Physical knowledge on historical data matters more than enforcing physical constraints on the forecast

arXiv cs.AI Papers
physics-informed time-series-forecasting recurrent-neural-networks physical-constraints groundwater-modeling deep-learning ablation-study

Summary

The paper proposes a Physics Informed Recurrent Neural Network (PIRNN) that predicts unobservable physical variables to improve time series forecasting in physical processes, demonstrated through groundwater level predictions.

arXiv:2609.19871v1 Announce Type: new Abstract: Time series forecasting has seen signicant advancements with the emergence of new deep learning models. However, forecasting time series in applications involving physical processes remains a major challenge. Despite the apparition of Physics Informed Neural Networks (PINN), recent models do not estimate unobservable intermediate physical variables, which are important for domain experts to understand the target behavior. To this end, we propose a Physics Informed Recurrent Neural Network (PIRNN) which predicts, along the target, unobservable variables on both historic data and forecast target. This approach enhances the model robustness and results interpretation using domain knowledge. Our method is easily adaptable to any physical model using several equations, each having its own set of unobservable variables, to describe it-self. As a case study, we incorporate physical equations used for groundwater levels predictions by the physical model called Gardenia. This model uses transfers equations between reservoirs, optimized with data assimilation, to simulate the evolution of groundwater levels. Evaluation includes several well known neural network models and the Gardenia model compared on twelve real world datasets. In addition, we study the impact of each component through an ablation study. Our model outperforms other models on ve out of the twelve datasets and our ablation study underlines the importance of having a physical background in our time series forecasting task. Finally, the coherence of the physical variables predicted by our neural network is assessed by a domain expert.
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# Physical knowledge on historical data matters more than enforcing physical constraints on the forecast
Source: [https://arxiv.org/abs/2609.19871](https://arxiv.org/abs/2609.19871)
[View PDF](https://arxiv.org/pdf/2609.19871)

> Abstract:Time series forecasting has seen signicant advancements with the emergence of new deep learning models\. However, forecasting time series in applications involving physical processes remains a major challenge\. Despite the apparition of Physics Informed Neural Networks \(PINN\), recent models do not estimate unobservable intermediate physical variables, which are important for domain experts to understand the target behavior\. To this end, we propose a Physics Informed Recurrent Neural Network \(PIRNN\) which predicts, along the target, unobservable variables on both historic data and forecast target\. This approach enhances the model robustness and results interpretation using domain knowledge\. Our method is easily adaptable to any physical model using several equations, each having its own set of unobservable variables, to describe it\-self\. As a case study, we incorporate physical equations used for groundwater levels predictions by the physical model called Gardenia\. This model uses transfers equations between reservoirs, optimized with data assimilation, to simulate the evolution of groundwater levels\. Evaluation includes several well known neural network models and the Gardenia model compared on twelve real world datasets\. In addition, we study the impact of each component through an ablation study\. Our model outperforms other models on ve out of the twelve datasets and our ablation study underlines the importance of having a physical background in our time series forecasting task\. Finally, the coherence of the physical variables predicted by our neural network is assessed by a domain expert\.

## Submission history

From: Etienne Lehembre \[[view email](https://arxiv.org/show-email/bcc92548/2609.19871)\] \[via CCSD proxy\] **\[v1\]**Thu, 17 Sep 2026 08:23:08 UTC \(695 KB\)

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