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Learn how to combine prediction-error method (PEM) techniques with universal differential equations (UDEs) to tackle the challenging problem of fitting chaotic dynamical systems, particularly in computational neuroscience applications. Explore the fundamental difficulties in applying UDEs to chaotic systems, including numerical simulation challenges and experimental data noise issues that make traditional approaches intractable for biological systems. Discover how the PEM-UDE approach smooths the harsh parameter landscape that typically prevents successful UDE fitting, demonstrated through validation on classical chaotic systems like the Rössler system and Chua's circuit. Examine the method's robustness to noisy parameter estimates and its ability to learn dynamics even when entire system states are unobserved. Investigate the application of this technique to next-generation neural mass models (NGNMMs) that extend beyond the traditional Ott-Antonsen ansatz assumptions, enabling simulation of spiking neuron populations in more realistic, non-fully-connected network configurations. Gain insights into how symbolic regression can be used to extract analytical forms from the learned UDE representations, and understand the broader implications for scientific machine learning in dynamical systems where traditional physics-informed approaches fall short.
