Physical law ecology finds multiple governing laws in complex systems
Physical Law Ecology: mapping multi-mechanism ecologies as the zeroth step of data-driven scientific discovery
Symbolic Computation
Summary
Many scientific methods assume complex systems follow just one main rule, but this paper shows that often several rules work together. The authors introduce Physical Law Ecology, a way to figure out how many different rules are at play before trying to find the rules themselves. They tested this on four very different systems and in all cases found three main rules working together. This approach also improved predictions in engineering tasks and challenged some existing theories about how gravity works in galaxies.
What this means in practice
- •For engineering design teams: Improve predictions in materials and fluid systems by combining multiple physical laws for more accurate and interpretable models.
- •For astrophysics data analysts: Analyze galaxy data with multi-law models to better distinguish gravitational behaviors beyond single-law assumptions like MOND.
Authors
Xiongheng Bian, Xiangyu Cui, Ma Feng, Xiaoyan Shen
Abstract
Every data-driven equation discovery method assumes (implicitly and without verification) that the target system obeys a single governing law ($K{=}1$). Here we show that this assumption is the primary bottleneck limiting scientific discovery in multi-mechanism systems, and introduce Physical Law Ecology, a framework that makes $K^*$ (the number of coexisting independent mechanisms) itself the first quantity to be determined from data. The framework automatically mines a pool of topologically distinct candidate equations, constructs a continuous dominance weight field across parameter space, and discovers analytic evolution laws governing mechanism succession---with optional monotonicity constraints encoding irreversible physics. Across four unrelated systems (elastomer mechanics, pool boiling, galactic dynamics, and droplet evaporation), BIC consistently identifies $K^*{=}3$ independent governing topologies. Applied to 163 SPARC galaxies (3,269 spatially resolved measurements), the framework autonomously recovers three gravitational laws whose coexistence provides evidence against the single-universal-acceleration hypothesis of MOND ($p<10^{-34}$). In engineering applications, multi-law weighted prediction reduces error by 67-72\% over single-equation baselines while retaining full interpretability. By establishing the determination of $K^*$ as the zeroth step of scientific discovery-prior to and independent of equation search---this work opens a direction orthogonal to existing symbolic regression: not finding better equations, but mapping the ecology of mechanisms that govern complex systems.