# Multi-Step Prediction of Dynamic Systems with Recurrent Neural Networks

arXiv: https://arxiv.org/abs/1806.00526 · HTML: https://mlsanity.com/paper/1806.00526/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2018.

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

- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
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- [DroneDiffusion: Robust Quadrotor Dynamics Learning with Diffusion Models](https://mlsanity.com/paper/2409.11292/) (2024)
- [Hybrid Physics and Deep Learning Model for Interpretable Vehicle State Prediction](https://mlsanity.com/paper/2103.06727/) (2021)
- [Temporal Convolutions for Multi-Step Quadrotor Motion Prediction](https://mlsanity.com/paper/2110.04182/) (2021)
- [TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems](https://mlsanity.com/paper/2412.07010/) (2024)
- [Predictions Based on Pixel Data: Insights from PDEs and Finite Differences](https://mlsanity.com/paper/2305.00723/) (2023)
- [PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting](https://mlsanity.com/paper/2605.08935/) (2026)
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- [Deep Long-Short Term Memory networks: Stability properties and Experimental validation](https://mlsanity.com/paper/2304.02975/) (2023)
- [Learning Long-Horizon Predictions for Quadrotor Dynamics](https://mlsanity.com/paper/2407.12964/) (2024)
- [A Novel Neural Filter to Improve Accuracy of Neural Network Models of Dynamic Systems](https://mlsanity.com/paper/2409.13654/) (2024)
- [A Deep-Learning Iterative Stacked Approach for Prediction of Reactive Dissolution in Porous Media](https://mlsanity.com/paper/2503.08410/) (2025)
- [Assumed Density Filtering and Smoothing with Neural Network Surrogate Models](https://mlsanity.com/paper/2511.09016/) (2025)
- [Dreaming Falcon: Physics-Informed Model-Based Reinforcement Learning for Quadcopters](https://mlsanity.com/paper/2511.18243/) (2025)
- [Am I fit for this physical activity? Neural embedding of physical conditioning from inertial sensors](https://mlsanity.com/paper/2103.12095/) (2021)
- [Towards lifelong learning of Recurrent Neural Networks for control design](https://mlsanity.com/paper/2208.03980/) (2022)
- [Robust Recurrent Neural Network to Identify Ship Motion in Open Water with Performance Guarantees -- Technical Report](https://mlsanity.com/paper/2212.05781/) (2022)
- [One-shot backpropagation for multi-step prediction in physics-based system identification -- EXTENDED VERSION](https://mlsanity.com/paper/2310.20567/) (2023)
- [Forecasting the Forced van der Pol Equation with Frequent Phase Shifts Using Reservoir Computing](https://mlsanity.com/paper/2404.14651/) (2024)
- [Lipschitz-Based Robustness Certification for Recurrent Neural Networks via Convex Relaxation](https://mlsanity.com/paper/2509.17898/) (2025)
- [Unlocking Out-of-Distribution Generalization in Dynamics through Physics-Guided Augmentation](https://mlsanity.com/paper/2510.24216/) (2025)
- [CaLiSym: Learning Symplectic Dynamics of Real-World Systems through Structured Canonical Lifts](https://mlsanity.com/paper/2607.06824/) (2026)

## Abstract (arXiv)

Recurrent Neural Networks (RNNs) can encode rich dynamics which makes them suitable for modeling dynamic systems. To train an RNN for multi-step prediction of dynamic systems, it is crucial to efficiently address the state initialization problem, which seeks proper values for the RNN initial states at the beginning of each prediction interval. In this work, the state initialization problem is addressed using Neural Networks (NNs) to effectively train a variety of RNNs for modeling two aerial vehicles, a helicopter and a quadrotor, from experimental data. It is shown that the RNN initialized by the NN-based initialization method outperforms the state of the art. Further, a comprehensive study of RNNs trained for multi-step prediction of the two aerial vehicles is presented. The multi-step prediction of the quadrotor is enhanced using a hybrid model which combines a simplified physics-based motion model of the vehicle with RNNs. While the maximum translational and rotational velocities in the quadrotor dataset are about 4 m/s and 3.8 rad/s, respectively, the hybrid model produces predictions, over 1.9 second, which remain within 9 cm/s and 0.12 rad/s of the measured translational and rotational velocities, with 99% confidence on the test dataset