Introduction: Tabula Geometrica—Let Neural Networks 'Invent' Spacetime Geometry from Scratch

Tabula Geometrica (Geometric Whiteboard) is a bold research project: Without giving neural networks any prior knowledge of spacetime (such as metric tensors, Einstein equations), through training only on raw observational data, it explores whether they can independently discover Minkowski intervals, light cones, gravitational wells, spacetime curvature, and even extra dimensions in the Kaluza-Klein theory. This project is not only a technical experiment but also an exploration of the essence of scientific discovery, challenging whether AI can rediscover the physical laws established by humans over centuries in a data-driven way.

Original author/maintainer: sumit7194 Source platform: GitHub Original link: https://github.com/sumit7194/tabula-geometrica Publication time: June 12, 2026

Core Question: Can Geometry Be 'Rediscovered' from Data?

Tabula Geometrica raises a philosophically profound question: If neural networks are not given any prior knowledge of spacetime (without mentioning metric tensors, Einstein equations, or concepts of 'distance' or 'time'), can they independently 'invent' the structural framework of spacetime geometry through raw observational data alone?

This is not just a machine learning experiment but also an exploration of the essence of scientific discovery—human physicists spent centuries building the framework of general relativity; can neural networks rediscover these laws in a data-driven way within hours of training?

Technical Approach: SciNet Architecture and Information Bottleneck

The project uses a SciNet-style architecture: an autoencoder with an information bottleneck (the encoder compresses high-dimensional observations into a low-dimensional latent space, and the decoder reconstructs the original data), forcing the network to learn the most compact and universal representation (often corresponding to physical laws).

Key techniques include:

• Adjacent observations: Only receiving local data of spatiotemporally neighboring events, requiring independent inference of global structure
• Unsupervised training: No 'correct answer' labels, only using reconstruction error as feedback
• Competition between geometry and force: Designing experiments to let the two interpretations compete and observing the network's preference

Experimental Results: From Minkowski Intervals to Kaluza-Klein Dimensions

The project is gradually built through the 'dimension ladder' method:

1. 1+1-dimensional spacetime: The network emerges with a structure similar to the Minkowski interval (ds²=-dt²+dx²), where the negative sign distinguishes timelike/lightlike/spacelike separations—core features of special relativity
2. Light cone structure: The internal representation shows light cones (dividing past, future, and unreachable regions), with causal structure naturally emerging
3. Gravitational wells and curvature: After introducing mass distribution, the network develops a gravitational well structure, and intrinsic curvature is confirmed via Gaussian curvature (correlation coefficient 0.99)
4. Kaluza-Klein miracle: When processing data on charged particle motion, the network spontaneously encodes electric charge as an extra internal dimension (correlation coefficient 0.9998), corresponding to the five-dimensional geometric unification of electromagnetism and gravity

Scientific Significance: Exploring the Boundaries of Data-Driven Discovery

The project touches on deep questions:

1. 'Necessity' of physical laws: If neural networks can rediscover the core structures of relativity/electromagnetism from data, does this mean these laws are the optimal compression method for describing data? This echoes Wheeler's 'It from Bit'
2. Machine learning version of the equivalence principle: The network cannot distinguish between real geometric curvature and apparent coordinate effects unless through global topological probes
3. Emergence and reduction: The 'blank slate' approach allows high-level concepts to emerge naturally, opening new paths for AI-assisted scientific discovery

Limitations and Future Directions

Current limitations:

• Simplified models: Mainly in 2+1 dimensions (two spatial + one time), which is far from the real 3+1-dimensional world
• Toy data: Observational data are synthetic, not from real physical experiments
• Interpretation challenges: The network's internal geometric structure requires intervention from human physicists for understanding

Future directions: Phases G and H will attempt to train a general network to handle all types of 'worlds' (flat/curved, with/without gravity) and observe the organization of its concept space

Conclusion: The Possibility of AI as a Theoretical Discoverer

Tabula Geometrica is a beautiful thought experiment combining physics, machine learning, and epistemology, showing that neural networks can not only fit data but also discover deep structures (corresponding to physical theories established by humans over centuries).

This leads to a fascinating possibility: Future AI may act as independent theoretical discoverers, extracting conceptual frameworks that humans have not yet thought of from raw observations. The project's rigorous attitude (every claim is checked against literature) makes it one of the noteworthy projects in the AI for Science field.

Keywords: Neural networks, spacetime geometry, general relativity, machine learning, emergence, Kaluza-Klein theory, scientific discovery, information bottleneck, unsupervised learning