The scientific discipline of fluid dynamics is primarily concerned with the measurement, modelling and underlying physics and mathematics of how liquids and gases behave. Almost all natural and manufactured systems involve the flow of fluids, which are often complex. Consequently, an understanding of fluid dynamics is integral to addressing major societal challenges including industrial competitiveness, environmental resilience, the transition to net-zero and improvements to health and healthcare. Fluid dynamics is essential to the transition of the energy sector to a low-carbon future (for example, fluid dynamics simulations coupled with control algorithms can significantly increase wind farm efficiency). It is vital to our understanding and mitigation of climate change, including extreme weather events (for example in designing flood mitigation schemes). It is key to the digitisation of manufacturing through 3d printing/additive manufacturing and development of new greener processing technologies. In healthcare, computational fluid dynamics in combination with MRI scanning provides individualised modelling of the cardio-vascular system enabling implants such as stents to be designed and tested on computers. Fluid dynamics also shows how to design urban environments and ventilate buildings to prevent the build-up of pollutants and the transmission of pathogens. The UK has long been a world-leader in fluid dynamics research. However, the field is now advancing rapidly in response to the demand to address more complex and interwoven problems on ever-faster timescales. Data-driven fluid dynamics is a major area where there are rapid advances, with the increasing application of data-science and machine learning techniques to fluid flow data, as well as the use of Artificial Intelligence to accelerate computational simulations. For the UK to maintain its competitive position requires an investment in training the next generation of research leaders who have experience of developing and applying these new techniques and approaches to fluids problems, along with professional and problem-solving skills to lead the successful adoption of these approaches in industry and research. The University of Leeds is distinctive through the breadth, depth and unified structure of its fluid dynamics research, coordinated through the Leeds Institute for Fluid Dynamics (LIFD), making it an ideal host for this CDT. The CDT in Future Fluid Dynamics (FFD-CDT) will build on the experience of successfully running a CDT in Fluid Dynamics to address these new and exciting needs. Our students will carry out cutting-edge research developing new fluid dynamics approaches and applying them across a diverse range of engineering, physics, computing, environmental and physiological challenges. We will recruit and train cohorts of students with diverse backgrounds, covering engineering, mathematical, physical and environmental sciences, in both the fundamental principles of fluid dynamics and new data-driven methodologies. Alongside this technical training we will provide a team-based, problem-led programme of professional skills training co-developed with industry to equip our graduates with the leadership, team-working and entrepreneurial skills that they need to succeed in their future careers. We will build an inclusive, diverse and welcoming community that supports cross-disciplinary science and effective and productive collaborations and partnerships. Our CDT cohort will be at the heart of growing this capability, integrated with and within the Leeds Institute for Fluid Dynamics to deliver a dynamic and vibrant training and research environment with strong UK and international partnerships in academia, industry, policy and outreach.

The achievements of modern research and their rapid progress from theory to application are increasingly underpinned by computation. Computational approaches are often hailed as a new third pillar of science - in addition to empirical and theoretical work. While its breadth makes computation almost as ubiquitous as mathematics as a key tool in science and engineering, it is a much younger discipline and stands to benefit enormously from building increased capacity and increased efforts towards integration, standardization, and professionalism. The development of new ideas and techniques in computing is extremely rapid, the progress enabled by these breakthroughs is enormous, and their impact on society is substantial: modern technologies ranging from the Airbus 380, MRI scans and smartphone CPUs could not have been developed without computer simulation; progress on major scientific questions from climate change to astronomy are driven by the results from computational models; major investment decisions are underwritten by computational modelling. Furthermore, simulation modelling is emerging as a key tool within domains experiencing a data revolution such as biomedicine and finance. This progress has been enabled through the rapid increase of computational power, and was based in the past on an increased rate at which computing instructions in the processor can be carried out. However, this clock rate cannot be increased much further and in recent computational architectures (such as GPU, Intel Phi) additional computational power is now provided through having (of the order of) hundreds of computational cores in the same unit. This opens up potential for new order of magnitude performance improvements but requires additional specialist training in parallel programming and computational methods to be able to tap into and exploit this opportunity. Computational advances are enabled by new hardware, and innovations in algorithms, numerical methods and simulation techniques, and application of best practice in scientific computational modelling. The most effective progress and highest impact can be obtained by combining, linking and simultaneously exploiting step changes in hardware, software, methods and skills. However, good computational science training is scarce, especially at post-graduate level. The Centre for Doctoral Training in Next Generation Computational Modelling will develop 55+ graduate students to address this skills gap. Trained as future leaders in Computational Modelling, they will form the core of a community of computational modellers crossing disciplinary boundaries, constantly working to transfer the latest computational advances to related fields. By tackling cutting-edge research from fields such as Computational Engineering, Advanced Materials, Autonomous Systems and Health, whilst communicating their advances and working together with a world-leading group of academic and industrial computational modellers, the students will be perfectly equipped to drive advanced computing over the coming decades.

