Prediction of mode structure using a novel physics-embedded neural ODE method

We designed a new artificial neural network called Exposed latent state neural ordinary differential equation with physics (ExpNODE-p) by modifying the neural ordinary differential equation (NODE) framework to successfully predict the time evolution of the two-dimensional mode profile in nonlinear saturated stage. Starting from the magnetohydrodynamic equations, simplifying assumptions were applied based on physical properties and symmetry considerations of the energetic-particle-driven geodesic acoustic mode (EGAM) to reduce complexity. Our approach embeds known physical characteristics directly into the neural network architecture by exposing latent differential states, enabling the model to capture complex features in the nonlinear saturated stage that are difficult to describe analytically. ExpNODE-p was evaluated using a dataset generated from first-principles simulations of the EGAM instability, focusing on the nonlinear saturated stage where the mode properties (e.g. frequency) are quite difficult to capture. Compared to state-of-the-art models such as ConvLSTM, ExpNODE-p achieved superior performance in both accuracy and training efficiency for multi-step predictions. Additionally, the model exhibited strong generalization capabilities, accurately predicting mode profiles outside the training dataset and capturing detailed features and asymmetries inherent in the EGAM dynamics. Our results establish ExpNODE-p as a powerful tool for creating fast, accurate surrogate models of complex plasma phenomena, opening the door to applications that are computationally intractable with first-principles simulations.