  Supervised machine learning approaches require the formulation of a loss
functional to be minimized in the training phase. Sequential data are
ubiquitous across many fields of research, and are often treated with Euclidean
distance-based loss functions that were designed for tabular data. For smooth
oscillatory data, those conventional approaches lack the ability to penalize
amplitude, frequency and phase prediction errors at the same time, and tend to
be biased towards amplitude errors. We introduce the surface similarity
parameter (SSP) as a novel loss function that is especially useful for training
machine learning models on smooth oscillatory sequences. Our extensive
experiments on chaotic spatio-temporal dynamics systems indicate that the SSP
is beneficial for shaping gradients, thereby accelerating the training process,
reducing the final prediction error, increasing weight initialization
robustness, and implementing a stronger regularization effect compared to using
classical loss functions. The results indicate the potential of the novel loss
metric particularly for highly complex and chaotic data, such as data stemming
from the nonlinear two-dimensional Kuramoto-Sivashinsky equation and the linear
propagation of dispersive surface gravity waves in fluids.
