Data-Driven Equation Discovery for Nonlinear Liquid Film Flows
Over the past decade data-driven equation discovery emerged as a powerful alternative to first principles-based methodologies traditionally used in mathematical modelling cycles. The approach provides a promising path towards deep, physical insight into systems that have previously evaded rigorous mathematical derivation procedures, often due to intractable complexity. The strengths of such techniques have been successfully established for many problem classes described by systems of ordinary differential equations and continue to be extended, with their reach into partial differential equation systems gaining momentum, though comparatively nascent. Herein we tackle such a frontier: elucidating the dynamics of liquid film flows, a problem space providing a rich backdrop in terms of asymptotic analytical building blocks. By leveraging expert knowledge and the ability to carefully curate data, we establish a best-case scenario for identifying the underlying governing equations. Even here, multi-collinearity, stemming from the choice of monomial basis functions in our multi-scale flow configuration, introduces complex identifiability issues. Early-time transients compound this further, as the most dynamically rich behaviour carries the largest residuals in training data. Pinpointing such vulnerabilities allows us to better define the boundaries of current discovery techniques and paves the way for the next generation of more resilient, numerically stable algorithms.