  Physics-informed neural networks (PINNs) are effective in solving inverse
problems based on differential and integral equations with sparse, noisy,
unstructured, and multi-fidelity data. PINNs incorporate all available
information into a loss function, thus recasting the original problem into an
optimization problem. In this paper, we extend PINNs to parameter and function
inference for integral equations such as nonlocal Poisson and nonlocal
turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The
contribution of the paper is three-fold. First, we propose a unified nonlocal
operator, which converges to the classical Laplacian as one of the operator
parameters, the nonlocal interaction radius $\delta$ goes to zero, and to the
fractional Laplacian as $\delta$ goes to infinity. This universal operator
forms a super-set of classical Laplacian and fractional Laplacian operators
and, thus, has the potential to fit a broad spectrum of data sets. We provide
theoretical convergence rates with respect to $\delta$ and verify them via
numerical experiments. Second, we use nPINNs to estimate the two parameters,
$\delta$ and $\alpha$. The strong non-convexity of the loss function yielding
multiple (good) local minima reveals the occurrence of the operator mimicking
phenomenon: different pairs of estimated parameters could produce multiple
solutions of comparable accuracy. Third, we propose another nonlocal operator
with spatially variable order $\alpha(y)$, which is more suitable for modeling
turbulent Couette flow. Our results show that nPINNs can jointly infer this
function as well as $\delta$. Also, these parameters exhibit a universal
behavior with respect to the Reynolds number, a finding that contributes to our
understanding of nonlocal interactions in wall-bounded turbulence.

Analysis of PDEs;Machine Learning;Optimization and Control;Computational Physics