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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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Föppl–von Kármán Theory

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Dr. Rishabh Pathak

The Föppl–von Kármán theory is a fundamental theory in the field of solid mechanics, specifically in the study of the deformation of thin plates and shells. This theory provides a mathematical framework for analyzing the behavior of thin elastic structures subjected to bending and stretching loads. Here is an overview of the key aspects of the Föppl–von Kármán theory:


1.  Plate and Shell Deformation: The theory is commonly applied to analyze the deformation of thin plates and shells under various loading conditions. It considers the nonlinear effects of both bending and stretching in these structures.


2.  Nonlinear Elasticity: The theory accounts for the nonlinear elasticity of thin plates and shells, where the deformations are significant enough to warrant a nonlinear analysis. This is in contrast to linear elasticity theories that assume small deformations.


3.   Equilibrium Equations: The theory provides equilibrium equations that govern the deformation of thin plates and shells. These equations consider the balance of internal stresses, external loads, and geometric properties of the structure.


4.  Von Kármán Equations: The equations derived from the Föppl–von Kármán theory describe the equilibrium and compatibility conditions for thin plates and shells. These equations are essential for understanding the complex deformations that occur in these structures.


5.  Applications: The Föppl–von Kármán theory has applications in various fields, including aerospace engineering, civil engineering, and biomechanics. In the context of brain development, the theory is used to model the deformation of the cortical tissue during folding processes.


6.    Limitations: While the theory is powerful for analyzing the behavior of thin plates and shells, it has limitations, especially when dealing with highly nonlinear and complex deformations. In such cases, numerical methods like finite element analysis are often employed for more accurate predictions.


In the study of brain development, the Föppl–von Kármán theory is utilized to model the deformation of the cortical tissue and analyze the critical conditions at the onset of folding. By incorporating this theory into analytical and computational models, researchers can gain insights into the mechanical aspects of cortical folding and the formation of brain surface morphologies.

 

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