Mathematical modeling has facilitated and deepened our understanding of a myriad of phenomena that underlie most technological advances. Existing models and methods have led to very fast developments in the Sciences and Engineering, and the accompanying success has led to new challenges. Current methods lack versatility and often time-efficiency to be implemented on any new manufactured device. On the other hand, with the increasing amount of available data new methods have appeared. However, they often lack quantitative accuracy or certification, crucial issues for Health-related applications or billion-dollar instrumentation in astronomy. Thus, the main goal of this Nucleus is to find innovative solutions to real-world problems creating methods from mathematical modeling, inverse problems and control theory. To do that the use of physics-based and data-based methods to keep pushing the frontiers of knowledge is a key feature of this proposal. We aim at establishing a strong, well-known interdisciplinary group in Chile to address topics in modeling, identification, control theory and machine-learning (ML) techniques. Our applied work will focus on internationally renowned fields in Chile: astronomical instrumentation, neurosciences and medical imaging, which can mutually benefit from contributions in quantitative and theoretical analyses. Our hallmark is the use of both classical and ML methods from Mathematics and Engineering.
Our scientific proposal is structured in three research axes of transverse interest for the considered applications.
MATHEMATICAL MODELING AND IDENTIFICATION METHODS
Continuous models have already demonstrated to improve accuracy compared to discrete approximations, but they are scarcely used because of their complexity. Some processes require, however, an accurate physics-based mathematical approach, and uncertainty on model parameters remains unsolved in various problems. In such problems, the development of model or parameter identification techniques are therefore critical. In this axis of research, we will develop mathematical modeling and model identification techniques for neuro-stimulation, light-sheet microscopy and adaptive optics for astronomy.
DEEP LEARNING (DL) TECHNIQUES FOR SELF-CALIBRATION AND RECONSTRUCTION
Model-based identification methods usually lead to memory- and time-demanding non-convex optimization methods. Recently, it has emerged the idea of translating the complexity and slowness of the complete inverse problem solving to the offline training of Neural Networks (NNs). We plan to use DL-based methods and big data (from both our models and experimental measurements) to learn the key parameters during an up-front training procedure, enabling the system to further calibrate itself implicitly with the newly-acquired data. We will use NNs of different types depending on the application -light-sheet microscopy, magnetic resonance imaging, adaptive optics- and the specific component of the problem being addressed.
OPTIMIZED CONTROL WITH IDENTIFIED MODELS
Controlling a process means that we search for an action that will have a specific consequence, excitation or correction. Having a simple process model may help guarantee robustness with respect to uncertainties, but degrades the achievable performance. In Adaptive Optics (AO), performance decay can have strong economic consequences for the observatories, since the systems should provide well-corrected observations in any conditions. We will investigate optimized control methods and techniques for neuro-stimulation therapies and AO in astronomy.
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