Adaptive Neural Tensor Networks for parametric PDEs

该项目的重点是在分层张量网络支持的适当模型不可知的深度神经网络(NN)体系结构中验证高维参数偏微分方程解的自适应表示。从现有的表现力结果中我们知道，原则上这样的方程可以用神经网络表示解，但瓶颈在于定义网络拓扑、使其适应解以及在训练过程中找到合适的参数。该项目旨在为参数偏微分方程正问题和反问题的自适应和收敛的神经网络逼近奠定理论和实践基础。基础的两个支柱是1)可靠和可计算的后验误差估计器，导致(深度)神经网络逼近的收敛算法，以及2)通过局部单层张量网络表示寻找(深度)神经网络拟合参数的可靠算术框架。我们的愿景是使深度神经网络成为这些方程的可靠和高效的计算方法，其最终性能优于当前最好的方法，并成为解决其他棘手问题的通用工具。此外，该项目有助于更好地理解和利用深层网络表示和层次张量网络之间的关系。张量网络更简单的多线性结构将有助于我们为深度神经网络的高度非线性结构指明方向。

The focus of the project lies on the certified adaptive representation of high-dimensional parametric Partial Differential Equation (PDE) solutions in an adequate model agnostic deep Neural Network (NN) architecture supported by hierarchical tensor networks. From existing expressivity results we know that in principle such equations would allow for an NN representation of the solution but the bottleneck lies in defining the network topology, adapting it to the solution and finding the fitting parameters during training. The project aims to develop the theoretical and practical foundation for adaptive and converging NN approximations of parametric PDEs, both in forward and inverse problems. The two pillar stones of the foundation are1) a reliable and computable a posteriori error estimator leading to a convergent algorithm for the (deep) NN approximation, and2) a reliable arithmetic framework for finding the fitting parameters of the (deep) NN by help of local herarchical tensor network representations.Our vision is to make deep NNs a reliable and efficient computational approach for these equtions, which eventually outperforms current best-of-class methods and becomes a versatile tool for otherwise intractable problems. Moreover, the project contributes to a better understanding and leveraging of the relation of deep network representations and hierarchical tensor networks. The simpler multilinear structure of tensor networks will help us guide the way for the highly non-linear structure of deep NNs.