DeSyR: A decoupled symbolic recovery framework for PDEs with PINN-guided structure discovery and physics-based coefficient refinement
Pancheng Niu, Jun Guo, Qiaolin He, Jingcai Guo, Yanchao Shi
Recovering explicit symbolic solutions of partial differential equations (PDEs) from physics-informed neural networks (PINNs) is attractive, yet selecting and fitting expressions against neural predictions can lock the recovered solution into the teacher's approximation error. We propose DeSyR (Decoupled Symbolic Recovery) to break this dependence. DeSyR uses PINNs only to discover candidate structures and initial coefficient estimates; once a structure is fixed, the governing equations and constraints alone determine the refined coefficients. Multiple candidates with different fitting-error--complexity trade-offs are retained, evaluated after physics-only refinement, and reduced to the simplest expression within a physically near-optimal set. Joint refinement extends the same principle to coupled fields. For fixed candidate parameterizations, affine error identities and local nonlinear analysis show how teacher errors enter fitted coefficients and establish conditions under which physical residuals control coefficient and solution errors. Across seven benchmarks covering scalar evolution equations, multidimensional elliptic problems and coupled incompressible flows, DeSyR attains mean relative solution errors on the order of $10^{-14}$ or smaller and lower reported mean physical residuals than PINNs, physics-informed symbolic networks and teacher-fitted symbolic distillation. With the expression structure held fixed, estimating coefficients from physical constraints further yields more accurate frequency estimates and solutions than fitting teacher predictions. DeSyR therefore turns neural approximations into guides for structure discovery, while physical constraints determine the final explicit solution.
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