In many scientific applications, measured time series are corrupted by noise or distortions. Traditional denoising techniques often fail to recover the signal of interest, particularly when the signal-to-noise ratio is low or when certain assumptions on the signal and noise are violated. In this work, we demonstrate that deep learning-based denoising methods can outperform traditional techniques while exhibiting greater robustness to variation in noise and signal characteristics. Our motivating example is magnetic resonance spectroscopy, in which a primary goal is to detect the presence of short-duration, low-amplitude radio frequency signals that are often obscured by strong interference that can be difficult to separate from the signal using traditional methods. We explore various deep learning architecture choices to capture the inherently complex-valued nature of magnetic resonance signals. On both synthetic and experimental data, we show that our deep learning-based approaches can exceed performance of traditional techniques, providing a powerful new class of methods for analysis of scientific time series data.