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. 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%. Energy landscapes are inherent in many stochastic dynamical systems in nature, such as the potential energy surface of protein conformations (Norn et al., 2021), the fitness landscape of species evolution (Papkou et al., 2023; Poelwijk et al., 2007), and the fractal energy landscapes of soft glassy materials. The evolution of these systems can be modeled as particles moving on the landscape under the combined effect of energy-driven drift and noise-induced diffusion. When multiple low-energy regions exist in the landscape, the combined effect of the energy gradient and noise induces high-frequency movement within individual regions and low-frequency transitions between different regions (Lin et al., 2024). In this context, energy landscapes have been applied to guide the generation of stable molecular structures (No e et al., 2019) and direct the evolution of proteins (Packer & Liu, 2015; Greenbury et al., 2022), and more recently, they have been incorporated as physical knowledge into deep learning for predicting system evolution (Guan et al., 2024; Wang et al., 2024b; Ding et al., 2024). Couce et al. (2024) cultivate 50,000 generations of bacteria to measure the fitness effects of mutations, while Sarkisyan et al. (2016) measure tens of thousands of luminescent protein genotypic sequences to construct the functional landscape. These manual experimental approaches are not only costly but also heavily reliant on expert knowledge. With the success of deep learning in numerous disciplines (Jumper et al., 2021; Han et al., 2023; Wang et al., 2023; Chen et al., 2024), several deep learning models have been proposed to estimate energy or equivalent quantities based on molecular spatial structures (Zhang et al., 2018), species genotypes (Tonner et al., 2022), or population compositions (Skwara et al., 2023). These methods still require high-cost annotations to provide supervisory signals for energy, which limits their practicality.

For example, cells are described as complex networks of chemicals linked by chemical reactions [7]; ecological networks link populations together through food chains [64]; and the World Wide Web is a vast virtual network of web pages and hyperlinks [47]. These complex networks are just a few of many examples. The local microscopic behavior of these complex networks often shows disorder. However, at the macroscopic scale, they show simple and even symmetrical structures. In order to understand the transition and evolution of complex systems from microscopic disorder to macroscopic order, current complex network studies mainly fall into the following paradigm: the combination of graph theory and statistical mechanics [3]. They construct the core principle of complex network science, that is, simple random rules and network dynamics together drive the emergence of non-trivial topological structures. Early works mainly focused on the topology of the interactions between the components, i.e., the birth-death process of edges on the graph. The two representative works, the Watts-Strogatz (WS) model and the scale-free model [11, 252], embody this principle and successfully generate graphs that approach real-world complex networks with high clustering coefficients and small average paths or power-law degree distribution. Despite their success in certain domains [17, 221, 222, 235], they do not provide a way to model the dynamics of the nodes, i.e., the change in the node's features.

Diffusion models, a family of generative models based on deep learning, have become increasingly prominent in cutting-edge machine learning research. With a distinguished performance in generating samples that resemble the observed data, diffusion models are widely used in image, video, and text synthesis nowadays. In recent years, the concept of diffusion has been extended to time series applications, and many powerful models have been developed. Considering the deficiency of a methodical summary and discourse on these models, we provide this survey as an elementary resource for new researchers in this area and also an inspiration to motivate future research. For better understanding, we include an introduction about the basics of diffusion models. Except for this, we primarily focus on diffusion-based methods for time series forecasting, imputation, and generation, and present them respectively in three individual sections. We also compare different methods for the same application and highlight their connections if applicable. Lastly, we conclude the common limitation of diffusion-based methods and highlight potential future research directions.