  Modeling sequential patterns from data is at the core of various time series
forecasting tasks. Deep learning models have greatly outperformed many
traditional models, but these black-box models generally lack explainability in
prediction and decision making. To reveal the underlying trend with
understandable mathematical expressions, scientists and economists tend to use
partial differential equations (PDEs) to explain the highly nonlinear dynamics
of sequential patterns. However, it usually requires domain expert knowledge
and a series of simplified assumptions, which is not always practical and can
deviate from the ever-changing world. Is it possible to learn the differential
relations from data dynamically to explain the time-evolving dynamics? In this
work, we propose an learning framework that can automatically obtain
interpretable PDE models from sequential data. Particularly, this framework is
comprised of learnable differential blocks, named $P$-blocks, which is proved
to be able to approximate any time-evolving complex continuous functions in
theory. Moreover, to capture the dynamics shift, this framework introduces a
meta-learning controller to dynamically optimize the hyper-parameters of a
hybrid PDE model. Extensive experiments on times series forecasting of
financial, engineering, and health data show that our model can provide
valuable interpretability and achieve comparable performance to
state-of-the-art models. From empirical studies, we find that learning a few
differential operators may capture the major trend of sequential dynamics
without massive computational complexity.
