A minimal experiment showing that extrapolation is governed by structure—not data volume.
- A linear model fails immediately outside the training domain
- A resonance-based model preserves structure and continues to track the true function
Extrapolation is not a data problem.
It is a structure problem.
Traditional machine learning models rely on:
- large datasets
- interpolation
- optimization
However, this paradigm breaks down when:
- data is limited
- the domain is unseen
- the structure is nonlinear
This repository demonstrates a minimal but fundamental result:
When structural resonance is encoded, extrapolation becomes possible—even under low-data conditions.
[ y = \sin(x) + 0.3 \sin(3x) ]
[ x \in [0,6] ]
[ x \in [6,10] ]
Linear regression:
[ y = ax + b ]
- captures only linear trend
- fails immediately outside training domain
Feature transformation:
[ \phi(x) = [ \sin x,\cos x, \sin 2x,\cos 2x, \sin 3x,\cos 3x ] ]
- encodes periodic structure
- preserves behavior beyond training region
- Baseline diverges outside training domain
- Resonance model follows the true function
- Baseline accumulates large error
- Resonance model maintains low, stable error
pip install numpy matplotlib scikit-learn
python experiment.pyfigure1.png– behavior comparisonfigure2.png– error comparison- reproducible minimal experiment
This experiment is grounded in the I2OS principle:
[ I = (\nabla M) \otimes R ]
Where:
- ( \nabla M ): structural gradient
- ( R ): resonance
Meaning:
Intelligence = structure × resonance
Baseline:
Data fitting → interpolation
Resonance:
Structure encoding → extrapolation
This work suggests a shift in AI design:
From:
Data-driven optimization
To:
Structure-driven generalization
This direction connects:
- Quantum Machine Learning
- Low-data scientific modeling
- Structure-based intelligence (I2OS)
Full paper:
Masayuki Ando
MIT License