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...
Finite growth theory is a
mathematical framework used to describe the growth and deformation of
biological tissues over time. In the context of brain development and cortical
folding, finite growth theory plays a crucial role in understanding how the
brain tissue evolves and changes shape during development. Here are key points
related to finite growth theory:
1. Definition: Finite growth theory considers biological tissues as
deformable continua that undergo growth and remodeling processes. It accounts
for changes in tissue shape, size, and structure over time due to cellular
activities such as proliferation, differentiation, and migration.
2. Continuum Mechanics: Finite growth theory is often formulated within the
framework of continuum mechanics, which describes the behavior of continuous
media subject to external forces and deformations. By applying principles of
continuum mechanics, researchers can model the growth and deformation of
tissues at different length scales.
3. Growth Kinematics: In finite growth theory, growth kinematics describe how
tissue elements deform and change size as a result of growth processes. This
includes defining how growth rates vary spatially and temporally within the
tissue, influencing its overall morphology.
4. Material Growth: The concept of material growth in finite growth theory
refers to the changes in tissue properties such as stiffness, density, and
composition as the tissue grows. Material growth is essential for capturing the
evolving mechanical behavior of tissues undergoing growth and remodeling.
5. Growth Laws: Finite growth theory often incorporates growth laws that
govern how tissue properties evolve in response to growth stimuli. These growth
laws can be based on experimental observations or biological principles,
providing a mathematical framework for simulating tissue growth and
deformation.
6. Computational Modeling: Finite growth theory is frequently implemented using
computational models, such as finite element analysis, to simulate the growth
and deformation of tissues. Computational simulations allow researchers to
predict how tissues will deform under different growth conditions and study the
underlying mechanisms driving tissue morphogenesis.
7. Biological Applications: In the context of brain
development, finite growth theory helps researchers understand the mechanisms
underlying cortical folding, gyrification, and brain morphogenesis. By
incorporating growth dynamics into mathematical models, researchers can simulate
the complex processes that shape the structure of the developing brain.
In summary, finite growth theory
provides a mathematical framework for studying the growth and deformation of
biological tissues, including the brain, and plays a key role in elucidating
the mechanisms driving tissue morphogenesis during development.
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