A Prior-Information-Guided Residual Diffusion Model for Multi-Modal PET Synthesis from MRI

Abstract

Geometric problem solving has always been a long-standing challenge in the fields of mathematical reasoning and artificial intelligence. We built a neural-symbolic system, called FGeo-HyperGNet, to automatically perform human-like geometric problem solving. The symbolic component is a formal system built on FormalGeo, which can automatically perform geometric relational reasoning and algebraic calculations and organize the solution into a hypergraph with conditions as hypernodes and theorems as hyperedges. The neural component, called HyperGNet, is a hypergraph neural network based on the attention mechanism, including an encoder to effectively encode the structural and semantic information of the hypergraph and a theorem predictor to provide guidance in solving problems. The neural component predicts theorems according to the hypergraph, and the symbolic component applies theorems and updates the hypergraph, thus forming a predict-apply cycle to ultimately achieve readable and traceable automatic solving of geometric problems. Experiments demonstrate the correctness and effectiveness of this neural-symbolic architecture. We achieved state-of-the-art results with a TPA of 93.50% and a PSSR of 88.36% on the FormalGeo7K dataset.