Machine learning models offer the capability to forecast future energy production or consumption and infer essential unknown variables from existing data. However, legal and policy constraints within specific energy sectors render the data sensitive, presenting technical hurdles in utilizing data from diverse sources. Therefore, we propose adopting a Swarm Learning (SL) scheme, which replaces the centralized server with a blockchain-based distributed network to address the security and privacy issues inherent in Federated Learning (FL)'s centralized architecture. Within this distributed Collaborative Learning framework, each participating organization governs nodes for inter-organizational communication. Devices from various organizations utilize smart contracts for parameter uploading and retrieval. Consensus mechanism ensures distributed consistency throughout the learning process, guarantees the transparent trustworthiness and immutability of parameters on-chain. The efficacy of the proposed framework is substantiated across three real-world energy series modeling scenarios with superior performance compared to Local Learning approaches, simultaneously emphasizing enhanced data security and privacy over Centralized Learning and FL method. Notably, as the number of data volume and the count of local epochs increases within a threshold, there is an improvement in model performance accompanied by a reduction in the variance of performance errors. Consequently, this leads to an increased stability and reliability in the outcomes produced by the model.

Equation discovery is aimed at directly extracting physical laws from data and has emerged as a pivotal research domain. Previous methods based on symbolic mathematics have achieved substantial advancements, but often require the design of implementation of complex algorithms. In this paper, we introduce a new framework that utilizes natural language-based prompts to guide large language models (LLMs) in automatically mining governing equations from data. Specifically, we first utilize the generation capability of LLMs to generate diverse equations in string form, and then evaluate the generated equations based on observations. In the optimization phase, we propose two alternately iterated strategies to optimize generated equations collaboratively. The first strategy is to take LLMs as a black-box optimizer and achieve equation self-improvement based on historical samples and their performance. The second strategy is to instruct LLMs to perform evolutionary operators for global search. Experiments are extensively conducted on both partial differential equations and ordinary differential equations. Results demonstrate that our framework can discover effective equations to reveal the underlying physical laws under various nonlinear dynamic systems. Further comparisons are made with state-of-the-art models, demonstrating good stability and usability. Our framework substantially lowers the barriers to learning and applying equation discovery techniques, demonstrating the application potential of LLMs in the field of knowledge discovery.

Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge, especially when encountering noisy observations and no prior knowledge available. This study proposes R-DISCOVER, a framework designed to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with superior fits are utilized to iteratively optimize the generator via the RL method and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.