2 research outputs found
Learning Efficient Surrogate Dynamic Models with Graph Spline Networks
While complex simulations of physical systems have been widely used in
engineering and scientific computing, lowering their often prohibitive
computational requirements has only recently been tackled by deep learning
approaches. In this paper, we present GraphSplineNets, a novel deep-learning
method to speed up the forecasting of physical systems by reducing the grid
size and number of iteration steps of deep surrogate models. Our method uses
two differentiable orthogonal spline collocation methods to efficiently predict
response at any location in time and space. Additionally, we introduce an
adaptive collocation strategy in space to prioritize sampling from the most
important regions. GraphSplineNets improve the accuracy-speedup tradeoff in
forecasting various dynamical systems with increasing complexity, including the
heat equation, damped wave propagation, Navier-Stokes equations, and real-world
ocean currents in both regular and irregular domains.Comment: Published as a conference paper in NeurIPS 202
Spectral properties of kernel matrices in the flat limit
Kernel matrices are of central importance to many applied fields. In this
manuscript, we focus on spectral properties of kernel matrices in the so-called
"flat limit", which occurs when points are close together relative to the scale
of the kernel. We establish asymptotic expressions for the determinants of the
kernel matrices, which we then leverage to obtain asymptotic expressions for
the main terms of the eigenvalues. Analyticity of the eigenprojectors yields
expressions for limiting eigenvectors, which are strongly tied to discrete
orthogonal polynomials. Both smooth and finitely smooth kernels are covered,
with stronger results available in the finite smoothness case.Comment: 40 pages, 8 page