Data Enrichment for Symbolic Regression Using Diffusion Models
For practitioners of symbolic regression in scientific discovery, this work provides a data enrichment method that enhances reliability under sparse/noisy conditions, though it is incremental as it applies existing diffusion models to a known bottleneck.
The paper tackles the problem of sparse, noisy, or physically incomplete spatiotemporal measurements degrading symbolic regression (SR) performance. It introduces a physics-guided latent diffusion framework for data enrichment that consistently improves equation recovery across multiple physical dynamics and SR backends, without requiring additional domain expertise.
Symbolic regression (SR) offers a route to scientific discovery by converting observations into interpretable governing equations. However, despite its promise, its reliability degrades sharply when spatiotemporal measurements are sparse, noisy, or physically incomplete, as commonly occurring in practice. Data enrichment (DE) has been shown to be able to mitigate this limitation, yet additional samples can mislead equation discovery unless they preserve the physical structure of the target system. Such implication of DE requires narrow domain expertise as well as technical fluidity, highly limiting its practical usefulness. In this study, we introduce a physics-guided latent diffusion framework for DE for down the line SR models. The proposed framework combines a variational autoencoder, a conditional latent diffusion model, and a physics-informed residual corrector to complete sparse observations with synthetic fields constrained by governing relations. We evaluate the approach on heat conduction, incompressible Navier-Stokes flow, and a moving single-mass Newtonian gravitational potential, using GPLearn, DEAP, and PySR as downstream SR backends. Our results reveal that physics-corrected enrichment consistently improves recovery in sparse regimes across physical dynamics and SR models. These results show that generative enrichment can strengthen equation discovery without additional domain expertise.