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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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What is Mechanics?

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Dr. Rishabh Pathak

Mechanics is a branch of physics that deals with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of these bodies on their environment. It is divided into two main subfields:


  1. Classical Mechanics: This branch of mechanics deals with the motion of macroscopic objects at speeds much lower than the speed of light. It is further divided into two subfields:
    • Statics: Statics deals with objects at rest or in a state of constant motion. It involves the study of forces acting on objects in equilibrium.
    • Dynamics: Dynamics deals with the motion of objects under the influence of forces. It includes the study of acceleration, velocity, and displacement of objects.
  2. Quantum Mechanics: Quantum mechanics is a branch of physics that deals with the behavior of very small particles such as atoms and subatomic particles. It describes the wave-particle duality of matter and the probabilistic nature of particle interactions.

In the context of human movement analysis, mechanics plays a crucial role in understanding how forces act on the human body to produce various types of motion, such as walking, running, jumping, and throwing. By applying principles of mechanics, researchers and practitioners can analyze and optimize human performance, prevent injuries, and design effective rehabilitation programs.

Key concepts in mechanics relevant to human movement analysis include:

  • Force: A force is a push or pull that can cause an object to accelerate, change direction, or deform. In human movement, forces generated by muscles, gravity, and external factors play a significant role in determining motion.
  • Torque: Torque, also known as the moment of force, is a measure of the tendency of a force to rotate an object around an axis. In human movement, torque is essential for understanding joint movements and the generation of rotational motion.
  • Kinematics: Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces causing the motion. It involves concepts such as position, velocity, and acceleration.
  • Kinetics: Kinetics is the branch of mechanics that deals with the forces causing motion. It includes the study of how forces interact with the mass and geometry of objects to produce motion.

By applying the principles of mechanics to human movement analysis, researchers and practitioners can gain insights into the biomechanical basis of human motion, optimize athletic performance, and improve the quality of life for individuals through tailored interventions and rehabilitation programs.

 

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