Neural Percolation Model (NPM)

An Engineering Analogy Framework for Neural Network Information Propagation and Capability Emergence

With companion article series "AI Is Not Mysterious" (11 articles in Chinese), introducing four original concepts: NPM / EVM / Three-Layer Isomorphism / Learning from AI as a Teacher

Author: Tiexin Ding (丁铁新) · NeuralCAE · Independent Researcher, China

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Overview

NPM maps the physics of Pore Network Models (PNM) — widely used in reservoir engineering and materials science — onto neural network information propagation. Starting from a single conservation principle (steady-state nodal balance), the framework derives a unified master equation:

a_out = W_eff · a_in

where W_eff takes three forms covering all major architectures: feedforward networks (fixed weights), GNNs (graph Laplacian, exact PNM analog, verified with zero error), and Transformers (softmax attention as flow conservation).

Four Original Concepts

Concept	Nature	First Proposed	Core Content
NPM (Neural Percolation Model)	Mathematical framework	Article 2	GNN message passing = PNM nodal conservation, 4 isomorphisms, verification error = 0
EVM (Engineering Verification Model)	Application roadmap	Article 7	5-stage verification roadmap for Physical AI, V-model mapping
Three-Layer Isomorphism	Theoretical discovery	Article 8	Natural percolation / AI capability emergence / human cognitive emergence follow the same mathematics
Learning from AI as a Teacher	Methodology	Article 9	Lower your stance, learn AI by using AI; judgment is the essence of intelligence

Article Series: AI Is Not Mysterious — A CAE Engineer's Perspective

Eleven articles (in Chinese) systematically analyze AI from first principles using the language of CAE engineers. Full text available in 6_Articles_CN/ directory, also published on Zhihu and WeChat (NeuralCAE).

#	Title	Core Content
1	What Is AI — First Principles Reasoning	AI is a multimodal projection of nature's interaction processes
2	What Neural Networks Do — NPM	Projection forms through percolation connectivity
3	AI Is Not Mysterious — When Percolation Meets NN	GNN = PNM mathematical verification, error = 0
4	Neural Networks and CAE — A Panoramic Map	Four isomorphisms, shared mathematical foundation
5	Current AI Is Not Perfect — From Demystification to Debugging	MeshGraphNet hands-on, nine gaps all pointing to verification
6	Where Is the Road — Engineering World Models	Data + compute + verification: three legs
7	Physical AI Roadmap — Engineering Verification Model	EVM 5-stage roadmap, V-model mapping
8	Cognitive Percolation	Creative process documentary, three-layer isomorphism discovery
9	Insights Beyond Reasoning	Change vs. permanence in the AI era, learning from AI
10	Creator Interview	Five voices answer independently, human-AI collaboration in practice
11	Gratitude	To mentors, family, open-source community, and readers

Key Contributions

1. Unified master equation — three architecture instantiations from one conservation principle
2. Multi-threshold field view — emergence as a continuous capability spectrum, not a single phase transition
3. Participation Ratio (PR) method — quantifying data distribution breadth σ from covariance spectra
4. 17 LLM phenomena explained — under one physical mechanism, with 3 falsifiable predictions
5. Engineering framework positioning — in the tradition of Darcy's Law and the Reynolds number

Repository Structure

neural-percolation-model/
├── 1_Paper_CN/          # Chinese paper (xelatex + xeCJK)
├── 2_Paper_EN/          # English paper (pdflatex, arXiv-ready)
├── 3_Figures_CN/        # Chinese figure PNGs
├── 4_Figures_EN/        # English figure PNGs
├── 5_codes/             # Python implementation
│   ├── npm_core.py          # Core NPM equations
│   ├── sigma_estimation.py  # PR method for σ estimation
│   ├── npm_validation.py    # A-layer numerical checks
│   ├── npm_scaling_check.py # B-layer vs Kaplan/Chinchilla
│   ├── npm_dimensionless.py # Cn empirical analysis (14 models)
│   ├── npm_pore_analysis.py # MIS-based pore analysis
│   └── npm_param_fit.py     # B-layer parameter fitting
├── 6_Articles_CN/       # Chinese article series (9 articles)
│   ├── 01_AI_First_Principles.md
│   ├── 02_Neural_Percolation_Model.md
│   ├── 03_Percolation_Meets_NN.md
│   ├── 04_Panorama_Map.md
│   ├── 05_Demystify_to_Debug.md
│   ├── 06_Where_Is_The_Road.md
│   ├── 07_EVM_Roadmap.md
│   ├── 08_Cognitive_Percolation.md
│   ├── 09_Beyond_Reasoning.md
│   ├── 10_Creator_Interview.md
│   └── 11_Gratitude.md
├── README.md            # This file (English)
└── README_CN.md         # Chinese README

Compilation & Running

English paper (arXiv submission):

cd 2_Paper_EN/ && pdflatex npm_arxiv_paper.tex && pdflatex npm_arxiv_paper.tex

Chinese paper:

cd 1_Paper_CN/ && xelatex npm_paper_CN.tex && xelatex npm_paper_CN.tex

Python code:

cd 5_codes/
python -m venv .venv && source .venv/bin/activate
pip install -r requirements.txt
python npm_core.py

MIS Pore Analysis (Core Experiments)

Inspired by PoreSpy's MIS method, we mapped PNM pore analysis tools onto NN weight space. Three key findings:

1. NN weight distributions share shape with PNM pore-size distributions but belong to different families — rank correlation r=0.98 after scale removal; NNs develop more extreme heterogeneity than natural rock
2. Training = percolation channel optimization — GPT-2 training increases large-pore fraction from 0% to 41%, selectively amplifying dominant pathways
3. Three-phase training transition (Pythia-70m) — Pressurization → Breakthrough → Reorganization; CV surges from 0.82 to 1.51, corresponding to giant connected component formation in percolation theory

Current Status

Component	Status
Unified master equation (A-layer)	✓ Numerically verified, error = 0
GNN–PNM equivalence	✓ Exact mathematical equivalence
B-layer parameters	✓ Fitted: α=0.57, β=0.068, δ=2.76, k=68.5 (Spearman=0.70)
MIS pore analysis	✓ 6 experiments on GPT-2 & Pythia-70m
Training phase transition	✓ Three-phase pattern identified
Article series (Chinese)	✓ All 11 articles published
Capability emergence ordering	△ Predicted, awaiting BIG-Bench validation

Collaboration

Open invitation for collaboration: researchers with large model checkpoints, theorists, engineers, and experimentalists are all welcome.

Contact: GitHub Issues or Pull Request.

Citation

@article{ding2026npm,
  title={Neural Percolation Model (NPM): An Engineering Analogy Framework
         for Neural Network Information Propagation and Capability Emergence},
  author={Ding, Tiexin},
  journal={arXiv preprint},
  year={2026}
}

Acknowledgments

The author used Claude (Anthropic) as a writing assistance tool. All theoretical content, conceptual framework, equations, and conclusions are the sole intellectual contribution of the author.

Links

• Zenodo Archive: https://zenodo.org/records/19209722
• arXiv Paper: [to be added after acceptance]
• Zhihu Column: AI并不神秘——一个CAE工程师的视角
• WeChat: NeuralCAE
• GitHub: https://github.com/tiexinding/neural-percolation-model

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NeuralCAE · Tiexin Ding 20 years of simulation practice · Original concepts: NPM / EVM / Three-Layer Isomorphism / Learning from AI