# Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics

arXiv: https://arxiv.org/abs/2005.14617 · HTML: https://mlsanity.com/paper/2005.14617/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2020.

## Built on by (most cited first, 30 of 32)

- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control](https://mlsanity.com/paper/2106.12782/) (2021)
- [Extending Lagrangian and Hamiltonian Neural Networks with Differentiable Contact Models](https://mlsanity.com/paper/2102.06794/) (2021)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Learning Articulated Rigid Body Dynamics with Lagrangian Graph Neural Network](https://mlsanity.com/paper/2209.11588/) (2022)
- [Unravelling the Performance of Physics-informed Graph Neural Networks for Dynamical Systems](https://mlsanity.com/paper/2211.05520/) (2022)
- [Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2212.00893/) (2022)
- [Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2108.02318/) (2021)
- [Learning the Dynamics of Particle-based Systems with Lagrangian Graph Neural Networks](https://mlsanity.com/paper/2209.01476/) (2022)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [How to Learn and Generalize From Three Minutes of Data: Physics-Constrained and Uncertainty-Aware Neural Stochastic Differential Equations](https://mlsanity.com/paper/2306.06335/) (2023)
- [Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems](https://mlsanity.com/paper/2202.02710/) (2022)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Control of Dual-Sourcing Inventory Systems using Recurrent Neural Networks](https://mlsanity.com/paper/2201.06126/) (2022)
- [Guaranteed Conformance of Neurosymbolic Models to Natural Constraints](https://mlsanity.com/paper/2212.01346/) (2022)
- [SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision](https://mlsanity.com/paper/2111.05986/) (2021)
- [Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structures](https://mlsanity.com/paper/2207.07883/) (2022)
- [Lagrangian Neural Network with Differentiable Symmetries and Relational Inductive Bias](https://mlsanity.com/paper/2110.03266/) (2021)
- [ModLaNets: Learning Generalisable Dynamics via Modularity and Physical Inductive Bias](https://mlsanity.com/paper/2206.12325/) (2022)
- [Hamiltonian GAN](https://mlsanity.com/paper/2308.11216/) (2023)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)
- [Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solvers](https://mlsanity.com/paper/2106.02543/) (2021)
- [Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Control](https://mlsanity.com/paper/2301.03565/) (2023)
- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [Reinforcement Learning in System Identification](https://mlsanity.com/paper/2212.07123/) (2022)
- [Time-vectorized numerical integration for systems of ODEs](https://mlsanity.com/paper/2310.08649/) (2023)
- [Deep Learning with Physics Priors as Generalized Regularizers](https://mlsanity.com/paper/2312.08678/) (2023)
- [A Two-Stage Training Method for Modeling Constrained Systems With Neural Networks](https://mlsanity.com/paper/2403.02730/) (2024)

## Abstract (arXiv)

Identifying accurate dynamic models is required for the simulation and control of various technical systems. In many important real-world applications, however, the two main modeling approaches often fail to meet requirements: first principles methods suffer from high bias, whereas data-driven modeling tends to have high variance. Additionally, purely data-based models often require large amounts of data and are often difficult to interpret. In this paper, we present physics-informed neural ordinary differential equations (PINODE), a hybrid model that combines the two modeling techniques to overcome the aforementioned problems. This new approach directly incorporates the equations of motion originating from the Lagrange Mechanics into a deep neural network structure. Thus, we can integrate prior physics knowledge where it is available and use function approximation--e. g., neural networks--where it is not. The method is tested with a forward model of a real-world physical system with large uncertainties. The resulting model is accurate and data-efficient while ensuring physical plausibility. With this, we demonstrate a method that beneficially merges physical insight with real data. Our findings are of interest for model-based control and system identification of mechanical systems.