ICNCE 2026

Many real-life problems — from combinatorial optimization and constraint satisfaction to inference in energy-based AI models — share a common mathematical structure: finding configurations that minimize a complex, non-convex energy landscape. Boolean satisfiability SAT (or MaxSAT), canonical NP-complete (or NP-hard) problems, exemplify this class: the task is to find assignments of Boolean variables satisfying the maximum number of logical constraints. Despite their ubiquity, no efficient algorithm is known for these problems in the worst case. We show that this broad class of problems admits a unified treatment through continuous-time dynamical systems (CTDS) in the form of ordinary differential equations. Our analog solver operates as an energy-based reasoning machine; however, instead of performing static gradient descent, it generates a deterministic, history-weighted gradient flow on a dynamically evolving energy landscape. Unsatisfied constraints accumulate exponentially growing tension via auxiliary variables, continuously reshaping the effective landscape and enabling escape from spurious local minima — a mechanism fundamentally distinct from both standard gradient descent and stochastic Langevin dynamics. Hardness in this framework appears as transient chaos governed by a chaotic repeller. We will briefly discuss the mathematical foundations of analog, continuous-variable computation in continuous time — its advantages, and limitations. We then present CTDS variants, some inspired by experimentally observed many-body neuronal interactions and discuss implications for energy-based AI reasoning, where the same history-weighted dynamics may offer a principled alternative to stochastic sampling for inference in learned energy landscapes.