Data-Efficient Non-Gaussian Semi-Nonparametric Density Estimation for Nonlinear Dynamical Systems

arXiv stat.ML / 4/13/2026

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Key Points

The paper targets accurate density estimation for quantities of interest in nonlinear dynamical systems, focusing on non-Gaussian distributions that are hard to learn when forward simulation is computationally expensive.
It proposes a data-efficient seminonparametric (SNP/Gallant-Nychka) density estimation method using a probabilists' Hermite polynomial basis that is constructed to remain positive everywhere on the support.
Maximum-likelihood estimation of SNP coefficients is performed by approximating expectation integrals via Monte Carlo, while a convex relaxation is used to produce effective initial estimates.
The approach is evaluated on the chaotic Lorenz system for both density and quantile estimation, showing accurate recovery of non-Gaussian structure.
Results indicate that the method can compute quantiles with significantly fewer samples than raw Monte Carlo sampling.

Abstract

Accurate representation of non-Gaussian distributions of quantities of interest in nonlinear dynamical systems is critical for estimation, control, and decision-making, but can be challenging when forward propagations are expensive to carry out. This paper presents an approach for estimating probability density functions of states evolving under nonlinear dynamics using Seminonparametric (SNP), or Gallant-Nychka, densities. SNP densities employ a probabilists' Hermite polynomial basis to model non-Gaussian behavior and are positive everywhere on the support by construction. We use Monte Carlo to approximate the expectation integrals that arise in the maximum likelihood estimation of SNP coefficients, and introduce a convex relaxation to generate effective initial estimates. The method is demonstrated on density and quantile estimation for the chaotic Lorenz system. The results demonstrate that the proposed method can accurately capture non-Gaussian density structure and compute quantiles using significantly fewer samples than raw Monte Carlo sampling.