How to use dynamical system deep learning to model chaotic systems
Chaotic Systems are extremely hard to model. For the best results, you want to combine Deep Learning with strong rule based analysis.
An example of this done well is Dynamical System Deep Learning (DSDL), which uses time-series data to reconstruct the system’s attractor, the set of states the system tends towards. DSDL combines univariate (temporal) and multivariate (spatial) reconstructions to capture system dynamics.
“the current dynamical methods can only provide short-term precise predictions, while prevailing deep learning techniques with better performances always suffer from model complexity and interpretability. Here, we propose a new dynamic-based deep learning method, namely the dynamical system deep learning (DSDL), to achieve interpretable long-term precise predictions by the combination of nonlinear dynamics theory and deep learning methods. As validated by four chaotic dynamical systems with different complexities, the DSDL framework significantly outperforms other dynamical and deep learning methods. Furthermore, the DSDL also reduces the model complexity and realizes the model transparency to make it more interpretable.”
-“Interpretable predictions of chaotic dynamical systems using dynamical system deep learning”
Here is a sparknotes summary of the technique:
What DSDL does: DSDL utilizes time series data to reconstruct the attractor. An attractor is just the set of states that your systems will converge towards, even across a wide set of initial conditions.
DSDL combines two pillars to reconstruct the original attractor (A): univariate and multivariate reconstructions. Each reconstruction has its benefits. The Univariate way captures the temporal information of the target variable. Meanwhile, the Multivariate way captures the spatial information among system variables. Let’s look at how.
Univariate reconstruction (D) uses time-delayed samples of a single variable to capture its historical behavior and predict future trends. This is akin to using past temperature data to forecast future fluctuations, providing insights into the underlying dynamics of a single variable within a chaotic system.
Multivariate reconstruction (N) takes a more holistic approach, incorporating multiple variables such as temperature, pressure, and humidity to capture their complex relationships and understand the system’s overall dynamics. This method recognizes that these variables are interconnected and influence each other’s behavior within the chaotic system. DSDL employs a nonlinear neural network to model these intricate and often unpredictable interactions, enabling accurate predictions and a deeper understanding of the system’s behavior.
This approach identifies hidden patterns and relationships within the data, leading to more informed decision-making and effective control strategies for chaotic systems.
Finally, a diffeomorphism map is used to relate the reconstructed attractors to the original attractor. From what I understand, a diffeomorphism is a function between manifolds (which are a generalization of curves and surfaces to higher dimensions) that is continuously differentiable in both directions. In simpler terms, it’s a smooth and invertible map between two spaces. This helps us preserve the topology of the spaces. Since both N and D are equivalent (‘topologically conjugate’ in the paper), we know there is a mapping to link them.
This allows DSDL to make predictions on the system’s future states.
Here’s a simple visualization to see how the components links together-
For more techniques used in modeling chaotic systems check out our discussion, “Can AI be used to predict chaotic systems”-
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