用物理模型+神经网络预测土壤碳变化，抗噪能力强但需改进鲁棒性。
A study of Universal ODE approaches to predicting soil organic carbon
- 融合物理方程与神经网络，学习微生物活动对碳的影响
- 无噪声下预测误差极低（MSE=1.6e-5，R²=0.9999）
- 高噪声场景易过拟合，需引入概率建模提升可靠性
土壤有机碳（SOC）是土壤健康与全球气候韧性的基础，但其预测因复杂的物理、化学和生物过程而困难。本文探索基于通用微分方程（UDEs）的科学机器学习框架，用于预测土壤深度与时间维度上的SOC动态。UDE结合了如对流扩散传输等机制模型，以及学习非线性微生物产碳与呼吸的神经网络。通过合成数据集系统评估六种实验情形，从无噪声基准到含35%空间相关乘性噪声的强扰动测试。结果显示，在无噪声及中等噪声下，UDE能精准重建SOC动态；案例4（50年终端剖面）达到近完美拟合，MSE=1.6e-5，R²=0.9999；案例5（7%噪声）仍稳健，MSE=3.4e-6，R²=0.99998，可捕捉深度趋势并容忍真实测量不确定性。但在案例3（初始时刻35%噪声）出现明显过拟合，模型复现噪声输入但对真实值泛化能力下降（R²=0.94）；案例6（50年时35%噪声）则退化为过于平滑的均值剖面，丧失深度变异性，产生负R²，凸显标准训练在严重不确定性下的局限。结果表明，UDE适用于可扩展、抗噪的SOC预测，但迈向实际应用需引入噪声感知损失函数、概率建模与微生物动力学更紧密耦合。
原文摘要 · Abstract (English)
Soil Organic Carbon (SOC) is a foundation of soil health and global climate resilience, yet its prediction remains difficult because of intricate physical, chemical, and biological processes. In this study, we explore a Scientific Machine Learning (SciML) framework built on Universal Differential Equations (UDEs) to forecast SOC dynamics across soil depth and time. UDEs blend mechanistic physics, such as advection diffusion transport, with neural networks that learn nonlinear microbial production and respiration. Using synthetic datasets, we systematically evaluated six experimental cases, progressing from clean, noise free benchmarks to stress tests with high (35%) multiplicative, spatially correlated noise. Our results highlight both the potential and limitations of the approach. In noise free and moderate noise settings, the UDE accurately reconstructed SOC dynamics. In clean terminal profile at 50 years (Case 4) achieved near perfect fidelity, with MSE = 1.6e-5, and R2 = 0.9999. Case 5, with 7% noise, remained robust (MSE = 3.4e-6, R2 = 0.99998), capturing depth wise SOC trends while tolerating realistic measurement uncertainty. In contrast, Case 3 (35% noise at t = 0) showed clear evidence of overfitting: the model reproduced noisy inputs with high accuracy but lost generalization against the clean truth (R2 = 0.94). Case 6 (35% noise at t = 50) collapsed toward overly smooth mean profiles, failing to capture depth wise variability and yielding negative R2, underscoring the limits of standard training under severe uncertainty. These findings suggest that UDEs are well suited for scalable, noise tolerant SOC forecasting, though advancing toward field deployment will require noise aware loss functions, probabilistic modelling, and tighter integration of microbial dynamics.
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