Probabilistic AI involves the embedding of probability models, probabilistic reasoning and measures of uncertainty within AI methods. The ProbAI hub will develop a world leading, diverse and UK-wide research programme in probabilistic AI, that will develop the next generation of mathematically-rigorous, scalable and uncertainty-aware AI algorithms. It will have far-reaching impact across many aspects of AI, including: (1) The sudden and rapid growth of AI systems has led to a new impetus for businesses, governments and creators of AI tools to understand and convey the inherent uncertainties in their systems. A probabilistic approach to AI provides a framework to represent and manipulate uncertainty about models and predictions and already plays a central role in scientific data analysis, robotics and cognitive science. The consequential impact arising from from such developments has the potential to be wide-ranging and substantial: from utilising a probabilistic approach for effective resource allocation (healthcare), prioritisation of actions (infrastructure planning), pattern recognition (cyber security) and the development of robust strategies to mitigate risks (finance). (2) It is possible to gain important theoretical insights into AI models and algorithms through studying their, often probabilistic, limiting behaviour in different asymptotic scenarios. Such results can help with understanding why AI methods work, and how best to choose appropriate architectures - with the potential to substantially reduce the computational cost and carbon footprint of AI. (3) Recent breakthroughs in generative models are based on simulating stochastic processes. There is huge potential to both use these ideas to help develop efficient and scalable probabilistic AI methods more generally; and also to improve and extend current generative models. The latter may lead to more computationally efficient and robust methods, to generative models that use different stochastic processes and are suitable for different types of data, or to novel approaches that can give a level of certainty to the output of a generative model. (4) Models from AI are increasingly being used as emulators. For example, fitting a deep neural network to realisations of a complex computer model for the weather, can lead to more efficient approaches to forecasting the weather. However, in most applications for such methods to be used reliably requires that the emulators report a measure of uncertainty -- so the user can know when the output can be trusted. Also, building on recent generalisations of Bayes updates gives new approaches to incorporate known physical constraints and other structure into these neural network emulators, leading to more robust methods that generalise better outside the training sampler and that have fewer parameters and are easier to fit. Developing these new, practical, general-purpose probabilistic AI methods requires overcoming substantial challenges, and at their heart many of these challenges are mathematical. The hub will unify a fragmented community with interests in Probabilistic AI and bring together UK researchers across the breadth of Applied Mathematics, Computer Science, Probability and Statistics. The hub will promote the area of probabilistic AI widely, encouraging and facilitating cross-disciplinary mathematics research in AI, and has substantial flexibility to fund the involvement of researchers from across the breadth of the UK during its lifetime. ProbAI will draw on and benefit from the well-established world-leading strength in areas relevant to probabilistic AI across different areas of Mathematics and Computer Science, with the aim of making the UK the world-leader in probabilistic AI.

