# Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles

arXiv: https://arxiv.org/abs/1810.13075 · HTML: https://mlsanity.com/paper/1810.13075/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2018.

## Built on by (most cited first, 30 of 55)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [AirPhyNet: Harnessing Physics-Guided Neural Networks for Air Quality Prediction](https://mlsanity.com/paper/2402.03784/) (2024)
- [PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics](https://mlsanity.com/paper/2106.02993/) (2021)
- [MC-LSTM: Mass-Conserving LSTM](https://mlsanity.com/paper/2101.05186/) (2021)
- [Robust Inverse Framework using Knowledge-guided Self-Supervised Learning: An application to Hydrology](https://mlsanity.com/paper/2109.06429/) (2021)
- [Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems](https://mlsanity.com/paper/2403.12335/) (2024)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Knowledge-guided Machine Learning: Current Trends and Future Prospects](https://mlsanity.com/paper/2403.15989/) (2024)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
- [Multiscale Neural Operator: Learning Fast and Grid-independent PDE Solvers](https://mlsanity.com/paper/2207.11417/) (2022)
- [Physics-Guided Foundation Model for Scientific Discovery: An Application to Aquatic Science](https://mlsanity.com/paper/2502.06084/) (2025)
- [Knowledge-augmented Deep Learning and Its Applications: A Survey](https://mlsanity.com/paper/2212.00017/) (2022)
- [Adaptive Process-Guided Learning: An Application in Predicting Lake DO Concentrations](https://mlsanity.com/paper/2411.12973/) (2024)
- [Origin-Destination Network Generation via Gravity-Guided GAN](https://mlsanity.com/paper/2306.03390/) (2023)
- [Physics-Guided Generative Adversarial Networks for Sea Subsurface Temperature Prediction](https://mlsanity.com/paper/2111.03064/) (2021)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Enhancing predictive skills in physically-consistent way: Physics Informed Machine Learning for Hydrological Processes](https://mlsanity.com/paper/2104.11009/) (2021)
- [Reconstructing High-resolution Turbulent Flows Using Physics-Guided Neural Networks](https://mlsanity.com/paper/2109.03327/) (2021)
- [Koopman Invertible Autoencoder: Leveraging Forward and Backward Dynamics for Temporal Modeling](https://mlsanity.com/paper/2309.10291/) (2023)
- [Hybrid Physics and Deep Learning Model for Interpretable Vehicle State Prediction](https://mlsanity.com/paper/2103.06727/) (2021)
- [Towards Interpretable Physical-Conceptual Catchment-Scale Hydrological Modeling using the Mass-Conserving-Perceptron](https://mlsanity.com/paper/2401.14521/) (2024)
- [The Digital Twin Landscape at the Crossroads of Predictive Maintenance, Machine Learning and Physics Based Modeling](https://mlsanity.com/paper/2206.10462/) (2022)
- [Foundation Models for Environmental Science: A Survey of Emerging Frontiers](https://mlsanity.com/paper/2504.04280/) (2025)
- [A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems](https://mlsanity.com/paper/2208.04995/) (2022)
- [Task Aware Modulation using Representation Learning: An Approach for Few Shot Learning in Environmental Systems](https://mlsanity.com/paper/2310.04727/) (2023)
- [Densely Multiplied Physics Informed Neural Networks](https://mlsanity.com/paper/2402.04390/) (2024)

## Abstract (arXiv)

This paper proposes a physics-guided recurrent neural network model (PGRNN) that combines RNNs and physics-based models to leverage their complementary strengths and improve the modeling of physical processes. Specifically, we show that a PGRNN can improve prediction accuracy over that of physical models, while generating outputs consistent with physical laws, and achieving good generalizability. Standard RNNs, even when producing superior prediction accuracy, often produce physically inconsistent results and lack generalizability. We further enhance this approach by using a pre-training method that leverages the simulated data from a physics-based model to address the scarcity of observed data. The PGRNN has the flexibility to incorporate additional physical constraints and we incorporate a density-depth relationship. Both enhancements further improve PGRNN performance. Although we present and evaluate this methodology in the context of modeling the dynamics of temperature in lakes, it is applicable more widely to a range of scientific and engineering disciplines where mechanistic (also known as process-based) models are used, e.g., power engineering, climate science, materials science, computational chemistry, and biomedicine.