A trajectory-based loss function to learn missing terms in bifurcating dynamical systems.

Area of Science:

• Dynamical Systems Theory
• Machine Learning
• Scientific Modeling

Background:

• Identifying missing terms in dynamical systems is a significant challenge for accurate modeling.
• The integration of machine learning with dynamical system theory offers promising solutions.
• The Universal Differential Equation (UDE) framework combines physics-informed differential equations and machine learning.

Purpose of the Study:

• To modify the UDE framework for discovering missing terms in systems exhibiting bifurcations.
• To enhance the applicability of UDEs to a broader range of real-world problems.
• To address limitations of traditional loss functions like Mean Square Error (MSE) in capturing dynamical behavior.

Main Methods:

• Development of a novel trajectory-based loss function for UDEs.
• The new loss function optimizes two independent components: length and angle of state space vectors.
• Comparison of the proposed loss function against MSE using systems from neuroscience, chemistry, and biology.

Main Results:

• The trajectory-based loss function reliably discovers missing terms in systems undergoing bifurcations.
• Demonstrated success in modeling Saddle-Node, Pitchfork, Hopf, and Period-doubling bifurcations.
• The proposed method outperforms MSE in accurately reconstructing dynamical behavior.

Conclusions:

• The modified UDE approach with a trajectory-based loss function effectively identifies missing terms in complex dynamical systems.
• This method significantly expands the utility of UDEs for modeling systems with sudden changes in behavior.
• The findings have broad implications for scientific modeling in diverse fields like neuroscience, chemistry, and biology.