  Physics-informed neural networks (PINNs) are a promising approach that
combines the power of neural networks with the interpretability of physical
modeling. PINNs have shown good practical performance in solving partial
differential equations (PDEs) and in hybrid modeling scenarios, where physical
models enhance data-driven approaches. However, it is essential to establish
their theoretical properties in order to fully understand their capabilities
and limitations. In this study, we highlight that classical training of PINNs
can suffer from systematic overfitting. This problem can be addressed by adding
a ridge regularization to the empirical risk, which ensures that the resulting
estimator is risk-consistent for both linear and nonlinear PDE systems.
However, the strong convergence of PINNs to a solution satisfying the physical
constraints requires a more involved analysis using tools from functional
analysis and calculus of variations. In particular, for linear PDE systems, an
implementable Sobolev-type regularization allows to reconstruct a solution that
not only achieves statistical accuracy but also maintains consistency with the
underlying physics.
