Discovering Partial Differential Equations With Neural Cellular Automata

Area of Science:

• Artificial Intelligence
• Computational Science
• Dynamical Systems

Background:

• Neural Cellular Automata (NCAs) use neural networks for update rules, inspired by reaction-diffusion PDEs for texture synthesis.
• Training NCAs involves discretizing spacetime and simulating dynamics, but it's unclear if they learn continuous dynamics or overfit discretization.

Purpose of the Study:

• Investigate NCA behavior at the limit of continuous spacetime discretization.
• Address the overfitting of training discretization in existing NCA models, particularly near the initial condition.
• Propose and validate a method to enable NCAs to learn continuous dynamics.

Main Methods:

• Studied NCA models as spacetime discretization approaches continuity.
• Proposed using uniform noise as the initial condition to mitigate overfitting.
• Demonstrated consistency across various granularities and robustness to noise and stochastic updates.

Main Results:

• Existing NCA models were found to overfit training discretizations, especially near the initial seed.
• The proposed uniform noise initialization preserves NCA dynamics consistency across different spacetime granularities.
• The improved NCA model exhibits robustness to stochastic updates and additive Gaussian noise.

Conclusions:

• NCAs can learn continuous dynamics, moving beyond overfitting discretization.
• The new approach enables continuous control over pattern formation speed and scale during synthesis.
• This work opens new avenues for studying NCAs as PDEs and dynamical systems.