BLADE: Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systems

Traditional methods for system discovery frequently struggle with efficient data usage and uncertainty quantification. Identifying the governing equations of complex dynamical systems from data presents a significant challenge in scientific discovery, especially when high-quality measurements are scarce and expensive to obtain. To overcome these limitations, we propose Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systems (BLADE), a novel Bayesian framework that combines replica-exchange stochastic gradient Langevin Monte Carlo with active learning. By balancing gradient-driven exploration and exploitation in coefficient space, BLADE provides probabilistic parameter estimation and principled uncertainty quantification. Faced with data scarcity, the probabilistic foundation of BLADE further facilitates the integration of active learning through a hybrid acquisition strategy that combines predictive uncertainty with space-filling design, enabling efficient selection of informative samples. Across benchmark systems, BLADE reduces measurement requirements by roughly 60% for Lotka-Volterra and 40% for Burgers' equation relative to random sampling, demonstrating substantial data-efficiency gains. These results highlight BLADE as a general uncertainty-aware framework for discovering interpretable dynamical systems, particularly valuable when high-fidelity data acquisition is prohibitively expensive.