  Lumped parameter methods aim to simplify the evolution of spatially-extended
or continuous physical systems to that of a "lumped" element representative of
the physical scales of the modeled system. For systems where the definition of
a lumped element or its associated physics may be unknown, modeling tasks may
be restricted to full-fidelity simulations of the physics of a system. In this
work, we consider data-driven modeling tasks with limited point-wise
measurements of otherwise continuous systems. We build upon the notion of the
Universal Differential Equation (UDE) to construct data-driven models for
reducing dynamics to that of a lumped parameter and inferring its properties.
The flexibility of UDEs allow for composing various known physical priors
suitable for application-specific modeling tasks, including lumped parameter
methods. The motivating example for this work is the plunge and dwell stages
for friction-stir welding; specifically, (i) mapping power input into the tool
to a point-measurement of temperature and (ii) using this learned mapping for
process control.
