Building Trust in PINNs: Error Estimation through Finite Difference Methods

Aleksander Krasowski  ·  René P. Klausen  ·  Aycan Çelik  ·  Sebastian Lapuschkin  ·  Wojciech Samek  ·  Jonas Naujoks    arXiv:2603.15526  ·  GitHub

An interactive demo for our paper. Pick a benchmark problem, a model type, and an FDM grid size below — then hit Run Experiment to see how well our error estimation tracks the true PINN error, and how it compares to the FDM baseline.

No model selected
nx × nt (or nx × ny) for FDM solver
Heatmaps
PINN Prediction
True Solution
Estimated Error |eres| (our method)
Slice Plot —
PINN vs Exact Solution
Error Comparison
t =
Why is the error zero at the edges? The PINN is hard-constrained — initial and boundary conditions are built directly into the network's output, so they are satisfied exactly by construction. As a result, the error is always exactly zero at the domain boundaries (and at t = 0 for time-dependent problems), which you'll notice when moving the slider to the edges.
TypeMLP (hard-constrained)
Hidden layers3 × 20 neurons
Activationtanh
OptimizerAdam, lr = 10⁻³
Seed42
PINN Training
e_res (ours)
e_FDM
Problem
Model type
Iterations
Collocation points
PDE
Boundary cond.
Parameters
Max error
Mean error
L² relative error