# Physics-Informed Long Short-Term Memory for Forecasting and Reconstruction of Chaos

arXiv: https://arxiv.org/abs/2302.10779 · HTML: https://mlsanity.com/paper/2302.10779/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 18 references that are in the map.
- First posted on arXiv 3 Feb 2023.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [Physics-Informed Echo State Networks](https://mlsanity.com/paper/2011.02280/) (2020)
- [Automatic-differentiated Physics-Informed Echo State Network (API-ESN)](https://mlsanity.com/paper/2101.00002/) (2020)

## Built on by (most cited first, 3 of 3)

- [Reconstruction, forecasting, and stability of chaotic dynamics from partial data](https://mlsanity.com/paper/2305.15111/) (2023)
- [NeuroMemFPP: A recurrent neural approach for memory-aware parameter estimation in fractional Poisson process](https://mlsanity.com/paper/2512.05893/) (2025)
- [NeuroMem-FHP: A Likelihood-Free Deep Learning Framework for Parameter Estimation of Fractional Hawkes Process](https://mlsanity.com/paper/2607.11177/) (2026)

## Abstract (arXiv)

We present the Physics-Informed Long Short-Term Memory (PI-LSTM) network to reconstruct and predict the evolution of unmeasured variables in a chaotic system. The training is constrained by a regularization term, which penalizes solutions that violate the system's governing equations. The network is showcased on the Lorenz-96 model, a prototypical chaotic dynamical system, for a varying number of variables to reconstruct. First, we show the PI-LSTM architecture and explain how to constrain the differential equations, which is a non-trivial task in LSTMs. Second, the PI-LSTM is numerically evaluated in the long-term autonomous evolution to study its ergodic properties. We show that it correctly predicts the statistics of the unmeasured variables, which cannot be achieved without the physical constraint. Third, we compute the Lyapunov exponents of the network to infer the key stability properties of the chaotic system. For reconstruction purposes, adding the physics-informed loss qualitatively enhances the dynamical behaviour of the network, compared to a data-driven only training. This is quantified by the agreement of the Lyapunov exponents. This work opens up new opportunities for state reconstruction and learning of the dynamics of nonlinear systems.