Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes
2026-08-17 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce a new way to find important patterns in complex polynomial models using a method called coded Hankel polynomial chaos (CH-PC). Instead of the usual approach that relies on sparse regression, their method transforms the polynomial data into a special form that can be analyzed using linear algebra techniques on Hankel matrices. This helps to identify dominant modes even with noisy or limited data, and it can work efficiently without handling huge matrices. They show through tests that their method can accurately pick out key model features, even when the problem is complicated or involves equations from physics.
Polynomial Chaos Expansion (PCE)Sparse RegressionHankel MatricesSpectral MethodsLow-Rank ApproximationPhase EncodingLegendre PolynomialsStochastic PDEQuadratureDominant Mode Identification
Authors
Zhiliang Deng, Xiaomei Yang
Abstract
Identification of dominant polynomial-chaos modes is usually formulated as a sparse-regression problem on a sampled multivariate polynomial dictionary. We develop coded Hankel polynomial chaos (CH-PC), a complementary spectral formulation for dominant-mode identification. A finite generating transform converts PCE coefficients into a coefficient-generating polynomial, and evaluation along a geometric phase orbit produces a finite exponential sum. Its model order and spectral nodes are encoded by low-rank Hankel matrices, while coordinate phase shifts attach root-of-unity labels from which the full polynomial multi-indices are recovered. Coordinate-shifted probes are combined as common-node snapshots, and independent phase encodings provide redundant representations when a single spectral encoding is poorly conditioned. For finite observations, population, finite-data, and observed probes are kept distinct: sampling or quadrature error and observation error enter as separate Hankel perturbations, which are then connected to spectral stability, discrete decoding, and phase voting. For tensor-product candidate sets, the generating kernel factorizes into one-dimensional sums and can be evaluated without assembling the full multivariate PCE design matrix. Numerical experiments on sparse Legendre benchmarks and a stochastic Darcy problem illustrate exact recovery, noise stabilization, unknown-order identification by phase persistence, and dominant-mode recovery for a PDE-generated quantity of interest.