On the use of autoencoders to study the dynamics and the causality relations of complex systems with applications to nuclear fusion

Autoencoders are neural networks capable of learning compact representations of data through unsupervised learning. By encoding input data into a lower-dimensional space and subsequently reconstructing it, they enable efficient feature extraction, denoising, anomaly detection, and other applications. This work develops autoencoder-based methodologies tailored to time-dependent problems, specifically for reconstructing hidden dynamics, modelling governing equations, and detecting causal relationships. A physics-informed autoencoder (PIC-AE) is introduced to impose physical or mathematical constraints on the latent representation, allowing the discovery of fundamental dynamics and model parameters. The PIC-AE effectively reconstructs equivalent dynamical systems from indirect measurements, as exemplified by numerical tests based on the Lotka-Volterra system of equations. It has been applied to edge-localized modes (ELMs) in nuclear fusion plasmas to assess whether they follow a Lotka-Volterra model and the results indicate the need for alternative sets of equations. For causality detection, a novel autoencoder-based method has been developed to overcome the limitations of traditional techniques. This new approach accurately identifies causal relationships while providing a probabilistic measure of their strength. Applied to nuclear fusion data, it has confirmed the causal influence of ion cyclotron resonance heating (ICRH) on sawtooth crashes, reproducing previous findings obtained with different methodologies and extending the analysis to the spatio-temporal domain. Although initially designed for nuclear fusion applications, the proposed methodologies are broadly applicable to any scientific and technological domain, in which time series analysis is crucial. Indeed, the developed tools have the representational capabilities of deep learning networks but are much less prone to overfitting and can be accurate even whit sparse data. Future work will explore alternative representations for ELMs and further validate the causality detection method across different datasets.