Spatiotemporal processes commonly exist in manufacturing. Modeling and monitoring such processes are crucial for ensuring high-quality production. For example, ultrasonic metal welding is an important industrial-scale joining technique with wide applications. The surfaces of ultrasonic welding tools evolve in both spatial and temporal domains, resulting in a spatiotemporal process. Close monitoring of tool surface progression is imperative since degraded tools often lead to low-quality joints. However, it is generally expensive and time-consuming to acquire fine-scale surface measurement data, which is not economically viable. This paper develops a multi-task learning method to enable data-efficient spatiotemporal modeling. A Gaussian process-based hierarchical Bayesian inference structure is constructed to transfer knowledge among multiple similar-but-not-identical measurement tasks. Meanwhile, a spatiotemporal kernel is developed based on squared sine exponential damping (SSED) function to characterize the periodic trend of anvil surfaces. The proposed method is able to improve interpolation accuracy using limited measurement data compared with state-of-the-art techniques. Data collected from lithium-ion battery production are employed to demonstrate the effectiveness of the proposed method. Additionally, the influence of training data size and hyperparameter selection on the modeling performance is systematically investigated.

时空过程通常存在于制造业中。对此类过程进行建模和监视对于确保高质量生产至关重要。例如，超声波金属焊接是一种具有广泛应用的重要工业规模的连接技术。超声焊接工具的表面在时域和时域上均发生变化，从而导致时空过程。必须严格监控工具表面的进展，因为退化的工具通常会导致接头质量低下。然而，获取精细规模的表面测量数据通常是昂贵且费时的，这在经济上是不可行的。本文开发了一种多任务学习方法，以实现数据有效的时空建模。构造了基于高斯过程的层次贝叶斯推理结构，以在多个相似但不相同的测量任务之间传递知识。同时，基于平方正弦指数阻尼（SSED）函数开发了时空内核，以描述砧面的周期性趋势。与最新技术相比，该方法能够使用有限的测量数据来提高插值精度。从锂离子电池生产中收集的数据用于证明所提出方法的有效性。此外，系统地研究了训练数据大小和超参数选择对建模性能的影响。基于平方正弦指数阻尼（SSED）函数开发了时空内核，以表征砧面的周期性趋势。与最新技术相比，该方法能够使用有限的测量数据来提高插值精度。从锂离子电池生产中收集的数据用于证明所提出方法的有效性。此外，系统地研究了训练数据大小和超参数选择对建模性能的影响。基于平方正弦指数阻尼（SSED）函数开发了时空内核，以表征砧面的周期性趋势。与最新技术相比，该方法能够使用有限的测量数据来提高插值精度。从锂离子电池生产中收集的数据用于证明所提出方法的有效性。此外，系统地研究了训练数据大小和超参数选择对建模性能的影响。