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.

翻译：具有多峰孔径分布的多孔材料的非饱和行为建模面临重大挑战，因为标准水力模型通常难以捕捉其复杂的多尺度特性。一种常见的解决方案是将单峰持水函数叠加，每个函数专门针对特定的孔径范围；然而，这种方法需要为每个模态单独识别参数，这限制了模型的解释性和泛化能力，尤其在数据稀疏场景下。本研究提出一种根本性不同的方法：一个用于元建模的物理约束机器学习框架，能够直接从实验数据自动发现多峰持水曲线的闭合形式数学表达式。数学表达式以二叉树形式表示并通过遗传编程演化，同时将物理约束嵌入损失函数中，引导符号回归器向物理一致且数学稳健的解收敛。结果表明，该框架能够发现有效表征不同孔隙结构多孔材料持水特性的闭合形式方程。为支持第三方验证、应用与扩展，我们已在开源仓库中公开了完整的实现代码。