Failure-informed PINNs for a multi-strain SVEIR epidemic model

Tightly coupled multi-compartment epidemic models tend to expose a weakness of standard Physics-Informed Neural Networks (PINNs). When collocation points are placed uniformly, the network spends much of its capacity on smooth regions while leaving the sharp transients poorly resolved. We work around this by pairing Failure-Informed PINNs (FI-PINNs) with a Self-Adaptive Importance Sampling (SAIS) refinement strategy, and we apply the combination to a nine-equation Susceptible/Vaccinated/Exposed/Infected/Recovered (SVEIR) model that follows three viral strains together with a vaccination compartment. The idea behind the construction is simple. We build a residual-based limit-state function, estimate the failure probability associated with it, and let the network steer its own sampling toward the time intervals where the governing equations are not yet satisfied to within a prescribed tolerance. On the same temporal domain, with identical initial conditions and the same network architecture, SAIS-enhanced FI-PINNs reach a relative L₂ error of 1.04× 10⁻⁶, against 7.36× 10⁻³ for uniform sampling and 3.42× 10⁻⁴ for Residual-Based Adaptive Refinement (RAR) [Lu et. al,. 63:208-228, 2021], an improvement of two to three orders of magnitude. The benefits go beyond raw accuracy. The failure-probability estimate gives an interpretable stopping criterion, and the truncated Gaussian proposal of SAIS keeps the sampling-bias risk under control, a risk that affects purely greedy residual-based schemes. The framework is general enough to be transported to other coupled compartmental systems and to data-assimilation settings in which partial observations are available. Mathematics subject classification 92D30 , 65L05 , 68T07 , 00A71