arXiv:2603.03346v1 Announce Type: cross 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.

arXiv:2505.21668v3 Announce Type: replace Abstract: Practical guidance on training Large Language Models (LLMs) to leverage Code Interpreter across diverse tasks remains lacking. We present R1-Code-Interpreter, an extension of a text-only LLM trained via multi-turn supervised fine-tuning (SFT) and reinforcement learning (RL) to autonomously generate multiple code queries during step-by-step reasoning. Unlike prior RL + tool-use efforts focused on narrow domains such as math or retrieval, we curate 144 diverse reasoning and planning tasks and show that training a general-purpose Code Interpreter across them presents significant challenges due to task heterogeneity and scarcity of effective samples. To address this, we introduce a multi-stage curriculum learning approach that partitions training samples by measured improvement potential. The RL training prioritizes samples with higher potential and gradually shifts to lower-potential ones, increasing the average RL gains from merely +3.4% to +9.3% across Qwen-2.5 models (3/7/14B). Our final model, R1-CI-14B, improves average accuracy on the 37 test tasks from 44.1% to 72.4%, outperforming text-only GPT-4o (58.6%) and GPT-4o with Code Interpreter (70.9%). Notably, R1-CI-14B also exhibits emergent self-checking behavior through code generation. Datasets, Codes, and Models are available at https://github.com/yongchao98/R1-Code-Interpreter and https://huggingface.co/yongchao98.