Probabilistic Numeric SMC Sampling for Bayesian Nonlinear System Identification in Continuous Time

TL;DR

The paper tackles Bayesian identification of nonlinear continuous-time systems under measurement and numerical integration uncertainty. It fuses Sequential Monte Carlo with Probabilistic Numerics by solving ODEs as stochastic processes through an Integrated Wiener Process prior and enforcing ODE constraints via a discrete-time pseudo-measurement, yielding posterior distributions over both states and parameters that account for integration error. Key contributions include a method to approximate the marginal posterior $p(\theta | y_t, Z_t)$, a state-truncation strategy for computational efficiency, and benchmarking on three nonlinear dynamical datasets (Bouc-Wen, Silverbox, EMPS), demonstrating robust uncertainty quantification and potential for online estimation. The approach enhances reliability in engineering predictions by explicitly modeling numerical integration uncertainty, enabling risk-aware decisions in complex nonlinear systems.

Abstract

In engineering, accurately modeling nonlinear dynamic systems from data contaminated by noise is both essential and complex. Established Sequential Monte Carlo (SMC) methods, used for the Bayesian identification of these systems, facilitate the quantification of uncertainty in the parameter identification process. A significant challenge in this context is the numerical integration of continuous-time ordinary differential equations (ODEs), crucial for aligning theoretical models with discretely sampled data. This integration introduces additional numerical uncertainty, a factor that is often over looked. To address this issue, the field of probabilistic numerics combines numerical methods, such as numerical integration, with probabilistic modeling to offer a more comprehensive analysis of total uncertainty. By retaining the accuracy of classical deterministic methods, these probabilistic approaches offer a deeper understanding of the uncertainty inherent in the inference process. This paper demonstrates the application of a probabilistic numerical method for solving ODEs in the joint parameter-state identification of nonlinear dynamic systems. The presented approach efficiently identifies latent states and system parameters from noisy measurements. Simultaneously incorporating probabilistic solutions to the ODE in the identification challenge. The methodology's primary advantage lies in its capability to produce posterior distributions over system parameters, thereby representing the inherent uncertainties in both the data and the identification process.

Figures (9)

• Figure 1: Prior and posterior histograms and PDFs normalised by the true parameter values for the Bouc Wen system (denoted by $(\cdot)^*$) .
• Figure 2: Acceleration samples from the filtering distribution, EES and threshold for time instances 1600 to 3000.
• Figure 3: Samples from the posterior over the states for the Bouc Wen system plotted with the true states. An enlarge view of each state is shown to the left. Observations are plotted only for acceleration, as observations for the other states were not made available during the analysis.
• Figure 4: Measured input and response of the silver box benchmark.
• Figure 5: Prior and posterior histograms and PDFs for the silver box system.