How do physical phenomena give rise to formal mathematical systems?

Question

From Physics-Informed Neural Networks to Nature-Inspired Mathematics: How Do Observational Disciplines and Physical Phenomena Generate Formal Systems? How does the conceptual progression from Physics-Informed Neural Networks (SciML) to Physical Mathematics, Nature-Inspired Computing, and Mathematical Naturalism define the academic relationship between empirical observation and formal deductive mathematical systems?

Why it matters

It underpins why scientific ML works, and where learning from data meets deduction.

The full research is in the research notes.