Adaptive Neural Tensor Networks for parametric PDEs

• Applicants: Dr. Martin Eigel; Professor Dr. Lars Grasedyck
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 463293876

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2298: Theoretical Foundations of Deep Learning