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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
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Analytical Model: Growing Cortex on elastic subcortex

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Dr. Rishabh Pathak

In the analytical model of brain development, specifically focusing on cortical folding, the scenario of a growing cortex on an elastic subcortex is considered. Here are the key aspects of this analytical model:


1.   Model Description: The model involves interpreting the subcortex as an infinite half-space and imposing a sinusoidal deflection on its upper boundary. The deflection is considered as the sum of an elastic subcortical deflection and subcortical growth, reflecting the dynamic nature of the tissue.


2.  Deflection Analysis: The deflection of the growing cortex on the elastic subcortex is analyzed using the Föppl–von Kármán theory and the classical fourth-order plate equation. This analysis helps in understanding the deformation behavior of the cortical tissue as it grows and interacts with the underlying subcortical layer.


3.   Parameter Variation: The model explores the effects of varying parameters such as cortical thickness, stiffness ratios between the cortex and subcortex, and growth rates. By systematically changing these parameters, researchers can investigate how different mechanical properties influence the folding patterns and surface morphologies of the brain.


4. Sensitivity Studies: Sensitivity studies are conducted to analyze how changes in cortical thickness and stiffness ratios impact the wavelength of folding patterns. These studies provide insights into the relationship between mechanical properties and the resulting brain surface morphology.


5.  Computational Validation: The analytical estimates derived from this model are validated computationally using finite element analysis. Computational modeling allows for a more detailed exploration of the complex folding patterns and surface morphologies that arise from the interactions between the growing cortex and elastic subcortex.


6.     Implications: By studying the growth of the cortex on the elastic subcortex, researchers can gain a better understanding of the mechanical mechanisms underlying cortical folding in the brain. This model helps in predicting realistic surface morphologies and provides insights into the development of complex brain structures.


In summary, the analytical model of a growing cortex on an elastic subcortex provides a framework for investigating the mechanical interactions that drive cortical folding during brain development. By combining analytical and computational approaches, researchers can elucidate the role of growth, stiffness, and other factors in shaping the intricate surface morphologies of the mammalian brain.

 

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