Applications of Neural Networks in Theory and Simulation of Slow Dynamics

Kurzfassung

This thesis applies neural network methods to two types of problems in statistical and computational physics. The first concerns the construction of neural network potentials (NNPs) for particle-based simulations. The second involves the development of a neural network inverse Laplace transform (NNLT) that enables the calculation of memory kernels for slow dynamics in the context of generalized Langevin equations and, in particular, the mode-coupling theory of the glass transition (MCT).

In the first part, NNPs are evaluated on model systems of increasing complexity, including glass-forming binary mixtures. For Lennard-Jones systems and Kob–Andersen mixtures, trained NNPs reproduce structural and dynamical observables with high accuracy when the relevant regions of phase space are sampled. The Voronoi potential, based on a spatial tessellation, is then used as a demanding test case for applying NNPs to simulations of glassy dynamics. Different architectures are compared, with graph-based models outperforming simpler approaches and delivering accurate energies and forces. These models also enable stable simulations even in regimes with slow relaxation. Extensions to binary Voronoi mixtures reveal limitations as the system approaches dynamical arrest. The resulting training strategies transfer directly to elemental boron, where NNPs are trained on density-functional-theory data.

The second part introduces the NNLT, which is designed to address the ill-conditioned inverse Laplace transform of correlation functions that arises in MCT. Trained on superpositions of decaying exponentials, the NNLT generalizes to Laplace-domain correlation functions from both MCT and Brownian dynamics. Combined with the Laplace-domain Mori–Zwanzig equation, it yields accurate time-domain memory kernels. An iterative normalization scheme, refined using hydrodynamic scaling arguments, stabilizes the inversion and enables the calculation of memory kernels directly from simulation data. Comparisons of simulations with MCT reveal systematic deviations near dynamical arrest. Using the computed kernels, neural network functionals are constructed that approximate the memory kernel of the MCT-type generalized Langevin equation.

Overall, the results demonstrate that neural networks enhance theoretical and computational studies of many-body systems. Neural network potentials provide flexible models of potential energy surfaces, while the NNLT provides access to dynamical quantities that are otherwise numerically inaccessible. Together, these methods show how machine learning extends and complements established approaches in the study of complex fluids and glassy dynamics.