Data-driven discovery of sparse dimensionless numbers and nondimensional relations for low-temperature plasmas

Abstract Low-temperature plasmas (LTP) are increasingly used in medicine and biotechnology, materials processing, and environmental remediation, but their complex physics often render design and scale-up of LTP processes challenging. Scaling and similarity relations can aid in understanding and designing LTP processes, as they can be used to construct reduced-order models, relate design levers to discharge behavior, and predict discharge properties in novel systems. For example, the reduced electric field highlights the fundamental role of electron energy gained between collisions, and has been instrumental in scaling up LTP processes. Historically, similarity parameters and scaling laws are expressed as dimensionless relations. This paper introduces a learning-based approach to discover sparse dimensionless numbers (i.e. dimensionless numbers with as few process variables as possible while having an adequate predictive capability) and their relations. The method uses <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> -penalization in regression to find coefficients for null-space basis vectors of the dimension matrix, eliminating irrelevant process variables in discovery of dimensionless relationships. The method is demonstrated for learning nondimensional relations for electron number density and electron temperature in a nanosecond-pulsed packed and unpacked dielectric-barrier discharge in He. It is observed that process variables such as the electron density, the off-pulse electron temperature and the density of excited He can be related independently of reactor volume or operating conditions for the conditions studied. Further, it is shown that the proposed method can effectively discard input process variables that may not be relevant to describing a target dimensionless number. The resulting interpretable relations not only extrapolate, but also provide insights that are not readily apparent from conservation laws.