EulerNet: A Robust Discrete-Time Framework for Long-Term Soil Moisture Forecasting Under Significant Observational Noise

Soil moisture is a fundamental hydrological variable that governs groundwater recharge and agricultural productivity. Accurate long-term forecasting is essential for water resource management, yet it remains challenging due to significant observational noise in sensor data and the error propagation inherent in traditional deep learning models. While physics-based models struggle with site-specific calibration and Neural Ordinary Differential Equations (Neural ODEs) often fail to recover stable continuous dynamics from noisy, discretely sampled signals, there is a clear need for a more robust forecasting framework.

In this work, we propose EulerNet, a pragmatic discrete-time framework designed for high-fidelity soil moisture prediction. Instead of attempting to reconstruct complex latent continuous-time vector fields, EulerNet explicitly models the fixed-step mapping required for operational forecasting. The architecture integrates an Euler-style residual update to parameterize one-step tendencies, ensuring numerical stability through its incremental integration form. To mitigate the impact of sensor noise, we incorporate a Random Synthesizer feature mixer. By employing input-independent alignment matrices rather than dynamic self-attention, the Random Synthesizer acts as an implicit regularizer, preventing the model from overfitting to spurious, noise-induced correlations.

We evaluated EulerNet using high-noise in-situ observations. In a one-month autoregressive rollout, the model achieved exceptional performance with R² = 0.7977, RMSE = 0.0039, and RMAE = 0.0083. These results demonstrate that for fixed-step environmental forecasting, a specialized discrete-time formulation can effectively bypass the complexities of continuous-time modeling while maintaining high stability and accuracy under significant noise. Our findings provide a practical and efficient alternative for modeling complex Earth system dynamics from real-world observational data.