IsL2Physics-InformedLossAlwaysSuitablefor TrainingPhysics-InformedNeuralNetwork?

In particular, we leverage the concept of stability in the literature of partial differential equation tostudy the asymptotic behavior ofthe learned solution asthe loss approaches zero. Withthis concept, we study animportant class of high-dimensional non-linear PDEs in optimal control, the Hamilton-JacobiBellman (HJB) Equation, and provethat for generalLp Physics-Informed Loss, a wide class of HJB equation is stable only ifp is sufficiently large.