Summary
Many methods to discover scientific equations assume only one rule governs a system, but this paper shows that often multiple rules work together. The authors introduce a new approach called Physical Law Ecology that first figures out how many different rules (mechanisms) exist in a system, then finds those rules. They tested this method on systems as different as galaxy movement and boiling liquids, and found it could identify multiple governing laws that explain observations better than single-law models. This approach could help scientists better understand complex systems by mapping out all the rules involved, rather than trying to find just one.
governing lawmechanismsymbolic regressionBayesian information criterionmulti-mechanism systemphysical law ecologydata-driven discoverymonotonicity constraintstopologyNewtonian gravity
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.