Efficient Online Inference and Learning in Partially Known Nonlinear State-Space Models by Learning Expressive Degrees of Freedom Offline

TL;DR

This work tackles online inference and learning in partially known nonlinear state-space models by offline conditioning of a highly flexible Gaussian Process (GP) to a low-dimensional expressive subspace. The method first captures realistic target-function realizations with a Hilbert-GP using $N$ basis functions, then constructs $M<N$ expressive basis functions via data-driven conditioning (SVD) to enable online learning along a small set of degrees of freedom. A standard, noise-adaptive particle filter is employed for online inference in the reduced subspace, including an inverse-Wishart prior to adapt measurement noise and maintain robustness to rapid changes. Simulation on a nonlinear battery model shows rapid convergence with significantly fewer particles than baselines, highlighting the approach's potential for real-time operation without requiring expert prior structure and enabling nested nonlinear learning within first-principles models.

Abstract

Intelligent real-world systems critically depend on expressive information about their system state and changing operation conditions, e.g., due to variation in temperature, location, wear, or aging. To provide this information, online inference and learning attempts to perform state estimation and (partial) system identification simultaneously. Current works combine tailored estimation schemes with flexible learning-based models but suffer from convergence problems and computational complexity due to many degrees of freedom in the inference problem (i.e., parameters to determine). To resolve these issues, we propose a procedure for data-driven offline conditioning of a highly flexible Gaussian Process (GP) formulation such that online learning is restricted to a subspace, spanned by expressive basis functions. Due to the simplicity of the transformed problem, a standard particle filter can be employed for Bayesian inference. In contrast to most existing works, the proposed method enables online learning of target functions that are nested nonlinearly inside a first-principles model. Moreover, we provide a theoretical quantification of the error, introduced by restricting learning to a subspace. A Monte-Carlo simulation study with a nonlinear battery model shows that the proposed approach enables rapid convergence with significantly fewer particles compared to a baseline and a state-of-the-art method.

Figures (4)

• Figure 1: Unknown and varying effects $\boldsymbol{\Xi}$ in real-world systems that complicate operation (top) and approaches for learning of the underlying relationships (bottom). Without expert knowledge, online learning is difficult with state-of-the-art methods, e. g., Hilbert-gp Solin.2020, due to many inexpressive degrees of freedom (here: basis functions). In contrast, the proposed method enables efficient online learning by data-driven construction of few expressive basis functions.
• Figure 2: Proposed method to enable efficient online inference and learning without expert knowledge. First, "realistic" (i. e., actually occurring) shapes of the target function $\boldsymbol{\Xi}(\cdot,j)$, $j=1,\hdots,J$, are captured using the flexible Hilbert-gp formulation introduced in Solin.2020 with unspecific basis functions $\{\phi_n\}_{n=1}^{N}$. Second, in a data-driven conditioning step, a new set of few expressive basis functions $\{\rho_m\}_{m=1}^{M}$, $M<N$, is constructed from the most significant patterns in the Hilbert-gp coefficients $\boldsymbol{w}_j$ without expert knowledge. Based on the obtained low-dimensional approximation, efficient online inference and learning is accomplished using standard pf Ozkan.2013.
• Figure 3: Error between the true function $\Xi(x_k,j) = 10 \text{sinc}(j x_k /100)$ and the basis function expansions $\hat{\Xi}^{(i)}(x_k)$ for $i=N,M$ and for different realizations $j=1,\hdots,30$ on the domain $\Omega = \left[-15,15\right]$. Each error result corresponds to a dot, and the mean is drawn as a line. The number of required dof to achieve a certain approximation performance is significantly reduced using the proposed approach due to the choice of expressive basis functions.
• Figure 4: Online inference and learning in the simulation example \ref{['eq:batterymodel']} using the proposed Algorithm \ref{['alg:OnlineInferenceAndLearning']} for learning with the expressive basis function expansion $\hat{\Xi}^{(M)}$ and the pf described in Sec. \ref{['sec:OnlineLearningAndInference']}. Algorithm \ref{['alg:OnlineInferenceAndLearning']} converges rapidly from a wrong initial condition and after a sudden change in the true function $\Xi$ at $k=1,000$. The shown error is the mean for $50$ Monte-Carlo runs (standard deviation $\sigma$ shown semi-transparent), and the estimation results $\hat{\Xi}^{(M)}$ for the respective Monte-Carlo runs are presented in the bottom plots for the indicated iterations.

Theorems & Definitions (8)

• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4