DynDat – Dynamic data inversion, learning and control

Agile Team “Dynamic data inversion, learning and control”

Applied Mathematics is experiencing an unprecedented transformation in the last few years, pushed and motivated by the necessity, challenge and opportunity of integrating model based and data driven approaches. This integrative process of combining and merging model based and data driven approaches is more advanced and mature in some areas than in others, but the convenience and necessity of evolving in this direction is universally accepted in all scientific disciplines, industry, and technology sectors.

The above applies in most relevant areas of applications of mathematics, even in the most classical ones, like aeronautic optimal shape design, steel engineering processes, new composite materials, power grids and gas networks, etc. On the other hand, the need of developing and using the tools of Data Science and Machine Learning such as Reinforcement Learning (RL) and Federated Learning (FL) is even more obvious in other areas, such as the recommendation systems for internet services in which privacy is an added challenge that requires new paradigms.

One specific real-world example where the transition to data driven approaches currently ongoing is medical imaging, in particular the inverse problem of image generation (reconstruction from the acquired measurement data). More precisely, recently it has been proved that machine-learning-based image reconstruction for accelerated data acquisition leads to improvements in image quality over classic approaches like parallel imaging and compressed sensing. So far, most of the data driven developments were supervised approaches, where it is assumed that a ground truth is available for training. Unfortunately, this condition is violated in many real-world situations either because no training data sets are available, or because the inherent properties of the application make it impossible to obtain a ground truth. The latter is true for dynamic imaging problems or when motion occurs during data acquisition. Therefore, there is an unmet need to develop unsupervised machine learning methods for Magnetic Resonance (MR) reconstruction.

The possibilities of mutual fertilization and growth are huge. We expect to develop new mathematical methods, for instance in the context of FL and RL, that might have an impact in Imaging. But for achieving this goal research has to be conducted in a multidisciplinary atmosphere. The close contact of the mathematics young researchers with the true problems and challenges of Imaging in Medicine, together with the access to reliable and specialized data sets, will constitute a great opportunity to test the validity of the mathematical methods developed and to orient the mathematical efforts to most relevant issues. The combination of the various Machine Learning paradigms, including FL and RL mentioned above, strongly inspired in the theory of control of dynamical systems, and the traditional applied mathematics methods in imaging can substantially contribute to this endeavour. This constitutes the essence of the work plan of the AT-DynDat.

The main goal of the AT-DynDat is to generate an interdepartmental environment and context for joint research, aiming to foster and offer new scientific opportunities, in particular for young researchers. This will enable the team to address the main mathematical challenges in Machine Learning, strongly inspired in applications, and contribute to strengthen the positioning, leadership, and visibility of the newly launched Department of Data Science (DDS) of FAU in this competitive and vivid area.

The agenda of the AT in the initial phase focusses on two main grand topics/goals: Imaging and the dynamic aspects of Federated and Reinforced Learning. Of course, these two topics are of independent interest, but they merge in the most advanced applications, as we shall leading, leading to a new paradigm that can then be extrapolated to many other areas of application.

In imaging, our initial objectives are

• Develop an unsupervised machine learning image reconstruction methodology, based on recent developments in the literature.
• Use the existing fastMRI dataset (that F. Knoll makes accessible to the AT-DynDat members) for development and as a benchmark for the evaluation of the results.
• Compare the developed unsupervised approach to both a supervised machine learning approach like the variational network, as well as classic compressed sensing approaches based on L1-Wavelet sparsifying transforms or Total Variation.
• Improved learning of variational regularization methods for MR, which allow for detailed mathematical analysis and benefit from all techniques for variational methods.
• Develop uncertainty quantification for the arising reconstruction methods.
• Extend the methods for dynamic MRI.

Related to Federated and Reinforcement Learning, our initial objectives are

• To develop a transparent mathematical formulation of FL that could allow for the application of the extensive theory on operator splitting. This could aid the understanding of convergence properties of different training algorithms. A complete theory and a catalogue of numerical experiments regarding the possible efficiency of these algorithms to different ML models would be of interest.
• To explore the potential use of Model Predictive Control (MPC) methodologies, to enhance the efficiency of the existing algorithms and methods, based on our previous experience on combining splitting and MPC for large dimensional complex dynamic control problems.
• On the other hand, Reinforcement Learning (RL) is concerned with how intelligent agents design their actions in an environment to optimize a reward. RL differs from supervised learning because it does not require labelled input/output pairs. The strong links of RL with classical control theory are well known and lead to several challenging problems. In particular the adaptation of these methodologies to the PDE settings arising in Continuum Mechanics and Imaging. Our goal in this setting is to combine FL and RL strategies to complement and improve the unsupervised methodologies for dynamic MRI.
• One of the present limitations of current unsupervised deep learning approaches is that they come as black box methods without tools for error analysis and control. The AT-DynDat has the ambition to develop tools to overcome these limitations.

Although initially, the AT-DynDat focuses on imaging applications, as we mentioned above, the possible fields of applications are rather diverse, and will easily grow in cooperation with the industrial and technology partners of FAU.

Research groups

The team led by E. Zuazua in the DDS is one of the partners of the AT-DynDat and offers its experience and activity agenda in the rapidly growing interface of mathematical control theory and machine learning. The team is at present funded by an Alexander von Humboldt Professorship, resources that will contribute to the development of the AT.

The same can be said about the team in the Department of Mathematics (DM) led by Martin Burger, with a long tradition and internationally recognised expertise in areas such as Imaging, Inverse Problems, Optimal Transport and Data Science. His involvement in the AT, other than enriching its scientific potential, serves to bridge the DDS and the DM, which was one of the initial goals of DDS: widening the spectrum of mathematics developed within FAU, without losing the unity in the broad area of Mathematical Sciences. The team led by M. Burger is very actively funded through several programs and projects, in particular funded by the EU and the DFG and attracts a large number of young talented and dynamics researchers.

The team lead by F. Knoll, who recently joined the AIBE, is one of the leading international ones in the mathematics of medical Imaging. The team is at present funded by grants from the National Institute of Health (NIH), resources that will contribute to the development of the AT. His involvement in the AT established a second, very valuable bridge between DDS and AIBE.

The AT-DynDat team is complemented by G. Leugering, FAU Senior Fellow in Applied Mathematics, who will contribute to the development and success of the AT providing his long-term strategic vision of Science and advising the young fellows.

The triangle DDS-DM-AIBE at the foundations of this AT constitutes a strategic playground for the Mathematical Sciences and the HighTech Agenda within FAU. M. Burger, G. Leugering and E. Zuazua are also involved in the launching of the FAU Research Center for Mathematics of data (FAU MoD). The synergies between the DDS and FAU MoD, materialized in this AT, are also promising in all aspects of academic life. In particular, the link with FAU MoD will facilitate to extend the areas of interest of the of AT-DynDat to other FAU MoD researchers and fields, related to material sciences, brain science, immunology, linguistics, biomechanics, etc. This would be an extremely natural path for the evolution of the AT-DynDat since, as we have mentioned above, the mathematical and machine learning techniques we aim to develop are of potential application in all disciplines and technology sectors. The cooperation of MoD could also be reflected on the co-funding of some of the activities of AT-DynDat.