Improving deep learning techniques for sampling thermodynamic-equilibrium molecular dynamics with optimal transport

• Applicant: Professor Dr. Gero Friesecke
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 543965508

Molecular dynamics simulations play an important role in physics, chemistry, and biochemistry, but suffer from the curse of dimension. If the number of atoms or subsystems in a molecular dynamics simulation is denoted by N , then the Boltzmann distribution governing the thermodynamic-equilibrium behavior is a probability measure on the high-dimensional space R^3N. Recent work initiated in a pioneering paper by Noe et al has demonstrated the effectiveness of deep learning methods for sampling the Boltzmann distribution for large systems. More precisely, it was shown that normalizing flows, a type of neural network that approximate transport of measures, are effective at sampling from the Boltzmann distribution by transforming samples from a simpler probability measure, such as a Gaussian, into samples from the desired distribution. However, these techniques have not been mathematically justified for the actual systems of physical interest. Our main goal in this proposal is to put these normalizing flow techniques on a firm mathematical footing using methods from optimal transport which to our knowledge have not hitherto played a role in this context. A principal idea is to train normalizing flows to not just approximate some diffeomorphism pushing a simple reference measure forward to the Boltzmann distribution, but to train it to approximate the specific diffeomorphism provided by optimal transport. The latter is known to be unique and enjoys higher regularity. Thanks to recent advances in machine learning, due to its higher smoothness this map can be approximated by deep neural networks with a moderate number of parameters and layers. Moreover uniqueness of the target map and its smoothness is expected to allow more efficient training and more stable sampling.

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