Directing Open-Ended Evolution in Artificial Life via Multi-Scale Path Divergence

2026-06-12

AI Summary

This work addresses the lack of interpretable, theory-grounded drivers in open-ended evolution within artificial life, which often relies on opaque neural network–based metrics. The authors propose Multiscale Path Divergence (MSPD), an explicit scalar measure inspired by renormalization group theory, thereby introducing this physical framework into artificial life for the first time. MSPD enables gradient-free evolutionary guidance and quantifies temporal multiscale heterogeneity in population trajectories. Computed via windowed finite-resolution estimators, MSPD is applicable across diverse systems, including Flow-Lenia, Life-like cellular automata, and Particle Life++. Experiments demonstrate that MSPD-optimized parameters consistently outperform random baselines, enhancing complexity retention and revealing associations between high MSPD states and biological hallmarks such as dynamical instability and scale-dependent frustration, thus validating MSPD’s cross-system efficacy and its capacity to capture essential features of biological complexity.

Abstract

Open-ended evolution (OEE) in artificial life is typically driven by uninterpretable, black-box neural-network complexity metrics, leaving life-like systems disconnected from physical theories of complexity. We introduce MSPD (Multi-Scale Path Divergence, denoted DP ), a renormalization-group-inspired scalar that quantifies the temporal multiscale organization of heterogeneity in local transition laws. MSPD is defined at the population level as a functional of the realised trajectory and is computed as a windowed finite-resolution estimator, with consistency between the two stated as a proposition. The metric is an explicit formula and plays a dual role: as a gradient-free fitness function and as a post-hoc analytical lens on any simulation that exposes local transition laws. Empirically, MSPD-optimized parameters produce higher held-out complexity scores than matched random parameters from the same substrate. High-$H_{Delta_t}$ states correspond to states with higher instability to external interventions, so the metric tracks the biology of the underlying dynamics rather than noise. Higher MSPD corresponds to stronger scale-dependent frustration: high-complexity systems exhibit larger differences between the dynamics expressed at different spatial extents, linking MSPD directly to the frustration criterion of biological complexity in the sense of Vanchurin et al. [ 23 ]. The same protocol transfers beyond the primary Flow-Lenia substrate to Life-like cellular automata and Particle Life++, where C1, C2 and C5 all hold. A single explicit formula thus both directs open-ended evolution and provides a principled bridge to the physics of complexity that black-box drivers do not.

Problem

• open-ended evolution
• artificial life
• complexity metrics
• black-box models
• physical theories of complexity

Innovation

• Multi-Scale Path Divergence
• Open-Ended Evolution
• Renormalization Group
• Biological Complexity
• Local Transition Laws