Recovering closed-form PDE solutions from standard NNs via symbolic distillation

Establish whether symbolic distillation with SymTorch can recover the analytic solution u(x,t) = exp(−π^2 α t) sin(π x) of the 1-D heat equation from a standard neural network trained on the same sparse dataset, given that the method succeeds for a Physics-Informed Neural Network under identical conditions but failed for the standard neural network in the reported experiments.

Background

The study compares a standard neural network and a Physics-Informed Neural Network (PINN) trained on ten data points from the 1-D heat equation. Using SymTorch, the authors successfully distill the correct closed-form solution from the PINN.

However, they explicitly report being unable to distill the same solution from the standard neural network, leaving open whether such recovery is possible for non-physics-regularized models trained on the same data.