MFA I: Physics-informed neural networks

How do galaxies form and grow? Computer simulations of galaxy formation must capture processes ranging from single exploding stars to structures millions of light-years across – far too wide a range for any computer to resolve directly. This project develops a new mathematical method that combines classical physics equations with "physics-informed neural networks" (PINNs): artificial intelligence that is forced to respect the laws of nature, such as the conservation of energy and mass. The AI learns the fine details on small scales and feeds them back into the large-scale simulation. The project is genuinely multi-scale because it explicitly builds the bridge between small and large scales into the mathematics itself, and interdisciplinary because mathematicians specializing in optimization work hand in hand with astrophysicists – with methods that can later also be used in climate research and other fields.

When artificial neural networks learn from data, they can undergo abrupt changes – mathematically similar to water freezing into ice – and recent research shows these "phase transitions" reveal that the data contains structure on several distinct scales. This project turns that insight into a tool: neural networks as "scale detectors" for real-world data. The test case is the Earth's water cycle, where processes range from raindrops seeping into soil to entire river catchments responding to climate over decades – a range no single-scale theory can describe. By training networks on large hydrological datasets and analyzing where their learning jumps, the team aims to identify which physical processes matter on which scale. The project is multi-scale in both senses – the scales hidden in the data and those inside the network – and interdisciplinary, uniting the physics of machine learning, information theory, and hydrology, with links to astrophysics.