Discovering physical laws from data is a fundamental challenge in scientific research, particularly when high-quality data are scarce or costly to obtain. Traditional methods for identifying dynamical systems often struggle with noise sensitivity, inefficiency in data usage, and the inability to quantify uncertainty effectively. To address these challenges, we propose Langevin-Assisted Active Physical Discovery (LAPD), a Bayesian framework that integrates replica-exchange stochastic gradient Langevin Monte Carlo to simultaneously enable efficient system identification and robust uncertainty quantification (UQ). By balancing gradient-driven exploration in coefficient space and generating an ensemble of candidate models during exploitation, LAPD achieves reliable, uncertainty-aware identification with noisy data. In the face of data scarcity, the probabilistic foundation of LAPD further promotes the integration of active learning (AL) via a hybrid uncertainty-space-filling acquisition function. This strategy sequentially selects informative data to reduce data collection costs while maintaining accuracy. We evaluate LAPD on diverse nonlinear systems such as the Lotka-Volterra, Lorenz, Burgers, and Convection-Diffusion equations, demonstrating its robustness with noisy and limited data as well as superior uncertainty calibration compared to existing methods. The AL extension reduces the required measurements by around 60% for the Lotka-Volterra system and by around 40% for Burgers' equation compared to random data sampling, highlighting its potential for resource-constrained experiments. Our framework establishes a scalable, uncertainty-aware methodology for data-efficient discovery of dynamical systems, with broad applicability to problems where high-fidelity data acquisition is prohibitively expensive.

翻译：从数据中发现物理定律是科学研究中的一项基本挑战，尤其是在高质量数据稀缺或获取成本高昂的情况下。识别动力系统的传统方法通常面临噪声敏感性、数据利用效率低下以及无法有效量化不确定性等问题。为解决这些挑战，我们提出了朗之万辅助主动物理发现（LAPD），这是一个贝叶斯框架，它整合了副本交换随机梯度朗之万蒙特卡洛方法，以同时实现高效的系统辨识和稳健的不确定性量化。通过在系数空间中平衡梯度驱动的探索与在利用阶段生成候选模型集合，LAPD能够利用含噪数据实现可靠且具有不确定性感知的辨识。面对数据稀缺，LAPD的概率基础进一步促进了通过混合不确定性-空间填充获取函数集成主动学习。该策略顺序选择信息丰富的数据，以降低数据收集成本，同时保持准确性。我们在多种非线性系统（如Lotka-Volterra、Lorenz、Burgers和对流-扩散方程）上评估了LAPD，证明了其在含噪和有限数据下的鲁棒性，以及与现有方法相比更优的不确定性校准能力。与随机数据采样相比，主动学习扩展将Lotka-Volterra系统所需的测量减少了约60%，将Burgers方程所需的测量减少了约40%，突显了其在资源受限实验中的潜力。我们的框架建立了一种可扩展的、具有不确定性感知的方法，用于数据高效地发现动力系统，对于高保真数据获取成本过高的问题具有广泛的适用性。