Physics-constrained symbolic regression for discovering closed-form equations of multimodal water retention curves from experimental data

TL;DR

This paper presents a physics-constrained machine learning approach using genetic programming to automatically discover closed-form equations for multimodal water retention curves from experimental data, improving interpretability and robustness.

Contribution

It introduces a novel physics-constrained symbolic regression framework that directly derives physically consistent equations for complex water retention behaviors.

Findings

Successfully discovers closed-form equations for water retention curves

Outperforms traditional superposition methods in interpretability

Provides open-source implementation for validation and extension

Abstract

Modeling the unsaturated behavior of porous materials with multimodal pore size distributions presents significant challenges, as standard hydraulic models often fail to capture their complex, multi-scale characteristics. A common workaround involves superposing unimodal retention functions, each tailored to a specific pore size range; however, this approach requires separate parameter identification for each mode, which limits interpretability and generalizability, especially in data-sparse scenarios. In this work, we introduce a fundamentally different approach: a physics-constrained machine learning framework designed for meta-modeling, enabling the automatic discovery of closed-form mathematical expressions for multimodal water retention curves directly from experimental data. Mathematical expressions are represented as binary trees and evolved via genetic programming, while physical constraints are embedded into the loss function to guide the symbolic regressor toward solutions that are physically consistent and mathematically robust. Our results demonstrate that the proposed framework can discover closed-form equations that effectively represent the water retention characteristics of porous materials with varying pore structures. To support third-party validation, application, and extension, we make the full implementation publicly available in an open-source repository.