Neural lumped parameter differential equations with application in friction-stir processing

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 physics simulations. 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.