  Traditional data-driven deep learning models often struggle with high
training costs, error accumulation, and poor generalizability in complex
physical processes. Physics-informed deep learning (PiDL) addresses these
challenges by incorporating physical principles into the model. Most PiDL
approaches regularize training by embedding governing equations into the loss
function, yet this depends heavily on extensive hyperparameter tuning to weigh
each loss term. To this end, we propose to leverage physics prior knowledge by
``baking'' the discretized governing equations into the neural network
architecture via the connection between the partial differential equations
(PDE) operators and network structures, resulting in a PDE-preserved neural
network (PPNN). This method, embedding discretized PDEs through convolutional
residual networks in a multi-resolution setting, largely improves the
generalizability and long-term prediction accuracy, outperforming conventional
black-box models. The effectiveness and merit of the proposed methods have been
demonstrated across various spatiotemporal dynamical systems governed by
spatiotemporal PDEs, including reaction-diffusion, Burgers', and Navier-Stokes
equations.