Lancaster University (LU) proposes a Centre for Doctoral Training (CDT) to develop international research leaders in statistics and operational research (STOR) through a programme in which cutting-edge industrial challenge is the catalyst for methodological advance. Our proposal addresses the priority area 'Statistics for the 21st Century' through research training in cutting-edge modelling and inference for large, complex and novel data structures. It crucially recognises that many contemporary challenges in statistics, including those arising from industry, also engage with constraint, optimisation and decision. The proposal brings together LU's academic strength in STOR (>50FTE) with a distinguished array of highly committed industrial and international academic partners. Our shared vision is a CDT that produces graduates capable of the highest quality research with impact and equipped with an array of leadership and other skills needed for rapid career progression in academia or industry. The proposal builds on the strengths of an existing EPSRC-funded CDT that has helped change the culture in doctoral training in STOR through an unprecedented level of engagement with industry. The proposal takes the scale and scientific ambition of the Centre to a new level by: * Recruiting and training 70 students, across 5 cohorts, within a programme drawing on industrial challenge as the catalyst for research of the highest quality; * Ensuring all students undertake research in partnership with industry: 80% will work on doctoral projects jointly supervised and co-funded by industry; all others will undertake industrial research internships; * Promoting a culture of reproducible research under the mentorship and guidance of a dedicated Research Software Engineer (industry funded); * Developing cross-cohort research-clusters to support collaboration on ambitious challenges related to major research programmes; * Enabling students to participate in flagship research activities at LU and our international academic partners. The substantial growth in data-driven business and industrial decision-making in recent years has signalled a step change in the demand for doctoral-level STOR expertise and has opened the skills gap further. The current CDT has shown that a cohort-based, industrially engaged programme attracts a diverse range of the very ablest mathematically trained students. Without STOR-i, many of these students would not have considered doctoral study in STOR. We believe that the new CDT will continue to play a pivotal role in meeting the skills gap. Our training programme is designed to do more than solve a numbers problem. There is an issue of quality as much as there is one of quantity. Our goal is to develop research leaders who can innovate responsibly and secure impact for their work across academic, scientific and industrial boundaries; who can work alongside others with different skills-sets and communicate effectively. An integral component of this is our championing of ED&I. Our external partners are strongly motivated to join us in achieving these outcomes through STOR-i's cohort-based programme. We have little doubt that our graduates will be in great demand across a wide range of sectors, both industrial and academic. Industry will play a key role in the CDT. Our partners are helping to co-design the programme and will (i) co-fund and co-supervise doctoral projects, (ii) lead a programme of industrial problem-solving days and (iii) play a major role in leadership development and a range of bespoke training. The CDT benefits from the substantial support of 10 new partners (including Morgan Stanley, ONS Data Science Campus, Rolls Royce, Royal Mail, Tesco) and continued support from 5 existing partners (including ATASS, BT, NAG, Shell), with many others expected to contribute.

Understanding and characterising the behaviour of fluids is fundamental to numerous industrial and environmental challenges with wide-ranging societal impact. The CDT in Fluid Dynamics at Leeds will provide the next generation of highly trained graduates with the technical and professional skills and knowledge needed to tackle such problems. Fluid processes are critical to both economic productivity and the health and environmental systems that affect our daily lives. For example, at the microscale, the flow of liquid through the nozzle of an ink-jet printer controls the quality of the printed product, whilst the flow of a coolant around a microprocessor determines whether or not the components will overheat. At the large scale, the atmospheric conditions of the Earth depend upon the flow of gases in the atmosphere and their interaction with the land and oceans. Understanding these processes allows short term weather forecasting and long term climate prediction; both are crucial for industry, government and society to plan and adapt their environments. Fluid flows, and their interactions with structures, are also important to the performance of an array of processes and products that we take for granted in our everyday lives: gas and water flow to our homes, generation of electricity, fuel efficiency of vehicles, the comfort of our workplaces, the diagnosis and treatment of diseases, and the manufacture of most of the goods that we buy. Understanding, predicting and controlling Fluid Dynamics is key to reducing costs, increasing performance and enhancing the reliability of all of these processes and products. Our CDT draws on the substantial breadth and depth of our Fluid Dynamics research expertise at the University of Leeds. We will deliver an integrated MSc/PhD programme in collaboration with external partners spanning multiple sectors, including energy, transport, environment, manufacturing, consultancy, defence, computing and healthcare, who highlight their need for skilled Fluid Dynamicists. Through a combination of taught courses, team projects, professional skills training, external engagement and an in-depth PhD research project we will develop broad and deep technical expertise plus the team-working and problem-solving skills to tackle challenges in a trans-disciplinary manner. We will recruit and mentor a diverse cohort from a range of science and engineering backgrounds and provide a vibrant and cohesive training environment to facilitate peer-to-peer support. We will build strengths in mathematical modelling, computational simulation and experimental measurement, and through multi-disciplinary projects co-supervised by academics from different Schools, we will enable students to undertake a PhD project that both strengthens and moves them beyond their UG discipline. Our students will be outward facing with opportunities to undertake placements with industry partners or research organisations overseas, to participate in summer schools and study challenges and to lead outreach activities, becoming ambassadors for Fluid Dynamics. Industry and external engagement will be at the heart of the CDT: all MSc team projects will be challenges set and mentored by industry (with placements embedded); each student will have the opportunity for user engagement in their PhD project (from sponsorship, external supervision and access to facilities, to mentoring); and our partners will be actively involved in overseeing our strategic direction, management and professional training. Many components will be provided by or with our partners, including research software engineering, responsible innovation, commercial awareness and leadership.