Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers

TL;DR

The paper addresses propagating parametric uncertainty $\theta$ through probabilistic ODE solvers by marginalizing over $\theta$ rather than conditioning on it. It introduces a two-step approach that first approximates the marginal ODE solution $p(y(t))$ via numerical quadrature (producing a Gaussian mixture) and then computes the resulting moments, yielding a principled estimate of uncertainty in $y(t)$ that accounts for both numerical and parametric sources. The method leverages filtering-based ODE solvers with a $q$-times integrated Wiener process prior and spherical cubature, and it decomposes the propagated variance into solver (PN) and non-solver (non-PN) components, showing that solver uncertainty can mitigate overconfidence when step sizes are large. Empirical results on nonlinear systems such as the Logistic, FitzHugh–Nagumo, Lotka–Volterra, and Van der Pol equations confirm that the approach reproduces reference uncertainty well and provides practical benefits for uncertainty quantification in dynamical systems. This work demonstrates that probabilistic numerical methods can deliver meaningful uncertainty quantification for both numerical error and parametric propagation in ODEs.

Abstract

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical applications, however, the underlying dynamical system often contains uncertain parameters, requiring the propagation of this model uncertainty to the ODE solution. In this paper, we demonstrate that ODE filters, despite their probabilistic nature, do not automatically solve this uncertainty propagation problem. To address this limitation, we present a novel approach that combines ODE filters with numerical quadrature to properly marginalize over uncertain parameters, while accounting for both parameter uncertainty and numerical solver uncertainty. Experiments across multiple dynamical systems demonstrate that the resulting uncertainty estimates closely match reference solutions. Notably, we show how the numerical uncertainty from the ODE solver can help prevent overconfidence in the propagated uncertainty estimates, especially when using larger step sizes. Our results illustrate that probabilistic numerical methods can effectively quantify both numerical and parametric uncertainty in dynamical systems.

Figures (4)

• Figure 1: Difference between state estimation and uncertainty propagation. We consider two simple linear state-space models, both with transition model $\opbraces{p}(x_1 \mid x_0) = \opbraces{\mathcal{N}}(x_0, 0.01)$, observation model $\opbraces{p}(y_1 \mid x_1) = \opbraces{\mathcal{N}}(x_0, 0.01)$, and observation $y_1 = 2$, and with two different initial distributions $\opbraces{p}(x_0) = \opbraces{\mathcal{N}}(0, 1)$ and $\opbraces{p}(x_0) = \opbraces{\mathcal{N}}(0, 10)$. The top rows show the distributions computed by a regular Kalman filter applied to these models, which solves the corrensponding state estimation problems. The bottom rows instead show the solutions to the uncertainty propagation problem, in which the initial state should not be inferred but marginalized out. This aspect is visualized by the samples in gray, which are drawn from the initial distribution, propagated forward by running a separate Kalman filter for each sample, and then aggregated. In addition, the figure also demonstrates how the filter distribution barely changes when the initial uncertainty is increased, whereas the marginal filter distribution becomes significantly wider.
• Figure 2: Propagated uncertainties for the Logistic equation, FitzHugh--Nagumo system, Lotka--Volterra system, and Van der Pol oscillator. The figure shows both the mean (colored lines) and the 95% credible interval (shaded area) computed by our proposed method ("proposed") and by the reference solution ("reference") for each system. We observe that our method returns mean and variance estimates that largely align with the reference solution.
• Figure 3: Decomposition of the propagated variance into its non-PN and PN parts. For both the linear system and the Lotka--Volterra ODE, the non-PN variance decreases for larger step sizes and thereby diverges from the accurate reference variance. The PN variance on the other hand, which depends on the numerical uncertainty of the ODE filter, grows for increasing step sizes and helps fill the gap between the non-PN variance and the reference variance to reduce the resulting overconfidence.
• Figure 4: Propagated uncertainties for the Logistic equation, FitzHugh--Nagumo system, Lotka--Volterra system, and Van der Pol oscillator for non-Gaussian parameters. The figure shows both the mean (colored lines) and the 95% credible interval (shaded area) computed by our proposed method ("proposed") and by the reference solution ("reference") for each system. We observe that our method returns mean and variance estimates that largely align with the reference solution.

Theorems & Definitions (3)

• Remark 1: Initialization with higher-order derivatives
• Remark 2: Other numerical quadrature methods
• Remark 3: The role of numerical uncertainty and relation to non-probabilistic numerical simulators