Big data is transforming scientific progress by enabling the discovery of novel models, enhancing existing frameworks, and facilitating precise uncertainty quantification, while advancements in scientific machine learning complement this by providing powerful tools to solve inverse problems to identify the complex systems where traditional methods falter due to sparse or noisy data. We introduce two innovative neural operator frameworks tailored for discovering hidden physics and identifying unknown system parameters from sparse measurements. The first framework integrates a popular neural operator, DeepONet, and a physics-informed neural network to capture the relationship between sparse data and the underlying physics, enabling the accurate discovery of a family of governing equations. The second framework focuses on system parameter identification, leveraging a DeepONet pre-trained on sparse sensor measurements to initialize a physics-constrained inverse model. Both frameworks excel in handling limited data and preserving physical consistency. Benchmarking on the Burgers' equation and reaction-diffusion system demonstrates state-of-the-art performance, achieving average $L_2$ errors of $\mathcal{O}(10^{-2})$ for hidden physics discovery and absolute errors of $\mathcal{O}(10^{-3})$ for parameter identification. These results underscore the frameworks' robustness, efficiency, and potential for solving complex scientific problems with minimal observational data.

翻译：大数据通过支持新模型的发现、增强现有框架以及促进精确的不确定性量化，正在变革科学进步。与此同时，科学机器学习的进展为此提供了补充，通过提供强大的工具来解决反问题，以识别那些因数据稀疏或噪声而令传统方法失效的复杂系统。我们提出了两种创新的神经算子框架，专门用于从稀疏测量中发现隐藏物理并识别未知系统参数。第一种框架整合了流行的神经算子DeepONet与物理信息神经网络，以捕捉稀疏数据与底层物理之间的关系，从而能够准确发现一族控制方程。第二种框架侧重于系统参数识别，利用在稀疏传感器测量数据上预训练的DeepONet来初始化一个物理约束的反演模型。两种框架均擅长处理有限数据并保持物理一致性。在Burgers方程和反应-扩散系统上的基准测试展示了最先进的性能，在隐藏物理发现方面实现了平均$L_2$误差为$\mathcal{O}(10^{-2})$，在参数识别方面实现了绝对误差为$\mathcal{O}(10^{-3})$。这些结果凸显了该框架在利用最少观测数据解决复杂科学问题方面的鲁棒性、高效性和潜力。