None of these studies directly compares PINN solutions

The gap

Evidence profile

Sourced from the synthesized of the source papers, classified as general, drawn from work published between 2024 and 2026, spanning 4 journals. Those papers have been cited 154 times in total.

Supporting evidence — 4 representative gaps

• Advancements and Future Directions in Loss Function Designs for Physics-Informed Neural Networks: A Comprehensive Review (2026) · Journal of Scientific Computing · doi

  Existing PINN loss formulations for elliptic problems with Dirichlet boundary conditions use L² norms on both interior and boundary, but trace theory requires boundary values to be measured in fractional Sobolev spaces (H^(1/2) for H¹ solutions); none of these studies systematically address whether standard L² boundary losses are mathematically sufficient or whether stronger norms are necessary for well-posedness.

  generalsynthesizedevidence 5/5
  Keywords: existing pinn loss formulations elliptic problems dirichlet boundary
• A Variational Physics-Informed Neural Network Framework Using Petrov-Galerkin Method for Solving Singularly Perturbed Boundary Value Problems (2026) · International Journal of Computational Methods · doi

  Across this set, PINN loss function designs are evaluated primarily on smooth problems or low-regularity settings (singularly perturbed, advection-diffusion-reaction), but none systematically compares how different boundary loss norms (L², H^(1/2), or higher) affect convergence and accuracy for elliptic problems where the solution regularity is precisely H¹.

  generalsynthesizedevidence 5/5
  Keywords: across set pinn loss function designs evaluated primarily
• A Physics-Informed Neural Network Approach to Numerical Solution of Partial Differential Equations (2026) · Zenodo (CERN European Organization for Nuclear Research) · doi

  Training stability and convergence guarantees for PINNs remain unresolved across this set, particularly when boundary conditions are enforced via weak norms (H^(1/2)) rather than pointwise L² norms; existing studies report spectral bias and imbalance in loss terms but do not investigate how norm-theoretic correctness of boundary losses affects these phenomena.

  generalsynthesizedevidence 5/5
  Keywords: training stability convergence guarantees pinns remain unresolved across
• Can physics-informed neural networks beat the finite element method? (2024) · IMA Journal of Applied Mathematics · cited 154× · doi

  None of these studies directly compares PINN solutions for elliptic Dirichlet problems using standard L² boundary losses against solutions using trace-theoretically correct fractional Sobolev norms on the boundary, leaving open whether the mathematical gap translates to measurable differences in solution accuracy or convergence behavior.

  generalsynthesizedevidence 5/5
  Keywords: none studies directly compares pinn solutions elliptic dirichlet

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