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

TL;DR

The paper tackles state reconstruction in chaotic systems under partial observations by proposing a Physics-Informed Long Short-Term Memory (PI-LSTM) that imposes a physics-based regularization term enforcing $\frac{d}{dt}\tilde{\mathbf{y}}(t)\approx f(\tilde{\mathbf{y}}(t))$ during training. The method integrates an LSTM with a physics-informed loss and is validated on the Lorenz-96 model, analyzing long-term autonomous evolution and Lyapunov exponents. The results show that PI-LSTM accurately reconstructs unmeasured variables and preserves chaotic dynamics, achieving Lyapunov exponent estimates close to the reference, while a purely data-driven LSTM fails in many cases. This work demonstrates a practical approach for state reconstruction and dynamical learning in nonlinear chaotic systems, with potential applications to experimental data and broader chaotic systems.

Abstract

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.

Figures (4)

• Figure 1: Open-loop configuration.
• Figure 2: Closed-loop configuration.
• Figure 3: Statistics reconstruction of unmeasured variables. Comparison of the target (black line), PI-LSTM (red dashed line) and data-driven LSTM (blue line) probability density functions (PDF) of (i) $N_{\xi}=1$, (ii) $N_{\xi}=3$, (iii) $N_{\xi}=5$ unmeasured variables over a $1000 \tau_{\lambda}$ trajectory in closed-loop configuration.
• Figure 4: Comparison of the target (black squares), PI-LSTM (red dots) and data-driven LSTM (blue dots) LEs for (i) $N_{\xi}=1$, (ii) $N_{\xi}=3$, (iii) $N_{\xi}=5$ unmeasured variables. All the vertical axes are in logarithmic scale.