Time‑series forecasting has advanced dramatically with new deep‑learning models, yet forecasting processes governed by physics remains challenging. Existing Physics‑Informed Neural Networks (PINNs) incorporate physical constraints but do not estimate unobservable intermediate physical variables, which are essential for domain experts to interpret system behavior.
To address this gap, we introduce the Physics‑Informed Recurrent Neural Network (PIRNN). PIRNN predicts the target series while simultaneously estimating unobservable variables on both historical records and the forecast horizon. This design leverages domain knowledge to improve model robustness and interpretability, and it can be readily adapted to any physical model described by multiple equations, each with its own set of hidden variables.
As a case study, we embed the groundwater‑level physical model Gardenia into PIRNN. Gardenia uses transfer equations between reservoirs, calibrated via data assimilation, to simulate groundwater dynamics. We evaluate PIRNN against several well‑known neural‑network baselines and the original Gardenia model on twelve real‑world datasets. PIRNN achieves the best performance on five datasets, and an ablation study highlights the importance of incorporating physical background into time‑series forecasting.
Finally, a domain expert assesses the coherence of the physical variables predicted by the network, confirming their physical plausibility.
Review