(653f) M2E3d: Evolutionary Equation Discovery and Its Applications in the Powder-Handling Industries

Multiphase systems are vital to the development of a myriad of products spanning multiple industrial sectors, from the formulation of novel biofuels, chemicals and pharmaceuticals in stirred-tank reactors to plastic recycling in gas-fluidised beds. Numerical models of these and many other systems are used to better understand and optimise their internal dynamics and end-products. Modelling classical fluids is relatively simple: their properties may be directly measured using widely-available characterisation tools, and directly input to CFD models. Particulate media - powders, granulates - for which there exist no established continuum models, are significantly harder to simulate; they are typically modelled piece-wise using the Discrete Element Method (DEM), whose calibration is infamously problematic . Modelling multiphase systems is more complex still, requiring detailed properties of both fluid and particulate phases, and suitable drag models for coupling. There exists a dizzying array of (often imprecise) empirical models for this task (Luding S. Introduction to discrete element methods: basic of contact force models and how to perform the micro-macro transition to continuum theory. European journal of environmental and civil engineering. 2008).

Multiphase Materials Exploration via Evolutionary Equation Discovery M²E³D) is a novel evolutionary approach to determine the equations governing their interactions. In other words, M²E³D autonomously discovers the equations required to accurately reproduce a system’s full, three-dimensional dynamics. The resulting combinatorial optimisation problem of discovering analytical closed-form expressions modelling previously-measured data points is solved using a symbolic regression engine employing an evolutionary algorithm balancing the model's error and complexity.

M²E³D has been rigorously validated, using the technique to autonomously (re)discover well-known analytical laws such as Stokes drag on spherical particles and Darcy-Weisbach pressure drop in pipes using only 32 measurements. It has then been applied to derive entirely novel relations for complex industrial systems that are impossible to model analytically, such as:

• Predictive models for powder blending performance in continuous direct compression (paper in collaboration with AstraZeneca; Jones-Salkey O, Windows-Yule CR, Ingram A, Stahler L, Nicusan AL, Clifford S, de Juan LM, Reynolds GK. Using AI/ML to predict blending performance and process sensitivity for Continuous Direct Compression (CDC). International Journal of Pharmaceutics. 2024).
• Correlations predicting pharmaceutical tablet tensile strength from process conditions, with applications in online process control (paper in collaboration with GlaxoSmithKline, under review).
• Scaling laws for 3-phase fluidised beds as functions of liquid loading and viscosity.

In this talk we introduce the methodology underlying M²E³D and demonstrate its application through a series of case studies. The Julia, Python and C++ algorithms developed are open-source, portable and massively-parallel, and were used successfully from laptop-scale shared-memory machines to multi-node supercomputing clusters - facilitating high-throughput discovery and understanding of systems beyond those explored here.