Learning Physics from an Imperfect Ancestor
A three-stage method uses an imperfect neural operator to steer PINNs into the correct PDE basin.
The paper addresses neural operators that fail outside their training distribution and physics-informed neural networks that can settle in the wrong solution basin despite a small residual. A three-stage method freezes a physics-informed operator's spatial basis, extrapolates its solution branch with a polynomial continuation prior, and distills the field into a fresh PINN. On Allen-Cahn, distillation recovers the non-trivial branch that a standard PINN collapses away from; on lid-driven cavity flow at Reynolds number 3200 it reaches the physical state with fewer parameters and optimization steps than recent baselines.
- An imperfect neural operator identifies the basin; the PDE residual refines it
- Tested on Burgers, Allen-Cahn, and lid-driven cavity flow
- Recovers the non-trivial Allen-Cahn branch that a scratch PINN misses
- At Reynolds number 3200, converges with fewer parameters and steps
Full article274 words · extracted from arxiv.org · click to collapse
Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed neural networks avoid dependence on labeled data, yet their optimization can be basin-fragile: when the governing residual admits multiple solutions, a PINN trained from scratch may converge to a physically incorrect state despite achieving a small residual. We show that these failure modes can be addressed jointly: an imperfect NO provides the structural prior needed to place a PINN in the correct solution basin, while the PDE residual refines the solution beyond the operator's accuracy. We introduce a three-stage framework that freezes the spatial basis of a physics-informed NO, extrapolates its solution branch to an out-of-distribution parameter using a polynomial continuation prior, and distills the resulting field into a fresh PINN. The NO need not be accurate at the target; it transfers solution-branch information, while PDE residual minimization in the PINN governs convergence. We evaluate the framework on three nonlinear PDEs: 1D viscous Burgers, 2D steady Allen-Cahn near a pitchfork bifurcation, and 2D steady lid-driven cavity flow. For Allen-Cahn, where the trivial solution satisfies the PDE residual exactly, a standard PINN collapses to the trivial zero branch, whereas distillation from the crude extrapolated operator recovers the non-trivial branch that matches the finite-difference reference. For the lid-driven cavity, extrapolating to a Reynolds number of Re = 3200 accelerates convergence to the correct physical state, achieving competitive accuracy using fewer parameters and optimization steps than recent literature baselines. These results establish a simple principle: an NO need not accurately predict the solution to be useful; it only needs to identify the correct basin from which PINN optimization can recover it.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.24947