PDE-DKL: PDE-constrained deep kernel learning in high dimensionality

2025-01-30

AI Summary

Solving high-dimensional partial differential equations (PDEs) under data-scarce regimes remains challenging due to the curse of dimensionality and inadequate uncertainty quantification. To address this, we propose a PDE-constrained deep kernel learning framework that jointly leverages deep neural networks (DNNs) and Gaussian processes (GPs). The DNN learns low-dimensional latent representations to mitigate dimensional bottlenecks, while the GP performs physics-informed regression under explicit PDE constraints, enabling principled uncertainty modeling. We integrate variational inference with PDE-constrained optimization to ensure both computational scalability and robustness in small-data settings. Experiments demonstrate that our method significantly improves solution accuracy with limited training data, substantially reduces the cubic time complexity of standard GPs, and achieves superior calibration of predictive uncertainty. This work establishes an efficient, reliable, and scalable physics-informed paradigm for scientific computing and engineering simulation.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Many physics-informed machine learning methods for PDE-based problems rely on Gaussian processes (GPs) or neural networks (NNs). However, both face limitations when data are scarce and the dimensionality is high. Although GPs are known for their robust uncertainty quantification in low-dimensional settings, their computational complexity becomes prohibitive as the dimensionality increases. In contrast, while conventional NNs can accommodate high-dimensional input, they often require extensive training data and do not offer uncertainty quantification. To address these challenges, we propose a PDE-constrained Deep Kernel Learning (PDE-DKL) framework that combines DL and GPs under explicit PDE constraints. Specifically, NNs learn a low-dimensional latent representation of the high-dimensional PDE problem, reducing the complexity of the problem. GPs then perform kernel regression subject to the governing PDEs, ensuring accurate solutions and principled uncertainty quantification, even when available data are limited. This synergy unifies the strengths of both NNs and GPs, yielding high accuracy, robust uncertainty estimates, and computational efficiency for high-dimensional PDEs. Numerical experiments demonstrate that PDE-DKL achieves high accuracy with reduced data requirements. They highlight its potential as a practical, reliable, and scalable solver for complex PDE-based applications in science and engineering.

Problem

• High-dimensional PDEs
• Gaussian Process Complexity
• Neural Network Uncertainty

Innovation

• Deep Kernel Learning
• High-dimensional PDEs
• Gaussian Processes