Time-series generation has crucial practical significance for decision-making under uncertainty. Existing methods have various limitations like accumulating errors over time, significantly impacting downstream tasks. We develop a novel generation method, DT-VAE, that incorporates generalizable domain knowledge, is mathematically justified, and significantly outperforms existing methods by mitigating error accumulation through a cumulative difference learning mechanism. We evaluate the performance of DT-VAE on several downstream tasks using both semi-synthetic and real time-series datasets, including benchmark datasets and our newly curated COVID-19 hospitalization datasets. The COVID-19 datasets enrich existing resources for time-series analysis. Additionally, we introduce Diverse Trend Preserving (DTP), a time-series clustering-based evaluation for direct and interpretable assessments of generated samples, serving as a valuable tool for evaluating time-series generative models.