Probabilistic function-on-function nonlinear autoregressive model for emulation and reliability analysis of stochastic dynamical systems

2026-06-19 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Engineering & Applied Sciences, Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

The novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX) significantly enhances the emulation and reliability analysis of stochastic dynamical systems. This model reformulates conventional NARX from a function-on-function regression perspective, utilizing principal component analysis (PCA) for feature extraction and sparse Gaussian process (SGP) regression for learning. F2NARX employs a one-time-window-ahead prediction strategy and integrates the unscented transform for efficient probabilistic predictions. Benchmarking against state-of-the-art NARX models, F2NARX demonstrates orders-of-magnitude greater efficiency and higher accuracy. For instance, in a three-story steel frame example, it predicted 10,000 responses in 37 seconds with 50 training histories, achieving a normalized mean squared error below 1e-3, approximately 3,000 times faster than the finite element model. Its probabilistic capabilities also enable active learning for accurate first-passage failure probability estimation with only about 15 training trajectories.

Key takeaway

For Machine Learning Engineers and Research Scientists modeling complex dynamical systems, you should consider F2NARX to significantly enhance prediction efficiency and accuracy. Its function-on-function autoregressive approach, combined with sparse Gaussian processes and unscented transform, provides robust probabilistic predictions. This enables you to perform active learning for reliability analysis, accurately estimating first-passage failure probabilities with substantially fewer training trajectories, thereby reducing computational costs by orders of magnitude.

Key insights

F2NARX uses function-on-function autoregression with sparse Gaussian processes for efficient, probabilistic emulation of complex dynamical systems.

Principles

• Function-on-function autoregression significantly boosts predictive efficiency.
• PCA and SGP regression enable probabilistic predictions via unscented transform.
• Probabilistic predictions support active learning for reliability analysis.

Method

F2NARX employs a one-time-window-ahead strategy, using PCA for feature extraction from time windows. Sparse Gaussian Process regression learns these features, and the unscented transform provides probabilistic predictions.

In practice

• Emulate complex dynamical systems with significant speedup.
• Estimate first-passage failure probabilities using minimal training data.
• Integrate into uncertainty quantification and digital twin applications.

Topics

• F2NARX
• Dynamical Systems
• Surrogate Modeling
• Probabilistic Prediction
• Sparse Gaussian Processes
• Active Learning
• Reliability Analysis

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

Related on AIssential

• Dynestyx: A Probabilistic Programming Library for Dynamical Systems — Dynestyx unifies state-space model specification and Bayesian inference, making advanced methods accessible in probabilistic programming.
• Nonlinear function-on-function regression by RKHS — Nested Hilbert spaces and RKHS enable robust nonlinear function-on-function regression for complex functional data.
• AK-MCS-C2 : Active Kriging Monte Carlo Simulation method with conformal certification for failure probability estimation — AK-MCS-C2 combines active Kriging Monte Carlo simulation with conformal prediction for robust failure probability estimation.
• Federated Gaussian Process Learning via Pseudo-Representations for Large-Scale Multi-Robot Systems — pxpGP enables scalable, privacy-preserving Gaussian Process learning for large multi-robot systems via pseudo-representations and optimized ADMM.
• FreqLite: A Lightweight Frequency-Decomposed Linear Model with Adaptive Reversible Normalization for Robust Long-Term Time-Series Forecasting — FreqLite is a lightweight, frequency-decomposed linear model with adaptive normalization for robust, efficient long-term time-series forecasting.
• Stein’s method, learning and inference -or- how to really monitor convergence and thin chains — Stein's method, leveraging scores and operators, offers a robust approach for convergence monitoring and advanced probabilistic inference.

Open in AIssential →

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