Behavioural Partitions: Evolving Piecewise Laws from Data

Many empirical laws feature a symbolic formula that only holds within some region of the input space. Learning such laws from data is especially challenging, because different formulas apply in each region and the regions themselves are unknown. The challenge is to learn both at the same time: the regions and the formula that holds within each of them. We propose BePart, a model that learns a symbolic representation of the regions and of the law governing each region. To this end, BePart searches for a set of gates, one for each region, compared by an argmax, so that every input is routed to exactly one symbolic formula, the law of its region. In contrast to other mixture-of-experts approaches, BePart finds a symbolic representation of both: the gates and the law of each region. BePart evolves whole partitions by a genetic algorithm under Pareto selection on error and complexity. On a new suite of 38 regime problems with known partitions and laws, spanning synthetic recipes, piecewise physics and hybrid dynamical systems, BePart recovers the partition (ARI 0.91) and the complete piecewise law in 53% of the runs, where model trees and piecewise-affine regressors find isolated affine laws but never the partition, mixtures of experts recover the partition only roughly and no laws, and the best single-expression system with conditional primitives reaches an ARI of 0.63 and 17% piecewise law recovery. With a single region, the task becomes classical symbolic regression. BePart shows its strength in this case as well: under the SRBench evaluation cap it attains the best median test R2 and mean rank against the 21 published methods of the SRBench black-box suite, and on the SRBench ground-truth problems it recovers more laws symbolically than any published method at noise levels up to 1%.