Provides the first rigorous error certification for Physics-Informed Neural Networks (PINNs), bridging the gap between empirical residual loss and actual solution guarantees.
arXiv · March 20, 2026 · 2603.19165
The Takeaway
Neural PDE solvers have lacked the theoretical convergence guarantees of classical methods like FEM, limiting their use in safety-critical engineering. This paper provides deterministic and probabilistic bounds that translate residual errors into explicit solution space guarantees, making neural solvers viable for rigorous scientific computing.
From the abstract
Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains