Project Details
Projekt
Structure-preserving deep neural networks to accelerate the solution of the Boltzmann equation
Subject Area
Mathematics
Term
since 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 464075789
Our goal is to show that using deep neural networks and classical numerical methods together is possible and beneficial for solving PDEs. We will demonstrate this with the example of the Boltzmann equation, which is a challenging application problem which on the one hand has an intricate analytical structure that numerical methods should preserve, and on the other hand whose solution is still out of reach even on today’s largest supercomputers. In this second part of the project we focus in detail on substituting the collision operator by a deep neural network. More precisely, we use a DeepONet to approximate the nonlinear integral operator. The challenges are to enforce conservation, and entropy dissipation. Conservation is addressed by augmenting the basis in the DeepONet that is encoded in the trunk net. Entropy dissipation is addressed by using the recent interpretation of the Boltzmann equation as gradient flow, and by approximating the symmetric positive-definite metric tensor associated to the gradient flow. We validate our methods in realistic test cases, and provide all source code.
DFG Programme
Priority Programmes
Subproject of
SPP 2298:
Theoretical Foundations of Deep Learning
