(126f) Investigation of Fitness Criteria for Xai: Unlocking Process Dynamics with Symbolic Regression

Physics-informed symbolic regression (PISR) is emerging as a powerful tool for chemical engineers seeking interpretable solutions to partial differential equations (PDEs) inherent to process systems. The approach offers to facilitate transparent integration of learned PDE solutions into digital twins, process optimizers, and safety-critical simulations. Explicit solutions to PDEs enable domain experts to audit, validate, and adapt models to dynamic process environments, bridging the gap between machine learning advances and chemical engineering practices.

Finding a balance between accuracy, parameter identifiability, and parsimony in PDE solutions learned by PISR, remains challenging. Thus, we evaluated PISR with three fitness criteria – PDE residual based minimization, Bayesian information criterion (BIC), and an identifiability-focused criterion – across canonical PDEs central to chemical engineering: Laplace’s equation (steady-state diffusion), Burgers’ equation (nonlinear fluid flow), and a nonlinear wave equation (acoustics). Our findings reveal that BIC consistently outperforms other metrics, delivering high-fidelity, interpretable solutions that balance accuracy with parsimony. For Laplace’s equation, BIC enabled exact analytical discovery of steady-state solutions critical for reactor modeling. For the Burgers’ equation, BIC facilitated the discovery of a solution that achieved an R-squared value of 0.998 when evaluated against numerical solutions, while for the nonlinear wave equation, BIC yielded the best solution (R-squared=0.957) with only five parameters.

The identifiability-focused criterion, while enforcing parameter identifiability, imposed restrictive complexity constraints on solutions, often sacrificing accuracy. These results suggest a pragmatic two-step strategy: first leverage BIC to uncover candidate solutions capturing essential physics; second, perform post-hoc identifiability analysis to ensure robustness in process design and control. By expressing PDE solutions in symbolic form, PISR inherently contributes to explainable AI for process systems engineering and our findings provide actionable insights for advancing trustworthy AI in chemical process modeling and deploying symbolic machine learning in intelligent chemical plants.