Nektar++ is an open-source software framework designed to support the development of high-performance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains.Program summaryProgram title: Nektar++Catalogue identifier: AEVV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: MITNo. of lines in distributed program, including test data, etc.: 1052456No. of bytes in distributed program, including test data, etc.: 42851367Distribution format: tar.gzProgramming language: C++.Computer: Any PC workstation or cluster.Operating system: Linux/UNIX, OS X, Microsoft Windows.RAM: 512 MBClassification: 12.External routines: Boost, FFTW, MPI, BLAS, LAPACK and METIS (www.cs.umn.edu)Nature of problem:The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations.Solution method:Spectral/hp element methodRunning time:The tests provided take a few minutes to run. Runtime in general depends on mesh size and total integration time

Measurements of cardiac conduction velocity provide valuable functional and structural insight into the initiation and perpetuation of cardiac arrhythmias, in both a clinical and laboratory context. The interpretation of activation wavefronts and their propagation can identify mechanistic properties of a broad range of electrophysiological pathologies. However, the sparsity, distribution and uncertainty of recorded data make accurate conduction velocity calculation difficult. A wide range of mathematical approaches have been proposed for addressing this challenge, often targeted towards specific data modalities, species or recording environments. Many of these algorithms require identification of activation times from electrogram recordings which themselves may have complex morphology or low signal-to-noise ratio. This paper surveys algorithms designed for identifying local activation times and computing conduction direction and speed. Their suitability for use in different recording contexts and applications is assessed.

The simulation of fluid dynamics, typically by numerically solving partial differential equations, is an essential tool in many areas of science and engineering. However, the high computational cost can limit application in practice and may prohibit exploring large parameter spaces. Recent deep-learning approaches have demonstrated the potential to yield surrogate models for the simulation of fluid dynamics. While such models exhibit lower accuracy in comparison, their low runtime makes them appealing for design-space exploration. We introduce two novel graph neural network (GNN) models, multi-scale (MuS)-GNN and rotation-equivariant (RE) MuS-GNN, for extrapolating the time evolution of the fluid flow. In both models, previous states are processed through multiple coarsening of the graph, which enables faster information propagation through the network and improves the capture and forecast of the system state, particularly in problems encompassing phenomena spanning a range of length scales. Additionally, REMuS-GNN is architecturally equivariant to rotations, which allows the network to learn the underlying physics more efficiently, leading to improved accuracy and generalization. We analyze these models using two canonical fluid models: advection and incompressible fluid dynamics. Our results show that the proposed GNN models can generalize from uniform advection fields to high-gradient fields on complex domains. The multi-scale graph architecture allows for inference of incompressible Navier–Stokes solutions, within a range of Reynolds numbers and design parameters, more effectively than a baseline single-scale GNN. Simulations obtained with MuS-GNN and REMuS-GNN are between two and four orders of magnitude faster than the numerical solutions on which they were trained.

Over the last decade, deep learning (DL), a branch of machine learning, has experienced rapid progress. Powerful tools for tasks that have been traditionally complex to automate have been developed, such as image synthesis and natural language processing. In the context of simulating fluid dynamics, this has led to a series of novel DL methods for replacing or augmenting conventional numerical solvers. We broadly classify these methods into physics- and data-driven methods. Physics-driven methods, generally, tune a DL model to provide an analytical and differentiable solution to a given fluid dynamics problem by minimizing the residuals of the governing partial differential equations. Data-driven methods provide a fast and approximate solution to any fluid dynamics problem that shares some physical properties with the observations used when tuning the DL model’s parameters. Meanwhile, the symbiosis of numerical solvers and DL has led to promising results in turbulence modelling and accelerating iterative solvers. However, these methods present some challenges. Exclusively data-driven flow simulators often suffer from poor extrapolation, error accumulation in time-dependent simulations, as well as difficulties in training against turbulent flows. Substantial effort is, therefore, being invested into approaches that may improve the current state of the art.