Energy landscapes play a crucial role in shaping dynamics of many real-world complex systems. System evolution is often modeled as particles moving on a landscape under the combined effect of energy-driven drift and noise-induced diffusion, where the energy governs the long-term motion of the particles. Estimating the energy landscape of a system has been a longstanding interdisciplinary challenge, hindered by the high operational costs or the difficulty of obtaining supervisory signals. Therefore, the question of how to infer the energy landscape in the absence of true energy values is critical. In this paper, we propose a physics-informed self-supervised learning method to learn the energy landscape from the evolution trajectories of the system. It first maps the system state from the observation space to a discrete landscape space by an adaptive codebook, and then explicitly integrates energy into the graph neural Fokker-Planck equation, enabling the joint learning of energy estimation and evolution prediction. Experimental results across interdisciplinary systems demonstrate that our estimated energy has a correlation coefficient above 0.9 with the ground truth, and evolution prediction accuracy exceeds the baseline by an average of 17.65\%. The code is available at github.com/tsinghua-fib-lab/PESLA.

翻译：能量景观在塑造众多现实世界复杂系统的动力学行为中起着至关重要的作用。系统演化通常被建模为粒子在能量景观上的运动，该运动受能量驱动的漂移和噪声诱导的扩散共同影响，其中能量主导着粒子的长期运动。估算系统的能量景观一直是一个长期的跨学科挑战，受到高昂操作成本或难以获取监督信号的阻碍。因此，如何在缺乏真实能量值的情况下推断能量景观是一个关键问题。本文提出了一种基于物理信息的自监督学习方法，从系统的演化轨迹中学习能量景观。该方法首先通过一个自适应码本将系统状态从观测空间映射到离散的景观空间，然后明确地将能量整合到图神经Fokker-Planck方程中，从而实现能量估计与演化预测的联合学习。跨学科系统的实验结果表明，我们估计的能量与真实值之间的相关系数超过0.9，且演化预测精度平均超过基线17.65\%。代码可在 github.com/tsinghua-fib-lab/PESLA 获取。