Physics-based deep kernel learning for parameter estimation in high dimensional PDEs

2025-09-17

AI Summary

Solving high-dimensional partial differential equation (PDE) inverse problems under sparse observational data suffers from the curse of dimensionality and challenges in rigorous uncertainty quantification. Method: This paper proposes a physics-informed two-stage Bayesian framework. In Stage I, physics-constrained deep kernel learning (DKL) constructs an interpretable and generalizable surrogate model that explicitly encodes PDE priors. In Stage II, the DKL feature extractor’s weights are frozen, and Hamiltonian Monte Carlo (HMC) is employed to efficiently sample the posterior distribution over unknown parameters. Contribution/Results: The framework ensures computational scalability while significantly enhancing robustness to data sparsity and model complexity. Experiments demonstrate high-accuracy parameter estimation and reliable uncertainty quantification across canonical and high-dimensional PDE inverse problems. By unifying physical constraints with Bayesian inference in a modular architecture, the method establishes a novel paradigm for data-driven modeling of complex physical systems.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Inferring parameters of high-dimensional partial differential equations (PDEs) poses significant computational and inferential challenges, primarily due to the curse of dimensionality and the inherent limitations of traditional numerical methods. This paper introduces a novel two-stage Bayesian framework that synergistically integrates training, physics-based deep kernel learning (DKL) with Hamiltonian Monte Carlo (HMC) to robustly infer unknown PDE parameters and quantify their uncertainties from sparse, exact observations. The first stage leverages physics-based DKL to train a surrogate model, which jointly yields an optimized neural network feature extractor and robust initial estimates for the PDE parameters. In the second stage, with the neural network weights fixed, HMC is employed within a full Bayesian framework to efficiently sample the joint posterior distribution of the kernel hyperparameters and the PDE parameters. Numerical experiments on canonical and high-dimensional inverse PDE problems demonstrate that our framework accurately estimates parameters, provides reliable uncertainty estimates, and effectively addresses challenges of data sparsity and model complexity, offering a robust and scalable tool for diverse scientific and engineering applications.

Problem

• Estimating parameters in high-dimensional PDEs
• Addressing computational and inferential challenges
• Overcoming data sparsity and model complexity

Innovation

• Physics-based deep kernel learning integration
• Two-stage Bayesian framework with HMC
• Neural network feature extractor optimization