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Exact Inference for Continuous-Time Gaussian Process Dynamics

AAAI 2024 · Probabilistic and scientific modeling › Dynamical systems · approach: Switching and law-discovery dynamics · 2 citations (OpenAlex)

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Learns continuous-time GP ODE dynamics from discrete noisy data by training GPs with multistep and Taylor integrator residuals (exact inference), and samples consistent dynamics via decoupled sampling.

Problem

GP dynamics model learning is usually trained on one-step-ahead predictions (explicit Euler), which is inaccurate for continuous-time dynamics, irregularly sampled data, and structure preservation; higher-order integrators normally require intermediate evaluations that make exact GP inference intrac

Approach

Use multistep and Taylor integrators whose dynamics evaluations only involve past and current trajectory points, so the integrator residual becomes a linear functional of the GP and exact GP inference with derived kernels is possible; decoupled sampling (Matheron's rule with random Fourier features) then yields full dynamics functions that can be integrated with arbitrary solvers.

Key results

• On the DHO system, BDF order 3 achieves the lowest DS-prediction MSE (0.006) versus 0.187 for GP-ODE and 0.750 for explicit Euler baseline.
• On VDP with irregular steps, Taylor order 3 reaches MSE 0.005, while AM/BDF order 1 (implicit Euler) and AB 1 baseline perform poorly (34.578 and 1.843).
• On MoCap, the GP-ODE (DS MSE 20.71) outperforms the proposed integrators under DS sampling, while AB 2 with mean predictions gives the best mean result (10.12) versus 37.76 for AB 1.

Measured effects reported in the paper

Surprising findings

• On the MoCap data, the GP-ODE baseline outperforms the proposed methods under decoupled sampling, and increasing order does not clearly improve DS predictions there.
• Explicit AB 2 on the DHO system and AM/BDF order 1 on several systems show large errors, so the order-accuracy trend is not monotone for all integrators.

Limitations

• Approach fails when step sizes are too large or too irregular, and order must be chosen by numerical validation.
• Multistep/Taylor kernels are computed per dimension and Taylor components assumed independent, an approximation for error bounds; MoCap results are weaker under DS and rely on PCA projection losing information.
• Error bounds rely on RKHS assumptions and noiseless data with constant jitter.

Experimental setup

Datasets: simulated damped harmonic oscillator simulated Van-der-Pol oscillator real mass-spring air track measurements CMU MoCap subject 09 short

Benchmarks: damped harmonic oscillator Van-der-Pol oscillator real spring system CMU MoCap subject 09 short

Metrics: mean squared error standard deviation over five runs

Baselines: explicit Euler (AB 1) GP-ODE AM 1/BDF 1

Builds on: decoupled sampling (wilson2020efficiently) GP-ODE Adams-Bashforth/Adams-Moulton/BDF multistep methods

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