# Automatic-differentiated Physics-Informed Echo State Network (API-ESN)

arXiv: https://arxiv.org/abs/2101.00002 · HTML: https://mlsanity.com/paper/2101.00002/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2020.

## Built on by (most cited first, 8 of 8)

- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Reconstruction, forecasting, and stability of chaotic dynamics from partial data](https://mlsanity.com/paper/2305.15111/) (2023)
- [Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks](https://mlsanity.com/paper/2304.12865/) (2023)
- [Gradient-free optimization of chaotic acoustics with reservoir computing](https://mlsanity.com/paper/2106.09780/) (2021)
- [Modelling spatiotemporal turbulent dynamics with the convolutional autoencoder echo state network](https://mlsanity.com/paper/2211.11379/) (2022)
- [Physics-Informed Long Short-Term Memory for Forecasting and Reconstruction of Chaos](https://mlsanity.com/paper/2302.10779/) (2023)
- [Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems](https://mlsanity.com/paper/2407.18057/) (2024)
- [Quantum Reservoir Computing with Physics-Informed Correction for Reduced-Order PDE Forecasting](https://mlsanity.com/paper/2608.23119/) (2026)

## Abstract (arXiv)

We propose the Automatic-differentiated Physics-Informed Echo State Network (API-ESN). The network is constrained by the physical equations through the reservoir's exact time-derivative, which is computed by automatic differentiation. As compared to the original Physics-Informed Echo State Network, the accuracy of the time-derivative is increased by up to seven orders of magnitude. This increased accuracy is key in chaotic dynamical systems, where errors grows exponentially in time. The network is showcased in the reconstruction of unmeasured (hidden) states of a chaotic system. The API-ESN eliminates a source of error, which is present in existing physics-informed echo state networks, in the computation of the time-derivative. This opens up new possibilities for an accurate reconstruction of chaotic dynamical states.