BeliefFlow: Continuous-Time BP-Structured Neural Decoding for LDPC Codes

Belief propagation (BP) decodes low-density parity-check (LDPC) codes through sparse local message passing. Recent Transformer-based decoders achieve strong performance using parity-aware attention without explicitly implementing BP's update rules. This raises the question of whether strong performance can be achieved while retaining BP's computational structure. We introduce BeliefFlow, which evolves vector-valued Tanner-edge states in continuous time using a shared neural vector field. Its local operators combine attention-based aggregation over destination-excluded neighborhoods with gated, element-wise box-plus operations. Training uses duration-aware loss reweighting to emphasize difficult examples, while randomized perturbations generate diverse retry trajectories at inference. Across six LDPC codes, the 0.34M-parameter decoder leads the comparison in 17 of 18 code/SNR settings, based on reported $-\ln (\mathrm{BER})$ scores or lower bounds. These results show that expressive local updates and perturbation-based retries can deliver strong performance within a BP-structured architecture.