A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems

2026-06-12 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Emerging Technologies & Innovation · Depth: Expert, extended

Summary

A novel Quadratic Order Reduction - Gaussian Process Ordinary Differential Equation (QGPRODE) framework is introduced for forecasting complex dynamical systems and developing "Digital Shadows." This framework integrates Gaussian Process Ordinary Differential Equations (GPRODE), which approximates ODE drift using Gaussian Processes for accurate short-term forecasting with uncertainty quantification, with quadratic order reduced-order modelling and sphere projection. The QGPRODE extension enhances stability and efficiency in learning latent dynamics. Numerical experiments demonstrate that the full model outperforms traditional reduced-order modelling methods like Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition, and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. The framework is validated across diverse applications, including the Lorenz system, a synthetic quadratic latent system, the BV-α geophysical flow model, and global Earth air temperature forecasting using the ERA5 dataset, highlighting its potential as a robust and stable tool.

Key takeaway

For computational scientists developing robust forecasting models for large, nonlinear dynamical systems, especially for "Digital Shadows" applications, QGPRODE offers a compelling alternative to traditional reduced-order models. Its integration of Gaussian Processes with quadratic manifold projection provides superior accuracy, enhanced stability, and rigorous uncertainty quantification. You should evaluate QGPRODE when real-time updates and reliable predictive uncertainty bounds are critical for preventive decision-making in fields like numerical weather prediction or engineering.

Key insights

QGPRODE integrates Gaussian Processes with quadratic reduced-order modeling for stable, accurate, and uncertainty-quantified complex dynamical system forecasting.

Principles

• Reduced-order models face trade-offs in accuracy, stability, and interpretability.
• Gaussian Processes can approximate ODE drift with provable convergence and uncertainty.
• Sphere projection and quadratic reduction enhance stability and efficiency for latent dynamics.

Method

GPRODE approximates ODE drift using Gaussian Process Regression. QGPRODE extends this by constructing a quadratic reduced-order model, projecting dynamics onto a unit sphere for boundness, and applying GPRODE in the latent space.

In practice

• Implement Digital Shadows for near real-time anomaly detection in complex systems.
• Apply QGPRODE for robust forecasting in numerical weather prediction or geophysical models.

Topics

• Gaussian Processes
• Reduced Order Modeling
• Dynamical Systems Forecasting
• Uncertainty Quantification
• Digital Shadows
• Geophysical Flow Modeling

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.