Project Area C: Optimization with pH systems

In line with the analysis and discretization of pH systems treated in the other project areas, project
area C focuses on how to preserve and exploit pH structure in optimization and data-driven
modeling. Matching optimization with pH systems  has three aspects. First, the variety of pH systems (deter-
ministic / stochastic, (in)finite-dimensional, continuous / discrete) requires multiple optimization
approaches that fully exploit the particular characteristics. Second, the optimization strategy,
for example feedback or optimal control, needs to be well-suited for the problem or application
at hand. Third, viewing pH dynamics as generalized gradient dynamics allows to derive novel
optimization algorithms.

We develop and analyze new momentum-enhanced optimization methods in C01 for applica-
tions in multi-objective shape optimization problems, and adjoint-based optimization methods
tailored to preserve structure in C02 for pH systems on graphs. PH Gaussian process models based
on data or evaluations of black box simulation tools are developed in C03. This approach to
obtain surrogate models is complemented in project C04 by innovative reliable neural network
training approaches which involve multiple objectives and conserved quantities of pHS, balanc-
ing data and physics-inspired laws. Stochasticity is systematically and coherently introduced
into the pH formalism via Dirac structures in C05, and building on this, stochastic optimal con-
trol approaches are derived. The pH structure which arises naturally in optimizer dynamics is
leveraged in C06 to enable real-time optimization-based control-by-interconnection of complex
systems.