Nils Wildt | Search arXiv

arXiv2026

Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators

Gaussian random fields are a versatile tool used in the fields of stochastic PDEs, uncertainty quantification, and geostatistical simulation. One way to obtain them is to use a truncated Karhunen-Loève expansion (KLE). Computing the expansion requires the leading eigenpairs of an $N \times N$ covariance matrix, where $N$ is the total number of grid cells. These are usually computed with a Krylov eigensolver, which relies on the covariance operator only within matrix-vector products. Stored densely, the matrix takes $\mathcal{O}(N^{2})$ memory and each product $\mathcal{O}(N^{2})$ time. For a stationary kernel on an equispaced grid, the covariance matrix becomes (block-) Toeplitz and the product evaluates in $\mathcal{O}(N\log N)$ time using FFT-based circulant embedding, without the need to assemble the dense matrix. In a matched single-threaded comparison, the median speedup of the eigensolve grows from $18 \times$ at $N = 4096$ to $183 \times$ at $N = 2^{15}$. This makes it possible to compute discretized fields that would otherwise be infeasible to compute in the standard formulation. We show that the same construction carries over to non-separable kernels as well as $d$ dimensions, using block-Toeplitz matrices. We extend it to piecewise-constant fields on arbitrary domains given as subsets of a tensor grid. Computational savings grow with problem size, and storage drops from $\mathcal{O}(N^{2})$ to $\mathcal{O}(2^{d}N)$.

math.NA↗

arXiv2025

CODE: A global approach to ODE dynamics learning

Ordinary differential equations (ODEs) are a conventional way to describe the observed dynamics of physical systems. Scientists typically hypothesize about dynamical behavior, propose a mathematical model, and compare its predictions to data. However, modern computing and algorithmic advances now enable purely data-driven learning of governing dynamics directly from observations. In data-driven settings, one learns the ODE's right-hand side (RHS). Dense measurements are often assumed, yet high temporal resolution is typically both cumbersome and expensive. Consequently, one usually has only sparsely sampled data. In this work we introduce ChaosODE (CODE), a Polynomial Chaos ODE Expansion in which we use an arbitrary Polynomial Chaos Expansion (aPCE) for the ODE's right-hand side, resulting in a global orthonormal polynomial representation of dynamics. We evaluate the performance of CODE in several experiments on the Lotka-Volterra system, across varying noise levels, initial conditions, and predictions far into the future, even on previously unseen initial conditions. CODE exhibits remarkable extrapolation capabilities even when evaluated under novel initial conditions and shows advantages compared to well-examined methods using neural networks (NeuralODE) or kernel approximators (KernelODE) as the RHS representer. We observe that the high flexibility of NeuralODE and KernelODE degrades extrapolation capabilities under scarce data and measurement noise. Finally, we provide practical guidelines for robust optimization of dynamics-learning problems and illustrate them in the accompanying code.

cs.LG↗