Novel artificial intelligence (AI) technology has expedited various scientific research, e.g., cosmology, physics, and bioinformatics, inevitably becoming a significant category of workload on high-performance computing (HPC) systems. Existing AI benchmarks tend to customize well-recognized AI applications, so as to evaluate the AI performance of HPC systems under the predefined problem size, in terms of datasets and AI models. However, driven by novel AI technology, most of AI applications are evolving fast on models and datasets to achieve higher accuracy and be applicable to more scenarios. Due to the lack of scalability on the problem size, static AI benchmarks might be under competent to help understand the performance trend of evolving AI applications on HPC systems, in particular, the scientific AI applications on large-scale systems. In this paper, we propose a scalable evaluation methodology (SAIH) for analyzing the AI performance trend of HPC systems with scaling the problem sizes of customized AI applications. To enable scalability, SAIH builds a set of novel mechanisms for augmenting problem sizes. As the data and model constantly scale, we can investigate the trend and range of AI performance on HPC systems, and further diagnose system bottlenecks. To verify our methodology, we augment a cosmological AI application to evaluate a real HPC system equipped with GPUs as a case study of SAIH. With data and model augment, SAIH can progressively evaluate the AI performance trend of HPC systems, e.g., increasing from 5.2% to 59.6% of the peak theoretical hardware performance. The evaluation results are analyzed and summarized into insight findings on performance issues. For instance, we find that the AI application constantly consumes the I/O bandwidth of the shared parallel file system during its iteratively training model. If I/O contention exists, the shared parallel file system might become a bottleneck.