4 research outputs found
GNN-Assisted Phase Space Integration with Application to Atomistics
Overcoming the time scale limitations of atomistics can be achieved by
switching from the state-space representation of Molecular Dynamics (MD) to a
statistical-mechanics-based representation in phase space, where approximations
such as maximum-entropy or Gaussian phase packets (GPP) evolve the atomistic
ensemble in a time-coarsened fashion. In practice, this requires the
computation of expensive high-dimensional integrals over all of phase space of
an atomistic ensemble. This, in turn, is commonly accomplished efficiently by
low-order numerical quadrature. We show that numerical quadrature in this
context, unfortunately, comes with a set of inherent problems, which corrupt
the accuracy of simulations -- especially when dealing with crystal lattices
with imperfections. As a remedy, we demonstrate that Graph Neural Networks,
trained on Monte-Carlo data, can serve as a replacement for commonly used
numerical quadrature rules, overcoming their deficiencies and significantly
improving the accuracy. This is showcased by three benchmarks: the thermal
expansion of copper, the martensitic phase transition of iron, and the energy
of grain boundaries. We illustrate the benefits of the proposed technique over
classically used third- and fifth-order Gaussian quadrature, we highlight the
impact on time-coarsened atomistic predictions, and we discuss the
computational efficiency. The latter is of general importance when performing
frequent evaluation of phase space or other high-dimensional integrals, which
is why the proposed framework promises applications beyond the scope of
atomistics
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described