IN-DEEP is a European Doctoral Network composed of nine doctoral candidates (DCs) and top scientists with complementary areas of expertise in applied mathematics, artificial intelligence (AI), high-performance computing (HPC), and engineering applications. Its main goal is to provide high-level training to the nine DCs in designing, implementing, and using explainable knowledge-driven Deep Learning (DL) algorithms for rapidly and accurately solving inverse problems governed by PDEs. IN-DEEP will focus on real-life high-risk problems arising from applications related to geophysics, smart cities, and health. By doing so, we will boost Europe’s training and research activities in new DL technologies for inverse problems. IN-DEEP will develop fundamental research at universities and research institutes that will be validated and applied to real-world use cases in technology centers and companies.
Inverse problems in which the unknown parameters are connected to experimental measurements through partial differential equations (PDEs) cover from medical applications–like imaging of the human body or cancer growth assessment–to the safety of civil infrastructures (e.g. bridges and buildings). They also include green geophysical applications like underground hydrogen and CO2 storage and geothermal energy production. Their application value is measured in terms of human lives and society's well-being, which goes beyond any quantifiable amount of money. This is why equipping a new generation of specialists with highly-demanded skills for the upcoming transition toward safe and robust AI-based technologies is imperative.
All the aforementioned applications require a real-time interpretation of the experimentally acquired data, as critical decisions must be made rapidly, depending upon the found parameters. Until recently, inversion methods were based on repeatedly solving forward problems governed by PDEs, and, therefore, were computationally expensive. Now, AI-based approaches are the trend in numerical methods for inverse problems governed by PDEs. In particular, DL algorithms that incorporate the governing physics laws in the learning process arose recently as a promising alternative to traditional methods, as they execute very fast on GPUs, can approximate complex functions, and solve some nonlinear and high-dimensional problems that cannot be tackled with traditional numerical methods. Moreover, DL methods can deal with the difficulty entailed by the non-uniqueness of the inverse solution through various regularization techniques and the use of bayesian-based uncertainty quantification methods.
Despite these advantages and the promising results obtained for multiple problems, DL for PDEs has severe limitations. Among them, the most troublesome is its lack of a solid theoretical background and explainability, which prevents potential users from integrating them into high-risk applications and hinder the further development of robust methods.
IN-DEEP aims to remove these constraints to unleash the full potential of DL algorithms for PDEs, by: (a) focusing on emerging applications of DL for PDEs with immense societal and/or industrial value, (b) designing explainable advanced solvers to address them efficiently, and (c) involving, from the beginning, industrial and technological agents which can monitor, upscale, and exploit this knowledge.
The research and training agenda of IN-DEEP is driven by nine talented DCs distributed across the European consortium: Shima Baharlouei (UPV/EHU), Ivan Bioli (UniPV), Mikel Mendibe (Tecnalia), Meike Tütken (Siemens), Nicolás Zamorano (POLITO), Elias Al Ghazal (ENSAM), Elías Carú (BCAM), Askold Vilkha (AGH), and Emin Benny Chacko (UoN). Working at the intersection of applied mathematics and deep learning, they combine their diverse expertise to bridge the gap between academic theory and real-life applications.
