Gero Friesecke (Technical University of Munich)

High-dimensional optimal transport problems in the physical sciences

I present important examples of high-dimensional optimal transport problems arising in electronic structure, fluid dynamics, and molecular dynamics. In some cases the curse of dimensionality can be overcome via rigorous sparsity results as well as an ensuing efficient algorithm [F, Schulz, Voegler, SIAM JSC 2022; F, Penka, SIMODS 2023] which combines ideas from discrete mathematics and generative adversarial learning. In other cases the principal challenges remain currently unsolved. But interesting proxies which approximately solve the transport condition in high dimension without optimizing any cost, such as the Boltzmann generator method by Noë et al (developed partially within CRC 1114) and score matching, provide future perspectives.