Report on Current Developments in Cognidynamics

General Direction of the Field

The field of cognidynamics is witnessing a significant shift towards integrating advanced mathematical frameworks with neural network architectures to enhance the modeling and understanding of cognitive systems. This integration is primarily driven by the application of Hamiltonian dynamics and symplectic geometry, which are providing novel insights into the learning mechanisms of neural networks and their biological plausibility. The focus is not only on improving the computational efficiency and accuracy of these models but also on ensuring their robustness and interpretability.

Recent developments have emphasized the importance of dissipativity in dynamical systems, particularly in neural networks, to guarantee stability and energy conservation. This has led to the formulation of new learning methods that transform neural network dynamics into dissipative ones, thereby enhancing their applicability in real-world scenarios such as robotics and fluid dynamics.

Additionally, there is a growing interest in quantifying the behavioral distance between mathematical expressions to improve the efficiency of symbolic regression methods. This approach aims to smooth the error landscape in the search space by clustering expressions with similar errors, thereby facilitating the exploration by gradient-based methods.

The field is also making strides in discovering state variables in dynamical systems by leveraging physical principles, which is crucial for accurate and interpretable modeling. This approach ensures that the models not only fit the data but also adhere to the underlying physical laws, leading to more reliable predictions and controls.

Noteworthy Developments

• Symplectic Neural Networks Based on Dynamical Systems: This framework introduces symplectic neural networks that are universal approximators with non-vanishing gradient properties, significantly enhancing expressiveness and accuracy with lower training costs.
• Learning Deep Dissipative Dynamics: The proposed method ensures dissipativity in neural network-represented dynamical systems, guaranteeing stability and robustness against out-of-domain inputs, with applications demonstrated in robotics and fluid dynamics.
• Quantifying Behavioural Distance Between Mathematical Expressions: The introduction of a behavioral distance measure significantly improves the smoothness of the error landscape in symbolic regression, enhancing the efficiency of local, gradient-based methods.
• Physics-informed Discovery of State Variables in Second-Order and Hamiltonian Systems: This method leverages physical characteristics to constrain neural network models, outperforming baseline models in identifying minimal and interpretable state variables.

These developments highlight the innovative and transformative work being done in cognidynamics, pushing the boundaries of both theoretical understanding and practical applications.