Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard’s iteration—a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a class of PDEs which can be solved by Picard's iteration.

Business-to-business software systems are inherently specialized and operationally intricate, which make them crucial to develop accurate personas that reflect real end-user requirements throughout the development lifecycle. As user needs continuously evolve over time, it becomes imperative to establish a data-driven framework capable of persistently updating these personas and promptly integrating those changes into the development process to maintain long-term value delivery. Conventional persona generation techniques typically depend on clustering approaches applied to qualitative and quantitative data—a process that is time-intensive, expensive, and requires considerable domain expertise. This study introduces an automated method for continuous persona generation, extracting user requirements from an ongoing stream of user data. The approach utilizes large language models to interpret qualitative inputs and dynamically generate knowledge graphs, enabling real-time insights into shifting user needs. A case study involving inquiry call logs from a customer support center was conducted to validate the method. Results demonstrated that the proposed approach outperformed a traditional clustering-based baseline in approximately 86% of the cases in a question–answering task aimed at evaluating the structuring and retrieval of user requirements. Furthermore, clear insights into user pain points and requiring improvements were provided, reinforcing the effectiveness and practical utility of continuous, data-driven persona generation.

Machine-learning (ML) software engineering design patterns encapsulate reusable solutions to commonly occurring problems within the given contexts of ML systems and software design. These ML patterns should help develop and maintain ML systems and software from the design perspective. However, to the best of our knowledge, there is no study on the practitioners' insights on the use of ML patterns for design of their ML systems and software. Herein we report the preliminary results of a literature review and a questionnaire-based survey on ML system developers' state-of-practices with concrete ML patterns.