Euler-Net: A Tendency Learning Framework for Robust Long-Term Soil Moisture Forecasting

Accurate simulation of soil moisture is foundemental in hydrological modeling and water resource management. While increasingly widely used in soil moisture simulations, deep learning models often suffer from error accumulation in long-term recursive forecasting, leading to reduced prediction accuracy. In this study, we propose Euler-Net, a discrete-time neural dynamical framework for soil moisture prediction. Instead of directly predicting future states, the model learns the temporal tendency of soil moisture through an Euler-style update, which improves stability and aligns with the underlying system dynamics. This core architecture is coupled with a lightweight Random Synthesizer attention mechanism to further enhance feature representation and robustness. Evaluating the framework across 15 stations of the U.S. Climate Reference Network (USCRN) reveals that Euler-Net consistently outperforms LSTM and Transformer baselines in long-term recursive forecasting. It achieves a higher median R2 of 0.74 while utilizing an order of magnitude fewer parameters. In addition, Euler-Net exhibits significantly slower error growth over extended prediction horizons, maintaining stable performance throughout monthly forecasts. Interpretability analysis via SHAP reveals that by predicting moisture tendency, Euler-Net prioritizes dynamic hydrological forcing—attributing ~40% importance to precipitation (Pt)—whereas baselines over-rely on antecedent states (SMt-1) for naive persistence forecasting. Crucially, Euler-Net internalizes non-linear mechanisms like infiltration-excess runoff and threshold-based drainage, with the latter exhibiting clear texture-dependent thresholds. Furthermore, the model demonstrates exceptional robustness against observational noise, maintaining smooth and physically plausible prediction even when input data contain significant non-physical fluctuations. These results demonstrate that learning discrete-time dynamics provides an effective, efficient, and reliable solution for stable long-term soil moisture forecasting. Plain Language Summary Soil moisture is an important factor that affects hydrological processes, , agriculture production, and ecosystem services. Being able to predict how soil moisture changes over time is therefore very important. However, existing machine learning models become less accurate when used to make predictions over long periods of time, because small errors can accumulate step by step. In this study, we develop a new model called Euler-Net to improve long-term soil moisture prediction. Instead of directly predicting future values, the model learns how soil moisture changes from one time step to the next. This allows it to better capture the natural evolution of the system and reduces the growth of errors over time. Testing the model across multiple monitoring stations in the United States showed that Euler-Net consistently produces more reliable and stable predictions than commonly used methods, while requiring significantly less computational power. Crucially, our analysis reveals that Euler-Net is not just a "black box." It successfully learns real-world physical behaviors, such as how soil becomes harder to soak when saturated and how water drains away more quickly in sandy soils. Furthermore, the model is remarkably tough against "noisy" or incorrect data caused by sensor glitches. While traditional models often get confused by these errors or rely on simple mathematical patterns, Euler-Net maintains a smooth and realistic prediction by "understanding" the actual environmental processes. This work provides a simple, efficient, and trustworthy tool for predicting environmental changes, supporting better decision-making in water and agricultural management.