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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 growing subcortex

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Dr. Rishabh Pathak

In the analytical model of brain development, the scenario of a growing cortex on a growing subcortex is considered. Here are the key aspects of this analytical model:


1. Model Description: The model involves representing the cortex as a morphogenetically growing outer layer and the subcortex as a strain-driven growing inner core. This dual-layered approach captures the dynamic nature of both layers as they interact and influence the folding patterns of the brain.


2.  Mechanical Interactions: The model accounts for the mechanical interactions between the growing cortex and subcortex, considering how their respective growth rates and properties influence the deformation and folding of the brain tissue. This approach integrates both axonal tension-driven and differential growth-driven hypotheses of cortical folding.


3.  Continuum Theory of Finite Growth: The model is based on the continuum theory of finite growth, which describes the growth and deformation of biological tissues over time. By incorporating growth mechanisms into the model, researchers can simulate the evolving morphology of the brain surface during development.


4.  Parameter Exploration: The model explores the effects of varying parameters such as cortical thickness, stiffness ratios, and growth rates between the cortex and subcortex. By systematically varying these parameters, researchers can analyze how different growth dynamics impact the folding patterns and surface morphologies of the brain.


5. Analytical Estimates: The model provides analytical estimates for critical parameters such as the critical time, pressure, and wavelength at the onset of folding. These estimates offer insights into the conditions under which cortical folding initiates and how the growth dynamics of the cortex and subcortex contribute to this process.


6. Integration with Cellular Mechanisms: The model aims to connect the macroscopic mechanical behavior of the cortex-subcortex system with underlying cellular mechanisms such as axon elongation. By bridging the gap between macroscopic and microscopic scales, researchers can better understand the biological processes driving cortical folding.


In summary, the analytical model of a growing cortex on a growing subcortex offers a comprehensive framework for studying the mechanical and morphological aspects of brain development. By incorporating growth dynamics and mechanical interactions into the model, researchers can simulate the complex folding patterns observed in the developing brain and gain insights into the underlying mechanisms shaping brain morphology.

 

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