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Department of Computer Science

Solving PDEs via Deep Neural Nets: Underpinning Accelerated Cardiovascular Flow Modelling with Learning Theory

Primary supervisor

Additional supervisors

  • Anirbit Mukherjee

Additional information

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Other projects with the same supervisor


  • Directly Funded Project (Students Worldwide)

This research project has funding attached. Applications for this project are welcome from suitably qualified candidates worldwide. Funding may only be available to a limited set of nationalities and you should read the full department and project details for further information.

Project description

Simulating blood flow dynamics and tissue perfusion is about solving highly coupled and nonlinear Partial Differential Equations (PDEs). This system has multiple sources of complex couplings, like (a) fluid-structure interaction must be modeled when vessel distensibility is essential, and (b) apposition modeling may be required depending on the models of interactions between blood and any implanted devices. On top of this, the biochemical nature of blood might also need to be modeled if thrombosis is relevant.

Apart from this intrinsic multiphysics nature, there are also considerable variations in the length and time scales involved in cardiovascular flow problems. These systems include short-term processes such as aneurysmal hemodynamics and systemic autoregulation, alongside long-term processes such as aneurysm recanalization and growth. Vastly different length scales are also present, with thrombosis and endothelialization occurring at the molecular/cell/tissue level. In contrast, the main human arteries range from millimeters up to a few centimeters and whose boundary conditions depend on the broader systemic circulation. These diversities lead to varying flow regimes in different regions of the vasculature and at various times of the cardiac cycle.

Nonlinear effects complicating this flow modeling also come from various sources like (a) the nonlinearity in the Navier-Stokes equation, (b) the geometric complexity of blood vessels, (c) flow features that emerge in the presence of vascular pathologies such as stenosis, atherosclerosis, aneurysms, or valve defects, etc. The applicant can look up papers like (Sarrami-Foroushani et al., 2021) and (Shone et al., 2023) to get an understanding of the state-of-the-art attempts that have been made in Prof Frangi's group at solving
the above kind of dynamics - and often by using deep learning in the recent past.

A long-standing open question in deep-learning theory is to show provable training for any net via a stochastic algorithm while making no assumptions on the network width (size of the largest layer of activation functions) and the statistical distribution of the input data. Dr. Mukherjee and his intern (Gopalani & Mukherjee, 2022) have taken a step toward resolving this by giving the first proof that SGD-like algorithms can exploit a critical amount of weight decay in the loss function to find the global minima of sigmoid or tanh-activated depth-2 nets. Such guarantees in the existing literature require unrealistic assumptions about the input data or the width of the net. Progress is made in (Gopalani & Mukherjee, 2022) by making a fascinating new connection between deep learning and the interaction of optimization theory with the idea of "Villani functions" as was recently discovered in (Shi et al., 2020).

This project has a two-fold mission. First, the student will explore PINNs and DeepONets in the context of cardiovascular flows (e.g., intra-aneurysmal flows and flows through cardiac valves). Secondly, the student will build upon recent progress in mathematics to obtain provable gradient-based training on various neural losses at finite width - particularly for the neural loss functions used in the project's first part. This project will also explore new adaptive gradient algorithms developed by Dr. Mukherjee and his students and shown to have state-of-the-art performance and rigorous theoretical guarantees for usual deep learning settings. The student in this project will look at extending these insights to cutting-edge deep-learning settings, such as solving PDEs using neural nets.

The ideal student for this Ph.D. project would be someone very conversant with coding PDE solving by neural nets, has research experience with proving generalization theorems for such setups, and is, in general, deeply passionate about learning high-dimensional probability and its evolving interface with deep learning.

- Gopalani P, Mukherjee A. Global Convergence of SGD On Two Layer Neural Nets. (under review), 2022. URL
- Sarrami-Foroushani A, Lassila T, MacRaild M, Asquith J, Roes KCB, Byrne JV, Frangi AF. In-silico trial of intracranial flow diverters replicates and expands insights from conventional clinical trials. Nature Communications, 12(1):3861, 2021.
- Shi B, Weijie JS, Jordan MI. On learning rates and Schrodinger operators. arXiv preprint arXiv:2004.06977, 2020.
- Shone F, Ravikumar N, Lassila T, MacRaild M, Wang T, Taylor ZA, Jimack P, Dall'Armellina E, Frangi AF. Deep Physics-Informed Super-Resolution of Cardiac 4D-Flow MRI. In International Conference on Information Processing in Medical Imaging, pp. 511-522. Springer, 2023.

Person specification

For information


Applicants will be required to evidence the following skills and qualifications.

  • This project requires mathematical engagement and ability substantially greater than for a typical Computer Science PhD. Give evidence for appropriate competence, as relevant to the project description.
  • You must be capable of performing at a very high level.
  • You must have a self-driven interest in uncovering and solving unknown problems and be able to work hard and creatively without constant supervision.


Applicants will be required to evidence the following skills and qualifications.

  • You will have good time management.
  • You will possess determination (which is often more important than qualifications) although you'll need a good amount of both.


Applicants will be required to address the following.

  • Comment on your transcript/predicted degree marks, outlining both strong and weak points.
  • Discuss your final year Undergraduate project work - and if appropriate your MSc project work.
  • How well does your previous study prepare you for undertaking Postgraduate Research?
  • Why do you believe you are suitable for doing Postgraduate Research?