COFNET: Compositional functions networks - adaptive learning for high-dimensional approximation and uncertainty quantification

• Applicant: Privatdozent Dr. Martin Eigel
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 490877747

High-dimensional approximation tasks are ubiquitous in all areas of scientific computing and data science, including the solution of partial differential equations (PDEs), machine learning and uncertainty quantification (UQ). With the recent success of deep Neural Networks (NNs) and tree Tensor Networks (TNs), previously intractable problems have become feasible and the demand for reliable algorithms and a better understanding of theoretical aspects is greater than ever. This success caused a stark interest in the analysis of compositions of functions and motivates the introduction of new approximation tools preserving convenient mathematical structures of TNs while achieving a higher expressivity closer to NNs.The first main objective is to propose and analyze new families of approximation tools based on compositional functions networks (coined COFNETs) that are tree-structured compositions of functions. These new tools lie in between tree TNs and NNs and will combine the best of both worlds, namely (i) a beneficial mathematical structure for reliable learning, and (ii) a performance similar to NNs for many applications, including the approximations of dynamical systems. We aim to analyze the fundamental properties of COFNETs from approximation and statistical learning perspectives. Also, we aim at developing robust and efficient algorithms for these new tools, including compression techniques and adaptive learning procedures with limited data and low computational resources.We expect the results to have a major impact on the development of network-based learning methods, in particular tree TNs (a particular case of COFNETs) but also deep NNs.The second main objective concerns challenging problems in forward and inverse UQ. The focus lies on high-dimensional random PDEs interpreted as functions of the solutions of stochastic differential equations (SDEs). These naturally exhibit a compositional structure. Hence, COFNETs are expected to achieve a similar performance as state-of-the-art methods in UQ but also allow to address new classes of problems whose solutions do not possess any regularity in a usual sense. COFNETs should also make functional approaches to SDEs become tractable and provide an efficient alternative e.g. to Monte-Carlo methods. For Bayesian inverse problems, an interacting particle interpretation allows to determine the posterior distribution from the steady state of a dynamical system. We aim to analyse the approximation by COFNETs and will develop efficient reconstruction algorithms. In an optimal transport setting, we will consider the construction of transport maps with COFNETS, alleviating the curse of dimensionality for a wide range of high-dimensional problems.Beyond applications in UQ, the theoretical and practical outcomes of the project should be applicable to a wide range of problems and open the door for a promising new research direction which hopefully becomes beneficial for several scientific fields.

• DFG Programme: Research Grants
• International Connection: France
• Co-Investigator: Privatdozent Dr. Christian Bayer
• Cooperation Partner: Professor Dr. Anthony Nouy