Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's
Avetik Arakelyan, Rafayel Barkhudaryan
TL;DR
This work addresses the convergence of Physics-Informed Neural Networks (PINNs) for solving fully nonlinear second-order PDEs by framing training as the minimization of a Hölder-regularized empirical loss and leveraging viscosity-solution theory. It shows that, under mild data-distribution assumptions and a degeneracy/comparison framework, PINN minimizers converge to the unique viscosity solution $u^*$ of $F[u]=0$ as the training data grow, with the PINN loss decaying at a rate $O(m_r^{-α/d})$. The analysis uses probabilistic space-filling arguments, an Arzelà-Ascoli compactness argument, and a viscosity-solution characterization to transfer loss convergence into convergence to the PDE’s solution. The results provide a rigorous link between PINN training dynamics and nonlinear PDE theory, offering convergence guarantees for fully nonlinear problems in a data-driven setting.
Abstract
The present work is focused on exploring convergence of Physics-informed Neural Networks (PINNs) when applied to a specific class of second-order fully nonlinear Partial Differential Equations (PDEs). It is well-known that as the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We show that such sequence converges to a unique viscosity solution of a certain class of second-order fully nonlinear PDE's, provided the latter satisfies the comparison principle in the viscosity sense.