  Fractional and tempered fractional partial differential equations (PDEs) are
effective models of long-range interactions, anomalous diffusion, and non-local
effects. Traditional numerical methods for these problems are mesh-based, thus
struggling with the curse of dimensionality (CoD). Physics-informed neural
networks (PINNs) offer a promising solution due to their universal
approximation, generalization ability, and mesh-free training. In principle,
Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using
Monte Carlo methods and thus could lift CoD. However, this may cause
significant variance and errors, hence affecting convergence; in addition,
MC-fPINN is sensitive to hyperparameters. In general, numerical methods and
specifically PINNs for tempered fractional PDEs are under-developed. Herein, we
extend MC-fPINN to tempered fractional PDEs to address these issues, resulting
in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible
high variance and errors from Monte Carlo sampling, we replace the
one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to
both MC-fPINN and MC-tfPINN. We validate our methods on various forward and
inverse problems of fractional and tempered fractional PDEs, scaling up to
100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature
consistently outperforms the original versions in accuracy and convergence
speed in very high dimensions.
