Shape Arithmetic Expressions: Advancing Scientific Discovery Beyond
  Closed-Form Equations
Krzysztof Kacprzyk Mihaela van der Schaar
Symbolic regression has excelled in uncovering equations from physics,
chemistry, biology, and related disciplines. However, its effectiveness becomes
less certain when applied to experimental data lacking inherent closed-form
expressions. Empirically derived relationships, such as entire stress-strain
curves, may defy concise closed-form representation, compelling us to explore
more adaptive modeling approaches that balance flexibility with
interpretability. In our pursuit, we turn to Generalized Additive Models
(GAMs), a widely used class of models known for their versatility across
various domains. Although GAMs can capture non-linear relationships between
variables and targets, they cannot capture intricate feature interactions. In
this work, we investigate both of these challenges and propose a novel class of
models, Shape Arithmetic Expressions (SHAREs), that fuses GAM's flexible shape
functions with the complex feature interactions found in mathematical
expressions. SHAREs also provide a unifying framework for both of these
approaches. We also design a set of rules for constructing SHAREs that
guarantee transparency of the found expressions beyond the standard constraints
based on the model's size.