PS-PINN: Recovering Governing Equations and Readable Solutions from the Same Noisy Data

Recovering governing equations and symbolic solutions from the same noisy observations offers interpretable descriptions of both physical dynamics and their observed realizations. Existing approaches typically recover either an explicit PDE with a neural surrogate or a readable solution under a known equation. We present \ourmodel, a data-driven framework for PDE discovery and symbolic solution discovery. The reconstructed solution combines a sparsity-regularized analytic core with a neural corrector, $\hat{u}=\Phi \mathbf{\alpha }+r_{\theta }$. A symbolic-first curriculum fits the core before learning neural corrections, giving explicit structure priority during training. The framework delivers both outputs in identifiable settings. On Wave with a dictionary encoding the true wavenumber, the corrector becomes negligible and the solution is effectively closed form; on Burgers, the core captures the smooth bulk while the corrector concentrates near the steep front. Coupling an Allen–Cahn operator estimated within a prescribed structure reduces reconstruction RMSE by a factor of $2.5$ relative to uncoupled training. Matched-budget controls further show that Adam-based joint training yields symbolic-core RMSE values 2.3 to 5.7 times those of symbolic-first training on Allen–Cahn and Burgers, often with little change in prediction error. These results show how symbolic-first solution learning complements equation discovery, retaining an explicit governing law and an inspectable solution structure.