Beyond Accuracy: Can Physics-Informed Neural Networks Reproduce Root-Zone Soil Moisture Memory?

Soil moisture memory (SMM) is a primary driver of land-atmosphere coupling, hydrological predictability, and the response of ecosystems to climate variability. Although machine learning-based algorithms have recently been shown to predict soil moisture with a high degree of accuracy, it is unclear whether these models can predict SMM and SMR effectively. In this study, we assess whether better state estimation results in enhanced representation of SMM. To achieve this, we use six years (2013–2018) of daily grassland lysimeter observations from Rollesbroich, Germany (50°37'12" N, 6°18'15" E), including multi-depth soil moisture (10, 30, and 50 cm), through which a depth-averaged root-zone soil moisture is calculated (see Figure 1). In addition to the observational data, we also estimated soil moisture at different depths and then computed root zone soil moisture according to the upper and lower boundary fluxes (i.e., precipitation, drainage, and actual evapotranspiration), using two modelling methodologies: (i) a physics-based Richards equation model (HYDRUS-1D, calibrated against observations) and (ii) a physics-informed neural network (PINN), which was trained on the same dataset (see Figure 1). We analyze, then, the SMM structure in the simulated and observed time series using a Linear Integro-Differential Equations (LIDE) framework, which quantifies the accumulation of memory at different timescales, e.g., fast memory (τ_F), slow memory with short-term (τ_S), intermediate (τ_I), and long-term (τ_L) components, and memory saturation timescale (τ_∞). The results show that the PINN model is much more accurate than the HYDRUS-1D model at simulating observed soil moisture states (root mean square error, RMSE = 0.003 vs 0.018; Nash-Sutcliffe Efficiency, NSE = 0.997 vs 0.881). However, the fast memory timescale (τ_F) is slightly underpredicted by PINN (with τ_F ~ 4.5 days) and is slightly better approximated by HYDRUS-1D (with τ_F ~ 5.9 days) compared to observations (with τ_F ~ 7.6 days), reflecting stronger physical damping in HYDRUS-1D. While the short-term slow-memory timescale (τ_S) could not be identified using either measured or modeled data, the intermediate slow-memory timescale (τ_I) of measured data (with τ_I ≈ 4 months) could be robustly recovered using either model. The long-term slow-memory timescale (τ_L) and the saturation timescale (τ_∞) are, respectively, underestimated and overestimated by the PINN (with τ_L ~ 9 months and τ_∞ ~ 10.6 years), resulting in weaker persistence and a narrower window for re-emergence compared to the observed values (with τ_L ~ 9.5 months and τ_∞ ~ 8.96 years). In contrast, HYDRUS-1D better resolves the long-term memory dynamics (with τ_L ~ 9.7 months and τ_∞ ~ 8.26 years). These findings highlight that strong prediction skills for state variables do not necessarily equate to a good representation of their hidden memory structure.  According to these results, we suggest that memory-based diagnostics can probably serve as a complementary indicator to analyze the performance of simulated soil moisture dynamics alongside traditional performance measures and can provide a critical benchmark for evaluating physics-based and machine learning hydrological models.