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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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The Viscoelastic response of the tendons

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Dr. Rishabh Pathak

The viscoelastic response of tendons refers to their unique mechanical behavior characterized by a combination of viscous (time-dependent deformation) and elastic (time-independent deformation) properties. Tendons exhibit viscoelasticity due to the presence of collagen fibers and proteoglycans in their extracellular matrix, which contribute to their mechanical properties. Here is an explanation of the viscoelastic response of tendons:

Viscoelastic Response of Tendons:

1.    Elastic Behavior:

o    Elasticity:

§  Tendons exhibit elastic behavior, meaning they can deform under load and return to their original shape when the load is removed.

§  The elastic properties of tendons allow them to store and release energy during muscle contractions, contributing to the efficiency of movement.

2.    Viscous Behavior:

o    Viscosity:

§  Tendons also display viscous behavior, where deformation is time-dependent and involves energy dissipation.

§  The viscous component of tendon behavior is related to the flow of fluid within the tendon structure and the sliding of collagen fibers.

3.    Creep and Stress Relaxation:

o    Creep:

§  When a constant load is applied to a tendon over time, it gradually elongates due to creep, a viscoelastic phenomenon.

§  Creep behavior in tendons is influenced by factors such as collagen fiber orientation and the composition of the extracellular matrix.

o    Stress Relaxation:

§  Stress relaxation refers to the decrease in stress within a tendon when held at a constant length over time.

§  Tendons exhibit stress relaxation as the collagen fibers reorganize and adjust their alignment in response to sustained loading.

4.    Rate Dependency:

o    Strain Rate:

§  The mechanical properties of tendons can vary depending on the rate at which they are loaded.

§  Tendons may exhibit different stiffness and damping characteristics at different loading rates, reflecting their viscoelastic nature.

5.    Hysteresis:

o    Energy Dissipation:

§  Tendons exhibit hysteresis, where the energy input during loading is not fully recovered during unloading.

§  The hysteresis loop represents the energy dissipated as heat within the tendon structure due to viscous damping.

6.    Adaptation to Loading:

o    Mechanical Loading:

§  Tendons can adapt their viscoelastic properties in response to mechanical loading, such as exercise or training.

§  Regular loading can lead to changes in tendon stiffness, strength, and viscoelastic behavior to better withstand mechanical stresses.

Understanding the viscoelastic response of tendons is essential for biomechanical analyses, injury prevention strategies, and rehabilitation protocols. The complex interplay between the elastic and viscous components of tendon behavior influences their ability to transmit forces, absorb shock, and support joint movement effectively.

 

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