The Evolved Equation

Water retention curves describe how much water a porous material holds at different suction pressures. For materials with complex pore structures — multiple pore sizes, bimodal or multimodal distributions — the standard approach stacks equations: one per pore mode, each with its own parameters. The number of parameters multiplies with complexity, and each mode requires separate identification.

Kim and Suh let the equations evolve. Genetic programming represents mathematical expressions as binary trees and evolves them over generations, with physical constraints embedded as fitness penalties. The algorithm discovers closed-form equations directly from experimental data — equations that capture multimodal retention behavior in a single expression rather than a superposition of modal equations.

The through-claim: the closed form was always there; the human-designed parametric approach was an approximation to something simpler. Stacking single-mode equations is a modular strategy that mirrors how we think about pore distributions. But the data doesn’t care about our decomposition. The evolved equations find functional forms that cross modal boundaries, capturing the multimodal behavior without decomposing it into modes first.

The physics constraints matter: the evolved equations must satisfy monotonicity (more suction → less water), boundary conditions (saturation at zero suction, residual at infinite suction), and continuity. Without constraints, genetic programming produces accurate but unphysical expressions. The constraints don’t just improve accuracy — they make the results interpretable.

Evolution with constraints. The equations are discovered, not designed.