Graph Laplacian-based Bayesian Multi-fidelity Modeling

TL;DR

This work introduces a graph-Laplacian–based Bayesian framework to fuse dense low-fidelity data with sparse high-fidelity measurements for multi-fidelity modeling. By placing a graph spectral prior on low-to-high fidelity displacements and combining it with a Gaussian likelihood from limited high-fidelity data, the posterior is Gaussian with a MAP that can be computed via linear systems. Two scalable solvers—the truncated-eigenfunction expansion and Nyström-based low-rank approximation—enable efficient computation for high-dimensional QoIs and spatial fields. Across five physics-driven benchmarks (elasticity, Darcy flow, and Navier–Stokes–related heat flux), the method delivers 75–85% reductions in low-fidelity error using only 0.5–3.3% high-fidelity data, while also providing uncertainty quantification. The approach also offers theoretical convergence guarantees and attractive links to semi-supervised graph learning, making it practical for adaptive or active learning in large-scale simulations.

Abstract

We present a novel probabilistic approach for generating multi-fidelity data while accounting for errors inherent in both low- and high-fidelity data. In this approach a graph Laplacian constructed from the low-fidelity data is used to define a multivariate Gaussian prior density for the coordinates of the true data points. In addition, few high-fidelity data points are used to construct a conjugate likelihood term. Thereafter, Bayes rule is applied to derive an explicit expression for the posterior density which is also multivariate Gaussian. The maximum \textit{a posteriori} (MAP) estimate of this density is selected to be the optimal multi-fidelity estimate. It is shown that the MAP estimate and the covariance of the posterior density can be determined through the solution of linear systems of equations. Thereafter, two methods, one based on spectral truncation and another based on a low-rank approximation, are developed to solve these equations efficiently. The multi-fidelity approach is tested on a variety of problems in solid and fluid mechanics with data that represents vectors of quantities of interest and discretized spatial fields in one and two dimensions. The results demonstrate that by utilizing a small fraction of high-fidelity data, the multi-fidelity approach can significantly improve the accuracy of a large collection of low-fidelity data points.

Figures (16)

• Figure 1: Schematic representation of the low-fidelity approximation $\bar{\bm{u}}^{(i)}$, connected to the high-fidelity approximation $\hat{\bm{u}}^{(i)}$ and the true data point $\bm{u}^{(i)}$ through their respective displacement vectors, $\hat{\bm{\phi}}^{(i)}$ and $\bm{\phi}^{(i)}$, for $i \in \{1,\, \dots,\, M\}$.
• Figure 2: Schematic of the elastic body (light grey) with an elliptic stiffer inclusion (dark grey) for the elasticity problems (Case 1 and 2). The square is compressed on top with a uniform displacement $v=v_0$, while the bottom is fixed (Case 1).
• Figure 3: The figure shows orthogonal projections of the datasets for the low-fidelity (left column), multi-fidelity (center column), and high-fidelity (right column) models for the elasticity problem (Case 1). Each row shows a different orthogonal plane. The low-fidelity data points, in column 1, exhibit greater range compared to the high-fidelity point cloud. We note how the multi-fidelity points show better agreement with the target high-fidelity distribution.
• Figure 4: Histogram plots of the error distribution for the low- and multi-fidelity data for the elasticity problem (Case 1). Each plot shows the distribution of the error of the two models for one component of the vector of quantities of interest across the whole datasets. The values of mean and standard deviation of these distributions are reported in Table \ref{['tab:errors']}.
• Figure 5: Typical instances of the non-linear part of the vertical displacement fields from low- and multi-fidelity models for the elasticity problem (Case 2). The figure includes plots of point-wise error between each model and the high-fidelity field.

Theorems & Definitions (10)

• Remark
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• Lemma 4.1
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• Theorem 4.2
• proof