A Study of Universal ODE Approaches to Predicting Soil Organic Carbon

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 = 1.6 × 10−5, and
R2 = 0.9999. Case 5, with 7% noise, remained robust (MSE = 3.4×10−6, 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. Qualitatively, the
UDE framework consistently preserved broad SOC patterns, avoided overfitting in moderate noise,
and maintained physics-based plausibility even when data were corrupted. 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.