Learning to Solve PDEs on Neural Shape Representations

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural representations. This mismatch leaves no suitable method to solve surface PDEs directly within the neural domain, forcing explicit mesh extraction or per-instance residual training, preventing end-to-end workflows. We present a novel, mesh-free formulation that learns a local update operator conditioned on neural (local) shape attributes, enabling surface PDEs to be solved directly where the (neural) data lives. The operator integrates naturally with prevalent neural surface representations, is trained once on a single representative shape, and generalizes across shape and topology variations, enabling accurate, fast inference without explicit meshing or per-instance optimization while preserving differentiability. Across analytic benchmarks (heat equation and Poisson solve on sphere) and real neural assets across different representations, our method slightly outperforms CPM while remaining reasonably close to FEM, and, to our knowledge, delivers the first end-to-end pipeline that solves surface PDEs on both neural and classical surface representations. Code will be released on acceptance.

SCREENER: A general framework for task-specific experiment design in quantitative MRI

Quantitative magnetic resonance imaging (qMRI) is increasingly investigated for use in a variety of clinical tasks from diagnosis, through staging, to treatment monitoring. However, experiment design in qMRI, the identification of the optimal acquisition protocols, has been focused on obtaining the most precise parameter estimations, with no regard for the specific requirements of downstream tasks. Here we propose SCREENER: A general framework for task-specific experiment design in quantitative MRI. SCREENER incorporates a task-specific objective and seeks the optimal protocol with a deep-reinforcement-learning (DRL) based optimization strategy. To illustrate this framework, we employ a task of classifying the inflammation status of bone marrow using diffusion MRI data with intravoxel incoherent motion (IVIM) modelling. Results demonstrate SCREENER outperforms previous ad hoc and optimized protocols under clinical signal-to-noise ratio (SNR) conditions, achieving significant improvement, both in binary classification tasks, e.g. from 67% to 89%, and in a multi-class classification task, from 46% to 59%. Additionally, we show this improvement is robust to the SNR. Lastly, we demonstrate the advantage of DRL-based optimization strategy, enabling zero-shot discovery of near-optimal protocols for a range of SNRs not used in training. In conclusion, SCREENER has the potential to enable wider uptake of qMRI in the clinic.