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Building Trust in PINNs: Error Estimation through Finite Difference Methods

Aleksander Krasowski, René P. Klausen, Aycan Celik, Sebastian Lapuschkin, Wojciech Samek, Jonas Naujoks

Abstract

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

Building Trust in PINNs: Error Estimation through Finite Difference Methods

Abstract

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.
Paper Structure (14 sections, 11 equations, 3 figures, 1 table)

This paper contains 14 sections, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Theoretical Background
  4. Physics-Informed Neural Networks
  5. Finite Difference Methods
  6. Time-Stepping
  7. First-order Time Derivatives (Crank-Nicolson)
  8. Second-order Time Derivatives (Central Differences)
  9. Methodology
  10. Integrating the Residual using FDM
  11. Experiments & Discussion
  12. Discussion
  13. Limitations
  14. Conclusion

Figures (3)

  • Figure 1: Exact solutions for the benchmark problems. The horizontal line in \ref{['fig:poisson2d']} at $y=0.5$ represents the solution for the 1D Poisson problem.
  • Figure 2: Error estimates for a well-trained PINN on the heat equation, (a)--(d) at four different time-points across space and (e) $L_2$ error over time, including the bound $e_{\mathrm{bound}}$. The FDM-based methods use a $64 \times 64$ spatio-temporal grid (time slices shown at nearest grid point to specified values); $e_{\mathrm{bound}}$ uses $64$ spatial points with adaptive time integration (up to $165$ steps at $t=1$).
  • Figure 3: Accuracies of the error estimates across different discretizations, measured as $\left\lVert e_{\mathrm{true}} - e_{\mathrm{method}}\right\rVert_2$ on respective grid points, for all benchmark problems. Averaged over 10 runs, bands show one standard deviation.