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Traditional data-driven deep learning models often struggle with hightraining costs, error accumulation, and poor generalizability in complexphysical processes. Physics-informed deep learning (PiDL) addresses thesechallenges by incorporating physical principles into the model. Most PiDLapproaches regularize training by embedding governing equations into the lossfunction, yet this depends heavily on extensive hyperparameter tuning to weigheach loss term. To this end, we propose to leverage physics prior knowledge by``baking'' the discretized governing equations into the neural networkarchitecture via the connection between the partial differential equations(PDE) operators and network structures, resulting in a PDE-preserved neuralnetwork (PPNN). This method, embedding discretized PDEs through convolutionalresidual networks in a multi-resolution setting, largely improves thegeneralizability and long-term prediction accuracy, outperforming conventionalblack-box models. The effectiveness and merit of the proposed methods have beendemonstrated across various spatiotemporal dynamical systems governed byspatiotemporal PDEs, including reaction-diffusion, Burgers', and Navier-Stokesequations.

Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics