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Exact Inference for Continuous-Time Gaussian Process Dynamics

Katharina Ensinger, Nicholas Tagliapietra, Sebastian Ziesche, Sebastian Trimpe

TL;DR

This work tackles learning continuous-time ODE dynamics from discrete, noisy observations by marrying Gaussian process regression with higher-order numerical integrators. It develops exact GP inference for multistep and Taylor integrators, derives tailored kernels from time discretization, and introduces decoupled sampling to draw consistent dynamics from the posterior. The authors provide theoretical error bounds that combine GP uncertainty with integrator accuracy and demonstrate improved ODE representations on simulated and real data, including irregular sampling scenarios. The results show that higher-order integrators yield more accurate, structure-preserving dynamics and enable near-exact predictions when used with appropriate sampling, with potential applications in physics-informed learning and reinforcement learning.

Abstract

Physical systems can often be described via a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. This can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. Higher-order numerical integrators provide the necessary tools to address this problem by discretizing the dynamics function with arbitrary accuracy. Many higher-order integrators require dynamics evaluations at intermediate time steps making exact GP inference intractable. In previous work, this problem is often tackled by approximating the GP posterior with variational inference. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to make direct inference tractable, we propose to leverage multistep and Taylor integrators. We demonstrate how to derive flexible inference schemes for these types of integrators. Further, we derive tailored sampling schemes that allow to draw consistent dynamics functions from the learned posterior. This is crucial to sample consistent predictions from the dynamics model. We demonstrate empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

Exact Inference for Continuous-Time Gaussian Process Dynamics

TL;DR

This work tackles learning continuous-time ODE dynamics from discrete, noisy observations by marrying Gaussian process regression with higher-order numerical integrators. It develops exact GP inference for multistep and Taylor integrators, derives tailored kernels from time discretization, and introduces decoupled sampling to draw consistent dynamics from the posterior. The authors provide theoretical error bounds that combine GP uncertainty with integrator accuracy and demonstrate improved ODE representations on simulated and real data, including irregular sampling scenarios. The results show that higher-order integrators yield more accurate, structure-preserving dynamics and enable near-exact predictions when used with appropriate sampling, with potential applications in physics-informed learning and reinforcement learning.

Abstract

Physical systems can often be described via a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. This can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. Higher-order numerical integrators provide the necessary tools to address this problem by discretizing the dynamics function with arbitrary accuracy. Many higher-order integrators require dynamics evaluations at intermediate time steps making exact GP inference intractable. In previous work, this problem is often tackled by approximating the GP posterior with variational inference. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to make direct inference tractable, we propose to leverage multistep and Taylor integrators. We demonstrate how to derive flexible inference schemes for these types of integrators. Further, we derive tailored sampling schemes that allow to draw consistent dynamics functions from the learned posterior. This is crucial to sample consistent predictions from the dynamics model. We demonstrate empirically and theoretically that our approach yields an accurate representation of the continuous-time system.
Paper Structure (31 sections, 1 theorem, 15 equations, 3 figures, 4 tables)

This paper contains 31 sections, 1 theorem, 15 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Technical Background
  3. Gaussian Process Regression
  4. Training:
  5. Decoupled sampling (DS):
  6. Numerical Integration
  7. Varying step-size multistep integrators:
  8. Taylor integrators:
  9. Method
  10. Main idea and setup:
  11. Common setup:
  12. Data Processing and Kernels
  13. Multistep integrators:
  14. Taylor integrators:
  15. Training and inference:
  16. ...and 16 more sections

Key Result

Theorem 1

Consider a multistep method of order $P$ with coefficient matrices $A$ and $B$ (cf. Eq. eq:var_step). Assume $f_u \in H_u$ with kernel $k$ and RKHS norm $\Vert f_u \Vert_{k} \leq C$. Further, assume that $|f_u^{P+1}|_{\Omega} \leq L$ and $|f_u^{P+2}I|_{\Omega} \leq L$. Under mild assumptions and wit

Figures (3)

  • Figure 1: Overview over training and predicting for dimension $u$. Data $\hat{x}_n$, time steps $t_n$ and integrator details are used to generate the integrator coefficients (blue). This allows one to compute all necessary components for training, including transformed observations $Y$, corresponding noise $\lambda$ and kernels (orange). After training, we obtain $\tilde{f}_u$ via DS (green).
  • Figure 2: DS predictions (left, middle) and corresponding phase (right) for the VDP system with Taylor order 1 and 3. Shaded regions indicate the GP uncertainty. With increasing order, a clear improvement is visible.
  • Figure 3: DS predictions on a single seed for the DHO system (left) and the real spring system (middle). Uncertainty is illustrated via shaded regions. Mean predictions in PCA space for MoCap (right). Raising the integrator order yields an improvement.

Theorems & Definitions (1)

  • Theorem 1: Multistep error