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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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Continuum Model of Cortical growth

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Dr. Rishabh Pathak

In the context of brain development and cortical growth, a continuum model is used to describe the evolution of the brain's structure over time. Here are the key aspects of a continuum model of cortical growth:


1.  Representation of Brain Tissue: The continuum model represents the brain tissue as a continuous and deformable medium, allowing researchers to study the growth and deformation of the brain's cortical layers over developmental stages.


2.   Layered Structure: The model typically considers the brain tissue as a layered structure, with distinct regions such as the cortex and subcortex characterized by different mechanical properties and growth behaviors. This layered representation enables the simulation of interactions between different brain regions during growth.


3.  Growth Mechanisms: The continuum model incorporates growth mechanisms that drive changes in the brain's structure, including cell proliferation, differentiation, and migration. By integrating these growth processes into the model, researchers can simulate how the brain's morphology evolves over time.


4.  Mechanical Properties: The model accounts for the mechanical properties of brain tissue, such as stiffness, elasticity, and viscoelasticity. These properties influence how the brain responds to growth-induced stresses and strains, leading to changes in its shape and morphology.


5.  Continuum Mechanics: The model is often based on principles of continuum mechanics, which describe the behavior of continuous media under external forces and deformations. By applying continuum mechanics to the brain tissue, researchers can analyze how growth processes affect the tissue's mechanical response.


6. Computational Simulation: The continuum model is implemented using computational methods, such as finite element analysis, to simulate the growth and deformation of the brain tissue. Computational simulations enable researchers to predict how the brain's structure changes over time and investigate the underlying mechanisms of cortical growth.


7. Parameter Studies: Researchers can conduct parameter studies using the continuum model to explore the effects of different factors on cortical growth, such as growth rates, mechanical properties, and external stimuli. By varying these parameters, researchers can gain insights into the factors that influence cortical development.


8. Biological Relevance: The continuum model aims to capture the biological relevance of cortical growth processes, providing a framework for understanding how mechanical forces, growth dynamics, and cellular behaviors interact to shape the structure of the developing brain. This approach helps bridge the gap between biomechanics and developmental biology in studying cortical growth.


In summary, a continuum model of cortical growth offers a comprehensive framework for studying the mechanical and morphological aspects of brain development. By integrating growth mechanisms, mechanical properties, and computational simulations, researchers can gain valuable insights into the complex processes underlying cortical growth and the formation of the brain's intricate structure.

 

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