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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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Cancellous Bone

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Dr. Rishabh Pathak

Cancellous bone, also known as trabecular or spongy bone, is the other main type of bone tissue found in the human skeleton alongside cortical bone. Cancellous bone has a porous and lattice-like structure, providing flexibility, shock absorption, and a site for hematopoiesis (blood cell formation). Here are key features and characteristics of cancellous bone:

1.    Structure:

o    Trabeculae: Cancellous bone is composed of a network of thin, bony trabeculae that form an interconnected lattice structure.

o    Bone Marrow: The spaces between trabeculae contain red bone marrow, which is involved in the production of blood cells (hematopoiesis).

o    Less Compact: Cancellous bone is less dense and compact than cortical bone, with a higher surface area-to-volume ratio.

2.    Composition:

o    Trabecular Bone: The trabeculae are made up of lamellae, osteocytes, and canaliculi similar to cortical bone but with a more open and porous arrangement.

o    Collagen and Mineralization: Cancellous bone contains collagen fibers and mineral components like hydroxyapatite, providing a balance of strength and flexibility.

3.    Function:

o    Shock Absorption: The porous structure of cancellous bone allows it to absorb impact forces and distribute loads, reducing the risk of fractures.

o    Hematopoiesis: Red bone marrow within cancellous bone is a primary site for the production of red blood cells, white blood cells, and platelets.

o    Metabolic Activity: Cancellous bone is metabolically active and responds to mechanical stimuli, hormonal signals, and changes in bone density.

4.    Location:

o    Ends of Long Bones: Cancellous bone is typically found at the ends (epiphyses) of long bones, where it provides cushioning and support at joints.

o    Vertebrae and Pelvis: It also forms the internal structure of vertebrae, pelvis, and other flat bones, contributing to their strength and flexibility.

5.    Vascularization:

o    Cancellous bone has a higher vascular supply compared to cortical bone, with blood vessels penetrating the trabecular network to deliver nutrients and remove waste products.

o    The vascular network supports the metabolic needs of bone cells and facilitates the exchange of gases and nutrients.

6.    Mechanical Properties:

o    Cancellous bone is more flexible and compliant than cortical bone, making it well-suited for adapting to dynamic loads and maintaining bone health.

o    Its porous structure allows for rapid bone turnover, remodeling, and adaptation in response to mechanical stress and hormonal influences.

Understanding the unique characteristics and functions of cancellous bone is essential for comprehending the biomechanics of bone tissue, the role of bone marrow in hematopoiesis, and the structural adaptations of the skeleton to mechanical demands and metabolic requirements.

 

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