nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications
- Brown Univ., Providence, RI (United States)
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, 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 Laplace 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 also 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 $$\alpha$$, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, 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 $$\delta$$. More importantly, 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.
- Research Organization:
- Brown Univ., Providence, RI (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Defense Advanced Research Projects Agency (DARPA)
- Grant/Contract Number:
- SC0019453; W911NF-15-1-0562; HR00111990025; NA0003525
- OSTI ID:
- 2281995
- Alternate ID(s):
- OSTI ID: 1775911
- Journal Information:
- Journal of Computational Physics, Vol. 422; ISSN 0021-9991
- Publisher:
- ElsevierCopyright Statement
- Country of Publication:
- United States
- Language:
- English
Similar Records
Towards a Unified theory of Fractional and Nonlocal Vector Calculus
A Unified Theory of Fractional Nonlocal and Weighted Nonlocal Vector Calculus