The Nonlocal Neural Operator: Universal Approximation

• DOI: 10.48550/arxiv.2304.13221
• Publication Date: 2023-01-01
• Authors: Samuel Lanthaler; Zongyi Li; Andrew M. Stuart
• Keywords: Operator (biology)

Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both accelerate traditional numerical methods enable data-driven discovery. A popular variant neural is the Fourier (FNO). Previous analysis proving universal approximation theorems for FNOs resorts use an unbounded number modes limits basic form method problems with periodic geometry. Prior work relies on intuition from methods, interprets FNO as a nonstandard highly nonlinear spectral method. The present challenges this point view two ways: (i) introduces new broad class approximators, termed nonlocal (NNOs), which allow functions defined arbitrary geometries, includes special case; (ii) NNOs shows that, provided architecture computation spatial average (corresponding retaining only single mode case FNO) it benefits approximation. It demonstrated that theoretical result unifies wide range architectures. Furthermore, sheds light role nonlocality, its interaction nonlinearity, thereby paving way more systematic exploration through development learning existing