Optical monitoring of cardiac pulsations using near-infrared spectroscopy (NIRS), photoplethysmography (PPG), and diffuse correlation spectroscopy (DCS) is often hindered by motion artifacts and noise. We introduce a synthetic-data-driven framework using a long short-term memory (LSTM) network to trace and denoise pulsatile optical waveforms without reliance on annotated clinical datasets. Physiologically realistic pulsatile signals are generated, corrupted with parameterized artifacts, and used to train the LSTM model. Applied to experimental NIRS, PPG, and DCS signals, the model recovered beat-to-beat morphology more effectively than widely used wavelet and temporal derivative distribution repair (TDDR) filters. Heart rate (HR) extraction from LSTM-processed signals closely matched ECG-derived measurements (mean absolute error = 0.59 bpm, root mean square error = 0.74 bpm). This flexible approach shows potential for rapid adaptation across various devices and noise conditions.