# Bridging Physics-based and Data-driven modeling for Learning Dynamical Systems

arXiv: https://arxiv.org/abs/2011.10616 · HTML: https://mlsanity.com/paper/2011.10616/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2020.

## Built on by (most cited first, 25 of 25)

- [Domain Adaptation for Time Series Forecasting via Attention Sharing](https://mlsanity.com/paper/2102.06828/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts](https://mlsanity.com/paper/2210.03675/) (2022)
- [Learning continuous models for continuous physics](https://mlsanity.com/paper/2202.08494/) (2022)
- [EINNs: Epidemiologically-informed Neural Networks](https://mlsanity.com/paper/2202.10446/) (2022)
- [Data Augmentation vs. Equivariant Networks: A Theory of Generalization on Dynamics Forecasting](https://mlsanity.com/paper/2206.09450/) (2022)
- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
- [Data-Centric Epidemic Forecasting: A Survey](https://mlsanity.com/paper/2207.09370/) (2022)
- [MetaPhysiCa: OOD Robustness in Physics-informed Machine Learning](https://mlsanity.com/paper/2303.03181/) (2023)
- [Understanding the Difficulty of Solving Cauchy Problems with PINNs](https://mlsanity.com/paper/2405.02561/) (2024)
- [Discovering Physics Laws of Dynamical Systems via Invariant Function Learning](https://mlsanity.com/paper/2502.04495/) (2025)
- [Time Series Forecasting with Hypernetworks Generating Parameters in Advance](https://mlsanity.com/paper/2211.12034/) (2022)
- [PEMS: Pre-trained Epidemic Time-series Models](https://mlsanity.com/paper/2311.07841/) (2023)
- [Modeling the effect of the vaccination campaign on the Covid-19 pandemic](https://mlsanity.com/paper/2108.13908/) (2021)
- [Tailored Forecasting from Short Time Series via Meta-learning](https://mlsanity.com/paper/2501.16325/) (2025)
- [ST-PCNN: Spatio-Temporal Physics-Coupled Neural Networks for Dynamics Forecasting](https://mlsanity.com/paper/2108.05940/) (2021)
- [GOPHER: Categorical probabilistic forecasting with graph structure via local continuous-time dynamics](https://mlsanity.com/paper/2112.09964/) (2021)
- [Learning Feynman Diagrams using Graph Neural Networks](https://mlsanity.com/paper/2211.15348/) (2022)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [Rates and architectures for learning geometrically non-trivial operators](https://mlsanity.com/paper/2512.09376/) (2025)
- [Deep Learning-Based Spatiotemporal Multi-Event Reconstruction for Delay Line Detectors](https://mlsanity.com/paper/2306.09359/) (2023)
- [Uncertainty Informed Optimal Resource Allocation with Gaussian Process based Bayesian Inference](https://mlsanity.com/paper/2307.00032/) (2023)
- [Machine Learning for Infectious Disease Risk Prediction: A Survey](https://mlsanity.com/paper/2308.03037/) (2023)
- [A Wave is Worth 100 Words: Investigating Cross-Domain Transferability in Time Series](https://mlsanity.com/paper/2412.00772/) (2024)
- [Optimizing Hard Thresholding for Sparse Model Discovery](https://mlsanity.com/paper/2504.20256/) (2025)

## Abstract (arXiv)

How can we learn a dynamical system to make forecasts, when some variables are unobserved? For instance, in COVID-19, we want to forecast the number of infected and death cases but we do not know the count of susceptible and exposed people. While mechanics compartment models are widely used in epidemic modeling, data-driven models are emerging for disease forecasting. We first formalize the learning of physics-based models as AutoODE, which leverages automatic differentiation to estimate the model parameters. Through a benchmark study on COVID-19 forecasting, we notice that physics-based mechanistic models significantly outperform deep learning. Our method obtains a 57.4% reduction in mean absolute errors for 7-day ahead COVID-19 forecasting compared with the best deep learning competitor. Such performance differences highlight the generalization problem in dynamical system learning due to distribution shift. We identify two scenarios where distribution shift can occur: changes in data domain and changes in parameter domain (system dynamics). Through systematic experiments on several dynamical systems, we found that deep learning models fail to forecast well under both scenarios. While much research on distribution shift has focused on changes in the data domain, our work calls attention to rethink generalization for learning dynamical systems.