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.

This, under the Theory of Entropicity(ToE), we have that:

• Entropy is a real field, not just a probability measure.

• Every system has internal entropy tied to its state configuration.

• The entropy flows during interaction — and governs irreversibility, constraint, and evolution.

• Nothing ever happens instantly — because every change must pay an entropic cost.

This applies to neural weights, activation states, and even symbolic computation.

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 [LA] 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.

The above is the Obidi Learning Action [OLA] in the Theory of Entropicity(ToE) for building Artificial Intelligence Systems.

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. What Is Learning?

From the Theory of Entropicity(ToE), in any physical or computational system, we describelearning as:

The internal reconfiguration of a system in response to input stimuli, which increases the predictive coherence or stability of the system under future interactions.

Formally, in deep learning, we define this as the change in model parameters to reduce some loss or improve generalization.

But that’s still syntactic. So, the Theory of Entropicity(ToE) demands of us to ask: what physically drives this change?

First, let us give the following foundational statement on Learning from the principles of the Theory of Entropicity(ToE):

Learning is a change in internal entropy states of a system [a change in Self Referential Entropy[SRE]] towards a given [internal or external] reference entropy.

The above postulate lays the foundation of Learning in the Theory of Entropicity(ToE).

9. Why Learning Proceeds Along Entropy Gradients

Analogy with Heat and Thermodynamics:

In classical thermodynamics, we know that:

Heat flows from high temperature to low temperature.

This means that the rate of heat flow is proportional to the temperature gradient.

Equivalently, then, in ToE we can state as follows:

Learning proceeds from disordered uncertainty toward structured knowledge.

This corresponds to entropy flowing and reorganizing across the network, from the first layer (input layer) through the hidden layer(s) and then to the last layer (output layer).

Every learning step is then a redistribution of entropy within the layers of any given network.

10. Learning as Entropic Compensation

Now, imagine each weight is trapped in an entropy field.

To update, the system must overcome resistance in the entropy landscape.

That cost is paid through entropy flow.

Thus, in the Theory of Entropicity(ToE), if no entropy is generated or released, no learning occurs.

This leads to the formulation:

[math]\displaystyle{ \frac{d\phi_i}{dt} = \frac{1}{\mathcal{R}_i} \cdot \frac{\partial \Lambda}{\partial \phi_i} }[/math]

Where:

• [math]\displaystyle{ \frac{1}{\mathcal{R}_i} }[/math] = resistance to entropy flow (inertia of the weight)

• [math]\displaystyle{ \frac{\partial \Lambda}{\partial \phi_i} }[/math] = entropic reward for update

Thus, this equation tells us that:

The rate of change of a learning parameter is proportional to the entropy flow it induces, and inversely proportional to the entropic resistance of that parameter.

This is an entropy-constrained update rule, replacing traditional cost-gradient descent methods with a thermodynamic-like field law derived from the Theory of Entropicity(ToE).

Analogy with Newton's Law

The above equation is similar to Newton’s Second Law:

[math]\displaystyle{ F=ma }[/math]

Analogy with Ohm's Law

The above equation is also similar to Ohm’s Law:

[math]\displaystyle{ V=IR }[/math]

But here:

Entropy flow = Entropic resistance × Update speed

Or rearranged:

[math]\displaystyle{ \text{Update Speed} = \frac{\text{Entropy Flow}}{\text{Resistance}} }[/math]

This Learning framework from the Theory of Entropicity(ToE) implies:

1. Learning is not instantaneous — it’s constrained by the entropy environment.

2. Every weight update is entropically “paid for.”

3. The model optimally redistributes entropy to increase learning coherence under irreversibility.

11. Claude Shannon, Theory of Entropicity(ToE), and Entropic Learning

Claude Shannon showed that:

"Information is the reduction of uncertainty, quantified by entropy."

In ToE, we reverse the logic:

"Learning is the redistribution of entropy, constrained by irreversibility."

That is, learning is not about reducing entropy in a passive sense — it is about activating entropy flow in a structured, irreversible, and directional way.

Learning is now seen to be proportional to entropy because in the Theory of Entropicity(ToE), entropy is the driving physical quantity that enforces updates, mediates information, and constrains evolution in Artificial Intelligence(AI) systems — just like energy [force] drives motion and voltage drives current in classical physics [Newton and Maxwell].

12. Key Postulates for Entropic Learning

With all of the foregoing, we can now make the following axiomatic statements on Learning allowed by the Theory of Entropicity(ToE):

1. Irreversibility Principle:

Learning is irreversible due to entropy production. This irreversibility defines a directional “arrow of learning”.

2. Entropic Delay Principle:

Every update requires a minimal time interval Δt defined by the local entropy field, not just an arbitrary learning rate.

3. Entropic Cost Principle:

Each parameter update must overcome an entropy gradient barrier. The sharper the entropy gradient, the higher the “learning resistance”.

4. Maximum Entropic Flow Principle:

A learning system evolves such that it maximizes the entropy flow from misalignment (error) toward coherence (understanding), under constraint.

13. 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).