gaussian process regression for dynamics

Modeling non-conservative dynamical systems without velocity or momentum measurements remains challenging due to the difficulty of enforcing physical consistency in learned dynamics. Method: We propose the Non-Conservative Hamiltonian Gaussian Process (NCHGP) framework, which integrates Hamiltonian structure with Gaussian processes and incorporates an energy-conservation prior. It employs full Bayesian inference to jointly estimate latent states and heterogeneous hyperparameters, and leverages a low-rank approximation for scalable training and principled uncertainty quantification. Contribution/Results: To our knowledge, NCHGP is the first method enabling Hamiltonian-structured Bayesian modeling solely from input–output data—without requiring momentum observations. Experiments on nonlinear systems demonstrate substantial improvements in modeling accuracy and uncertainty calibration over state-of-the-art alternatives that rely on momentum measurements, while achieving superior robustness and data efficiency.

To address low prediction accuracy and poor uncertainty quantification in high-dimensional nonlinear solid mechanics simulations, this paper proposes a coupled surrogate model—Deep Autoencoder–Gaussian Process (DAE-GP). The method uniquely integrates the nonlinear dimensionality reduction capability of deep autoencoders with the Bayesian regression and probabilistic uncertainty modeling capacity of Gaussian processes, thereby overcoming the longstanding challenge of jointly achieving high accuracy and reliable uncertainty estimation for complex, high-dimensional nonlinear responses. Driven by nonlinear finite element simulation data, the DAE-GP model significantly improves both predictive accuracy and computational efficiency while delivering physically interpretable, pointwise uncertainty estimates. Experimental results demonstrate its strong generalization and robustness under challenging scenarios involving complex boundary conditions and material nonlinearity. This work establishes a new paradigm for high-fidelity real-time simulation and reliability analysis in computational solid mechanics.

Modeling mesh signals on geometrically variable graphs—where node counts, sizes, and adjacency structures differ across instances—remains a fundamental challenge in computational physics. Method: We propose the first unified framework integrating regularized optimal transport, graph embedding for dimensionality reduction, and Graph-Indexed Gaussian Processes (GIGP). Our approach maps variable-graph inputs onto a shared low-dimensional manifold and constructs a Gaussian process regression model directly on the graph-indexed space, with rigorous theoretical guarantees. Contribution/Results: The framework enables analytic node-level prediction intervals—the first such capability for graph-structured physical signals—while supporting uncertainty quantification and active learning. Evaluated on real-world fluid and solid mechanics engineering problems, it improves prediction accuracy (32% average error reduction) and uncertainty calibration (41% NLL improvement). This work establishes an interpretable, verifiable paradigm for physics-informed machine learning design.

Gaussian process regression (GPR) is often treated as a black-box surrogate model, limiting its pedagogical utility and interpretability in uncertainty quantification (UQ) for beginners. Core UQ tasks—including uncertainty propagation, risk estimation, Bayesian optimization, parameter inference, and sensitivity analysis—require deeper engagement with GPR’s inherent probabilistic structure. Method: This work develops a systematic, pedagogically grounded GPR-based UQ framework that integrates UQ-specific techniques: Bayesian quadrature, active learning, and surrogate-based sensitivity analysis. It emphasizes principled covariance kernel design, Bayesian hyperparameter estimation, and reproducible implementation. Contribution/Results: The framework lowers the barrier to applying GPR in complex UQ scenarios, enhances model transparency and decision reliability, and provides a theoretically rigorous yet practically actionable paradigm for UQ education and research across engineering and scientific disciplines.

This work addresses surrogate modeling for partial differential equation (PDE)-constrained systems with uncertain input locations. We propose a novel Bayesian framework that integrates Bayesian inference with Gaussian process regression (GPR). Our key contribution is the first explicit probabilistic modeling of input location uncertainty, achieved via joint Bayesian inversion to simultaneously infer the input distribution and the PDE solution function. During prediction, we analytically marginalize over the input uncertainty, thereby unifying treatment of deterministic boundary/initial conditions and stochastic observational data. The method is validated on the heat equation, Allen–Cahn equation, and diverse one-dimensional functions, demonstrating significantly reduced predictive variance and robust generalization. It provides a scalable, probabilistically rigorous framework for PDE surrogate modeling under input uncertainty.

This study addresses the limitations of traditional state-space models, which rely on predefined nonlinear dynamics and struggle with theoretically under-specified complex systems, as well as the high computational cost of Bayesian inference in Gaussian process state-space models for moderately long sequences. To overcome these challenges, the authors propose two enhanced Gibbs sampling strategies that substantially improve sampling efficiency and convergence reliability. By integrating confirmatory factor analysis to construct an identifiable and interpretable measurement structure, they develop a comprehensive framework for learning nonlinear latent dynamical systems. Simulation studies validate the accuracy of posterior inference, while two empirical applications demonstrate the method’s practical utility and interpretability. An open-source implementation is provided, offering researchers an efficient and feasible workflow for empirical analysis.

本文针对动力系统参数估计问题，提出了一种结合高斯过程学习与流映射精炼的两阶段方法，以提高在稀疏和噪声观测下的参数估计精度。

本文针对多物理场预测中的线性等式约束问题，提出了一种基于行向PCA和线性约束多输出高斯过程的新方法。

This study addresses the miscalibration of predictive uncertainty in Gaussian processes (GPs) arising from model misspecification by proposing the Prediction-Oriented Gaussian Process (PrO-GP). Moving beyond the limitations of conventional nonparametric Bayesian inference, this method explicitly treats the calibration of the predictive distribution as its core optimization objective. By leveraging a dimensionality reduction formulation coupled with a Markov chain Monte Carlo (MCMC) sampling algorithm, PrO-GP enables efficient, prediction-oriented Bayesian inference. Experiments conducted on both synthetic and real-world datasets demonstrate that PrO-GP substantially outperforms standard GPs, significantly improving the calibration accuracy of predictive uncertainty under model misspecification. These results establish PrO-GP as an effective framework for robust prediction in scenarios where standard GP assumptions are violated.

Traditional Gaussian process regression often yields physically inconsistent results when reconstructing full-field modal shapes from sparse sensor data. This work proposes a physics-constrained single-output Gaussian process framework (CONS-SOGP), which, for the first time, embeds mass orthogonality constraints directly into Gaussian process regression. By jointly optimizing independent modal kernels and an orthogonality penalty term, the method achieves high-fidelity modal expansion while preserving physical plausibility. Leveraging marginal likelihood derivation and gradient-based hyperparameter optimization, CONS-SOGP demonstrates significantly improved performance over existing Gaussian process approaches in numerical experiments on multi-degree-of-freedom structures, yielding more accurate and reliable reconstructions of modal shapes.