This work proposes a hybrid analytical-computational framework to characterize the non-linear thermal dynamics of a Bluefors dilution refrigerator. We combine classical cryogenic engineering principles with Kolmogorov-Arnold Networks (KANs) to develop a comprehensive mathematical model. The core of this methodology lies in using KANs not as a black-box predictor, but as a symbolic regression tool capable of discovering 'hidden variables' and missing physical terms in the heat transfer equations that are often overlooked in simplified lumped-element models.
The resulting model is rigorously analyzed from a dynamical systems perspective. We map the thermal evolution of the mixing chamber and still stages into phase space, identifying the underlying stability of the system. Specifically, the research characterizes the emergence of limit cycles under different cooling loads, providing a formal mathematical explanation for periodic thermal fluctuations. This deterministic approach allows for a deeper understanding of the coupling between the dilution process and external parasitic heat leaks.
By identifying these hidden governing equations in a well-characterized Bluefors system, we establish a robust diagnostic tool for the QUBIC experiment. The ability to extract explicit symbolic representations of the system’s phase portrait enables the design of advanced control laws that can suppress instabilities and optimize the duty cycle of sub-Kelvin astronomical instrumentation.

Keywords
Dilution Refrigeration|Dynamical Systems|KANs|Thermal Modeling|Cryocoolers