Discovering symbolic Partial Differential Equation (PDE) from data is one of the most promising directions of modern scientific discovery. Effectively constructing an expressive yet concise hypothesis space and accurately evaluating expression values, however, 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. The proposed consistency optimization process establishes a synergistic interaction between the knowledge base and the neural network learning module, achieving robust structure identification, accurate coefficient estimation, and enhanced stability against hyperparameter variation. Experimental results on three benchmarks across different noise levels demonstrate the effectiveness of our approach in PDE discovery.

Optical Character Recognition (OCR) of historical document images remains a challenging task because of the distorted input images, extensive number of uncommon characters, and the scarcity of labeled data, which impedes modern deep learning-based OCR techniques from achieving good recognition accuracy. Meanwhile, there exists a substantial amount of expert knowledge that can be utilized in this task. However, such knowledge is usually complicated and could only be accurately expressed with formal languages such as first-order logic (FOL), which is difficult to be directly integrated into deep learning models. This paper proposes KESAR, a novel Knowledge-Enhanced Document Segmentation And Recognition method for historical document images based on the Abductive Learning (ABL) framework. The segmentation and recognition models are enhanced by incorporating background knowledge for character extraction and prediction, followed by an efficient joint optimization of both models. We validate the effectiveness of KESAR on historical document datasets. The experimental results demonstrate that our method can simultaneously utilize knowledge-driven reasoning and data-driven learning, which outperforms the current state-of-the-art methods.