Physics-constrained robust learning of open-form PDEs from limited and
  noisy data
Mengge Du Longfeng Nie Siyu Lou Yuntian Chenc Dongxiao Zhang
Unveiling the underlying governing equations of nonlinear dynamic systems
remains a significant challenge, especially when encountering noisy
observations and no prior knowledge available. This study proposes R-DISCOVER,
a framework designed to robustly uncover open-form partial differential
equations (PDEs) from limited and noisy data. The framework operates through
two alternating update processes: discovering and embedding. The discovering
phase employs symbolic representation and a reinforcement learning (RL)-guided
hybrid PDE generator to efficiently produce diverse open-form PDEs with tree
structures. A neural network-based predictive model fits the system response
and serves as the reward evaluator for the generated PDEs. PDEs with superior
fits are utilized to iteratively optimize the generator via the RL method and
the best-performing PDE is selected by a parameter-free stability metric. The
embedding phase integrates the initially identified PDE from the discovering
process as a physical constraint into the predictive model for robust training.
The traversal of PDE trees automates the construction of the computational
graph and the embedding process without human intervention. Numerical
experiments demonstrate our framework's capability to uncover governing
equations from nonlinear dynamic systems with limited and highly noisy data and
outperform other physics-informed neural network-based discovery methods. This
work opens new potential for exploring real-world systems with limited
understanding.