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

高维逼近问题普遍存在于科学计算和数据科学的各个领域，包括偏微分方程的求解、机器学习和不确定性量化。随着最近深度神经网络(NNS)和树张量网络(TNS)的成功，以前难以解决的问题变得可行，对可靠算法和对理论方面的更好理解的需求比以往任何时候都更大。这一成功引起了人们对函数组合分析的浓厚兴趣，并促使了新的近似工具的引入，这些工具保留了TNS的方便的数学结构，同时获得了更接近于NN的更高的表现力。第一个主要目标是提出和分析基于组合函数网络的新的逼近工具家族(称为COFNETs)，这些工具是函数的树形组合。这些新的工具位于树TNS和NNS之间，将结合两个领域的最好，即(I)用于可靠学习的有益的数学结构，以及(Ii)对于许多应用，包括动态系统的近似，具有类似于NNS的性能。我们的目标是从近似和统计学习的角度分析COFNETs的基本性质。此外，我们的目标是为这些新工具开发健壮和高效的算法，包括压缩技术和具有有限数据和低计算资源的自适应学习过程。我们预计结果将对基于网络的学习方法的发展产生重大影响，特别是树TNS(COFNETs的特殊情况)，但也包括深度NN。第二个主要目标涉及UQ正反向中的挑战性问题。焦点集中在高维随机偏微分方程解的函数上。这些自然地呈现出一种成分结构。因此，COFNET有望实现与UQ中最先进的方法类似的性能，但也允许解决其解决方案在通常意义上不具有任何规律性的新问题类。COFNETs还应使对SDE的功能性方法变得易于掌握，并提供一种有效的替代方法，例如蒙特卡罗方法。对于贝叶斯反问题，相互作用的粒子解释允许从动力系统的稳态确定后验分布。我们的目标是分析COFNETs的逼近，并将开发有效的重建算法。在最优的交通环境下，我们将考虑使用COFNETS构建交通地图，以缓解一系列高维问题的维度诅咒。在UQ的应用中，该项目的理论和实践成果应该适用于广泛的问题，并为一个有前途的新研究方向打开大门，有望对几个科学领域有所裨益。

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.