Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification

TL;DR

This work introduces Physics-constrained Polynomial Chaos (PC$^2$), a surrogate modeling framework that embeds physical constraints into the polynomial chaos expansion to support both scientific machine learning (SciML) and uncertainty quantification (UQ). By enforcing PDE residuals, initial/boundary conditions, and inequality-type constraints at virtual collocation points, PC$^2$ achieves physically realistic predictions with reduced reliance on expensive model evaluations. A sparse variant using Least Angle Regression (LAR) further enables scalable performance for high-dimensional problems. The approach is validated on linear and nonlinear PDEs, data-driven equation-of-state modeling, and stochastic beam UQ, showing high accuracy, built-in UQ capability, and substantial computational efficiency gains. Overall, PC$^2$ offers a versatile, physics-informed surrogate framework that enhances predictive fidelity and uncertainty assessment across SciML and UQ tasks.

Abstract

We present a novel physics-constrained polynomial chaos expansion as a surrogate modeling method capable of performing both scientific machine learning (SciML) and uncertainty quantification (UQ) tasks. The proposed method possesses a unique capability: it seamlessly integrates SciML into UQ and vice versa, which allows it to quantify the uncertainties in SciML tasks effectively and leverage SciML for improved uncertainty assessment during UQ-related tasks. The proposed surrogate model can effectively incorporate a variety of physical constraints, such as governing partial differential equations (PDEs) with associated initial and boundary conditions constraints, inequality-type constraints (e.g., monotonicity, convexity, non-negativity, among others), and additional a priori information in the training process to supplement limited data. This ensures physically realistic predictions and significantly reduces the need for expensive computational model evaluations to train the surrogate model. Furthermore, the proposed method has a built-in uncertainty quantification (UQ) feature to efficiently estimate output uncertainties. To demonstrate the effectiveness of the proposed method, we apply it to a diverse set of problems, including linear/non-linear PDEs with deterministic and stochastic parameters, data-driven surrogate modeling of a complex physical system, and UQ of a stochastic system with parameters modeled as random fields.

Figures (15)

• Figure 1: Computational flowchart of the sparse PC$^2$ based on least angle regression (LAR) algorithm.
• Figure 2: 2D Heat Equation: Comparison of the PC$^2$ solution (right) with the FEM solution (left) at $t=1$, demonstrating very good agreement.
• Figure 3: 2D Heat Equation: Convergence plots of the MSEs, $\epsilon=\epsilon_u+\epsilon_p$, for PC$^2$, sparse PC$^2$ and sparse PCE for an increasing number of training points, where $\epsilon_u$ and $\epsilon_p$ are computed with respect to the FEM solution $u(x,\ t)$ and physics-based residuals associated with the PDE, BCs, and ICs, respectively. The dotted line indicates the average relative error, and the shaded area represents the minimum and maximum error across 10 repetitions.
• Figure 4: 2D Heat Equation: Decay of the regularized loss, $L_\textrm{PC$^2$}{}$, with respect to the number of polynomial basis functions $k$ for different training sample sizes ($n_t$), indicating an improvement in PC$^2$ sparsity as $n_t$ increases for the user-defined target error, $\tau=0.008$.
• Figure 5: UQ plots for the stochastic 2D heat equation: Top panel: Plots of the mean $\mu_u(x,\ y)$ at $t=1$ obtained by MCS and PC$^2$, along with their absolute error. Bottom panel: Plots of the standard deviation of $\sigma_u(x,\ y)$ at $t=1$ obtained by MCS and PC$^2$, along with their absolute error. The plots demonstrate the excellent performance of PC$^2$ in solving the stochastic 2D heat equation equation without requiring model evaluations.