-diffusion. Compared to conventional training methods like Adam, the proposed method achieves similar accuracy with 30% less parameters and a significantly smaller latent space.

Global Control for Local SO-Equivariant Scale-Invariant Vessel Segmentation

Personalized 3D vascular models can aid in a range of diagnostic, prognostic, and treatment-planning tasks relevant to cardiovascular disease management. Deep learning provides a means to automatically obtain such models. Ideally, a user should have control over the exact region of interest (ROI) to be included in a vascular model, and the model should be watertight and highly accurate. To this end, we propose a combination of a global controller leveraging voxel mask segmentations to provide boundary conditions for vessels of interest to a local, iterative vessel segmentation model. We introduce the preservation of scale- and rotational symmetries in the local segmentation model, leading to generalisation to vessels of unseen sizes and orientations. Combined with the global controller, this enables flexible 3D vascular model building, without additional retraining. We demonstrate the potential of our method on a dataset containing abdominal aortic aneurysms (AAAs). Our method performs on par with a state-of-the-art segmentation model in the segmentation of AAAs, iliac arteries and renal arteries, while providing a watertight, smooth surface segmentation. Moreover, we demonstrate that by adapting the global controller, we can easily extend vessel sections in the 3D model.

Learning a Sparse Representation of Barron Functions with the Inverse Scale Space Flow

This paper presents a method for finding a sparse representation of Barron functions. Specifically, given an $L^2$ function $f$, the inverse scale space flow is used to find a sparse measure $\mu$ minimising the $L^2$ loss between the Barron function associated to the measure $\mu$ and the function $f$. The convergence properties of this method are analysed in an ideal setting and in the cases of measurement noise and sampling bias. In an ideal setting the objective decreases strictly monotone in time to a minimizer with $\mathcal{O}(1/t)$, and in the case of measurement noise or sampling bias the optimum is achieved up to a multiplicative or additive constant. This convergence is preserved on discretization of the parameter space, and the minimizers on increasingly fine discretizations converge to the optimum on the full parameter space.