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

2026-02-24

AI Summary

Standard hydraulic models struggle to accurately capture the complex water retention behavior of multimodal porous media under unsaturated conditions. This work proposes a physics-informed symbolic regression framework that leverages genetic programming to automatically discover closed-form equations for multimodal soil water retention curves directly from experimental data, without assuming a predefined superposition structure. By embedding physical priors into the loss function, the approach ensures both physical consistency and mathematical robustness of the discovered models, which are represented as interpretable binary trees enabling strong generalization. Experimental results demonstrate that the identified equations accurately describe water retention characteristics across diverse pore structures. The implementation code has been made publicly available to facilitate reproducibility and further extension.

Technology Category

• Machine Learning: Multimodal Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• multimodal water retention curves
• porous materials
• symbolic regression
• unsaturated hydraulic behavior
• closed-form equations

Innovation

• physics-constrained symbolic regression
• genetic programming
• multimodal water retention curves
• closed-form equations
• porous materials