Deep Polynomial Chaos Expansion

2025-07-28

AI Summary

Traditional polynomial chaos expansion (PCE) suffers from the “curse of dimensionality,” where the number of basis functions grows exponentially with input dimensionality, severely limiting scalability in high-dimensional uncertainty quantification. To address this, we propose DeepPCE—a novel deep learning–enhanced PCE framework that for the first time integrates probabilistic circuit architectures into the PCE formalism. DeepPCE employs differentiable, hierarchical compositions of orthogonal polynomial bases, enabling depth-aware modeling without explicit construction of high-dimensional basis functions. Crucially, it preserves rigorous statistical interpretability and supports closed-form analytical computation of key uncertainty metrics—including mean, variance, covariance, and Sobol sensitivity indices—without requiring Monte Carlo sampling. Experiments demonstrate that DeepPCE matches MLPs in predictive accuracy while substantially improving modeling efficiency and scalability under high-dimensional inputs. This work establishes a new paradigm for efficient, interpretable uncertainty quantification in physics-based simulations.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Security and Privacy: Large-scale security measurements

Abstract

Polynomial chaos expansion (PCE) is a classical and widely used surrogate modeling technique in physical simulation and uncertainty quantification. By taking a linear combination of a set of basis polynomials - orthonormal with respect to the distribution of uncertain input parameters - PCE enables tractable inference of key statistical quantities, such as (conditional) means, variances, covariances, and Sobol sensitivity indices, which are essential for understanding the modeled system and identifying influential parameters and their interactions. As the number of basis functions grows exponentially with the number of parameters, PCE does not scale well to high-dimensional problems. We address this challenge by combining PCE with ideas from probabilistic circuits, resulting in the deep polynomial chaos expansion (DeepPCE) - a deep generalization of PCE that scales effectively to high-dimensional input spaces. DeepPCE achieves predictive performance comparable to that of multi-layer perceptrons (MLPs), while retaining PCE's ability to compute exact statistical inferences via simple forward passes.

Problem

• Scaling polynomial chaos expansion to high-dimensional problems
• Improving predictive performance while retaining exact statistical inference
• Combining PCE with probabilistic circuits for deep generalization

Innovation

• Combines PCE with probabilistic circuits
• Scales effectively to high-dimensional spaces
• Retains exact statistical inference capability