Self-Regeneration As a Search Signal For Diffusion Symbolic Regression

Symbolic regression seeks explicit mathematical expressions that describe relationships in numerical observations, but the vast discrete expression space makes it difficult to find high-quality solutions. Pretrained diffusion generators learn expression priors and directly produce candidates, while their continuous reverse process provides an interface for inference-time refinement before discrete decoding. Yet black-box fitness evaluates only completed expressions and does not directly indicate how to steer this process. We observe that time-aligned deviations between an original generation trajectory and its self-regeneration trajectories contain reusable directional information that can guide subsequent denoising toward useful symbolic transitions. Based on this insight, we introduce SelfRegeneration Controlled Diffusion Evolution (SR-CDE), which constructs such deviations when the target trajectory is unknown and assigns suitable donors to individual recipient contexts. SR-CDE injects the selected deviations during reverse diffusion and uses black-box fitness on the original regression task to select and evolve improved trajectory states. It thereby turns temporal information from past generations into trajectory states that can be reused and iteratively refined during inference-time search, while keeping the pretrained generator frozen. Across a 73-task symbolic regression benchmark, SR-CDE achieves competitive overall performance against leading methods. Controlled experiments on Hard-18 further show that full trajectory deviations consistently outperform endpoint-only transfer and random directions matched in summary statistics, confirming the joint importance of temporal application and self-regeneration-specific directional information.