Discovering Symbolic Partial Differential Equation by Abductive Learning
En-Hao Gao, Cunjing Ge, Yuan Jiang, Zhi-Hua Zhou
Discovering symbolic Partial Differential Equation (PDE) from data is one of the most promising directions of modern scientific discovery.    However, effectively constructing an expressive yet concise hypothesis space and accurately evaluating expression values remain challenging due to the exponential explosion with the spatial dimension and the noise in the measurements.    To address these challenges, we propose the ABL-PDE approach that employs the Abductive Learning (ABL) framework to discover symbolic PDEs.    By introducing a First-Order Logic (FOL) knowledge base, ABL-PDE can represent various PDEs, significantly constraining the hypothesis space without sacrificing expressive power, while also facilitating the incorporation of problem-specific knowledge.    Furthermore, the proposed consistency optimization process establishes a synergistic interaction between the knowledge base and the neural network, achieving robust structure identification, accurate coefficient estimation, and enhanced stability against hyperparameter variation.    Experimental results on three benchmarks across different noise levels demonstrate the state-of-the-art performance of our approach in PDE discovery.