Discover physical concepts and equations with machine learning (2412.12161v2)

Abstract: Machine learning can uncover physical concepts or physical equations when prior knowledge from the other is available. However, these two aspects are often intertwined and cannot be discovered independently. We extend SciNet, which is a neural network architecture that simulates the human physical reasoning process for physics discovery, by proposing a model that combines Variational Autoencoders (VAE) with Neural Ordinary Differential Equations (Neural ODEs). This allows us to simultaneously discover physical concepts and governing equations from simulated experimental data across various physical systems. We apply the model to several examples inspired by the history of physics, including Copernicus' heliocentrism, Newton's law of gravity, Schr\"odinger's wave mechanics, and Pauli's spin-magnetic formulation. The results demonstrate that the correct physical theories can emerge in the neural network.

• The paper presents an extended SciNet model integrating VAEs with Neural ODEs to uncover physical concepts and governing equations from data.
• The approach compresses observational data into latent representations while extracting dynamic laws, rediscovering classical and quantum principles.
• Validation through case studies, including heliocentric models and quantum equations, demonstrates the method’s ability to autonomously reveal known physical laws.

Discovering Physical Concepts and Equations with Machine Learning

The paper under consideration presents an innovative extension of the SciNet neural network, aimed at automating the process of discovering physical concepts and governing equations directly from experimental data. The framework integrates Variational Autoencoders (VAEs) with Neural Ordinary Differential Equations (Neural ODEs), exhibiting robustness across classical and quantum physics scenarios.

The proposed model addresses a long-standing challenge in theoretical physics: the often intricate coupling between physical concepts and governing equations. Traditional methods tend to treat these elements in isolation, missing the potential insights accessible through simultaneous analysis. By incorporating Neural ODEs, the extended SciNet model overcomes the limitation of uniform evolution inherent in its predecessor, achieving greater generalizability and applicability in diverse physical systems.

Methodology and Technical Implementation

The approach mimics the human physicist's reasoning process. Observational data, considered as evolving over time or space, are compressed into latent representations by VAEs, embodying the core physical concepts. These latent representations evolve according to differential equations, discovered through Neural ODEs, which infer the governing dynamical equations from continuous-time sequential data.

The encoder-decoder structure within VAEs effectively maps observational data to a multidimensional latent space and back. Key innovations in the methodology include employing KL divergence as a regularization term to reduce redundancy in latent representations, thereby ensuring that they capture the minimum essential features of the data.

Crucially, the model does not rely on strict prior knowledge about the system under paper. The order of differential equations and the number of latent space dimensions are treated as hyperparameters. This objectivity allows the model to be adapted through empirical experiments, optimizing these parameters to uncover the underlying physics.

Key Results and Case Studies

The authors validate their model through four illustrative examples fundamental to physics:

1. Copernicus' Heliocentric Solar System: The model autonomously identifies heliocentric angles and demonstrates their constant speed evolution, effectively rediscovering Copernican principles from geocentric angle data.
2. Newton's Law of Universal Gravitation: Utilizing trajectory data of celestial bodies, the model discerns the concepts of radial distance and velocity, and derives a second-order governing equation akin to the classical gravitational law.
3. Quantum Wave Function and Schrödinger Equation: The methodology uncovers the quantum wave function and Schrödinger-like equations without prior explicit guidance, showcasing its potential in quantum system modeling.
4. Spin-1/2 with the Pauli Equation: Even in scenarios where spin-related phenomena are obscured (due to a uniform magnetic field), the model reconstructs the Pauli equation, demonstrating the ability to identify fundamental quantum properties from limited experimental signatures.

Each case demonstrated the model's competence in extracting meaningful latent representations and governing dynamical equations, aligning the neural network's outputs with well-established physical theories.

Implications and Future Directions

This research underscores the potential of machine learning frameworks to serve as robust tools in extracting scientific knowledge, challenging current methodologies that heavily depend on human intuition and domain-specific expertise. The success in reconstructing known physical laws indicates a new avenue for scientific discovery, wherein AI augments traditional approaches by offering alternative pathways to understanding complex systems.

Future developments could focus on extending the model to include partial differential equations, broadening its scope to address a wider range of complex physical phenomena. Enhancements in the model architecture, questioning mechanisms, and use of symbolic regression to further refine equation discovery could result in AI systems that not only identify but also predict novel physical insights.

While challenges such as the "curse of length" in Neural ODEs and long-time dynamics remain, addressing these through methods like multiple shooting or integrating more scalable neural network architectures promises continued advancements. Overall, this work propels AI as an invaluable ally in the physicist's toolkit, potentially reshaping the landscape of scientific inquiry.