Physics-Informed Long Short-Term Memory for Forecasting and Reconstruction of Chaos
This work addresses state reconstruction in chaotic systems for applications like weather forecasting, but it is incremental as it builds on existing physics-informed neural networks and LSTM methods.
The paper tackles the problem of reconstructing and predicting unmeasured variables in chaotic systems by introducing a Physics-Informed Long Short-Term Memory (PI-LSTM) network, which uses a regularization term to enforce physical constraints, and demonstrates on the Lorenz-96 model that it correctly predicts statistics and aligns Lyapunov exponents, unlike data-driven-only training.
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.