Physics:Artificial Intelligence Formulated by the Theory of Entropicity(ToE)

Abstract

This introductory paper on the Theory of Entropicity(ToE) introduces the Entropy Learning Equation (ELE) as the foundational dynamical equation governing learning in artificial intelligence systems from the perspective of the Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi. ELE serves the same foundational role for intelligence as Shannon’s entropy does for information theory. Drawing on the Vuli–Ndlela Integral, the Entropic Time Limit (ETL), and entropy field dynamics, ELE provides a field-based, variational, and thermodynamic account of how learning occurs under entropy constraints. The equation captures the entropic flow of information across time, layers, and representational states in neural systems, both artificial and biological. Multiple formulations—static, dynamic, and field-theoretic—are presented along with their interpretations in deep learning contexts.

1. Motivation

Just as Claude Shannon derived the formula for information entropy to quantify uncertainty in communication, the Theory of Entropicity (ToE) aims to quantify learning as an entropic process. In this context:

Learning is modeled as the directed flow of entropy from the environment to the internal system.

Surprise, uncertainty, and error are expressions of entropy mismatch.

Intelligence is defined as the ability to steer entropy gradients under constraints such as time, noise, and energy.

The Entropy Learning Equation (ELE) formalizes this principle mathematically and physically.

2. Core Quantities

Let:

[math]\displaystyle{ \phi }[/math]: Internal state vector (e.g., weights, activations) of the AI.

[math]\displaystyle{ x \in \mathcal{X} }[/math]: Input data.

[math]\displaystyle{ y \in \mathcal{Y} }[/math]: Output or labels.

[math]\displaystyle{ P_\phi(y|x) }[/math]: Predictive distribution of the AI.

[math]\displaystyle{ S(\phi, x, y) }[/math]: Local entropy density.

[math]\displaystyle{ \mathcal{S}_{\text{irr}}[\phi] }[/math]: Irreversible entropy production during learning.

[math]\displaystyle{ \eta }[/math]: Entropic coupling constant (similar to a learning rate).

3. Entropy Learning Action

We define a Learning Action analogous to physical field actions:

[math]\displaystyle{ \mathcal{L}_{\text{ToE-AI}}[\phi] = \int_{\mathcal{T}} \left( - H[P_\phi(y|x)] + \eta \cdot \mathcal{S}_{\text{irr}}[\phi, x] \right) dt }[/math]

Where:

[math]\displaystyle{ H[P_\phi(y|x)] = - \sum_y P_\phi(y|x) \log P_\phi(y|x) }[/math] is the Shannon entropy of predictions.

The AI’s learning is governed by minimizing this action.

4. Entropy Learning Equation (ELE)

4.1 Static Form

[math]\displaystyle{ \frac{dS}{d\phi} = - \frac{\partial \mathcal{L}}{\partial \phi} }[/math]

The entropy change with respect to internal states is driven by the gradient of the entropic action.

4.2 Dynamic Form

[math]\displaystyle{ \frac{d\phi}{dt} = - \nabla_\phi H[P_\phi(y|x)] + \eta \cdot \nabla_\phi \mathcal{S}_{\text{irr}}[\phi, x] }[/math]

This governs the evolution of parameters in time during training.

4.3 Field-Theoretic Form

[math]\displaystyle{ \frac{\delta S[\phi(x)]}{\delta \phi(x)} = - \Box \phi(x) + \eta \cdot \nabla_\mu \nabla^\mu \mathcal{S}_{\text{irr}}(x) }[/math]

Where:

[math]\displaystyle{ \Box = \nabla^\mu \nabla_\mu }[/math] is the d'Alembertian.

This formulation supports large-scale, continuous AI systems.

5. Mapping to Deep Learning

In standard deep learning: [math]\displaystyle{ \phi_{t+1} = \phi_t - \alpha \cdot \nabla_\phi \mathcal{L}_{\text{cross-entropy}} }[/math]

Under ELE: [math]\displaystyle{ \phi_{t+1} = \phi_t - \eta \cdot \left[ \nabla_\phi H[P_\phi] - \nabla_\phi \mathcal{S}_{\text{irr}} \right] }[/math]

This version:

Includes irreversibility and entropy production explicitly.

Aligns learning with thermodynamic and entropic constraints.

Models intelligence as guided entropic flow.

6. Theoretical Significance

The Entropy Learning Equation is the first general-purpose learning equation grounded in the full field-theoretic framework of the Theory of Entropicity (ToE). It:

Explains learning delays via the Entropic Time Limit.

Allows for self-referential entropy computation for consciousness modeling.

Harmonizes deep learning with entropy conservation and production laws.

7. Toward Entropic Neural Networks

Using ELE, future architectures can:

Monitor entropy across layers ([math]\displaystyle{ S_\ell }[/math]).

Optimize entropic flow balance rather than only loss gradients.

Maintain entropic homeostasis, avoiding overfitting via entropy flattening.

Design psychentropic controllers that model their own uncertainty [math]\displaystyle{ S_{\text{self}} }[/math].

8. Future Work

This framework supports further development of:

EntropicNet: A neural architecture with embedded ELE constraints.

Simulations comparing ELE vs SGD in training convergence.

Thermodynamically-efficient learning systems for edge AI.

AI models of consciousness via Self-Referential Entropy (SRE).