Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning

Neural operators provide fast surrogates for PDEs but their deterministic predictions limit their use in tasks requiring uncertainty quantification (UQ), especially under geometric variability. Existing approaches primarily model uncertainty in network parameters, largely overlooking the geometry-aware representations learned by the operator itself. We propose REEF-GP (Residual on Embedded Features Gaussian Process), a post-hoc UQ framework that fits a GP to the residuals of a frozen neural operator whose internal embeddings define the kernel feature space. Rather than learning a separate feature map, REEF-GP adapts the operator's intrinsic coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability on unstructured domains, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training that avoids restrictive low-rank approximations. Across five PDE benchmarks with varying geometries, REEF-GP preserves predictive accuracy while achieving calibrated uncertainty estimates competitive with deep ensembles but at a fraction of their cost. Our approach remains robust under geometric distribution shift, with uncertainty concentrating in physically meaningful regions (e.g., shock fronts). Our results demonstrate that accurate and scalable post-hoc UQ for neural operators can be achieved directly in their learned feature space, offering a practical alternative to parameter-centric approaches.

On the Uncertainty Quantification Ability of Tabular Foundation Models

Foundation models (FMs) have achieved substantial success in generalizing across tasks without problemspecific training or fine-tuning. However, many critical applications in mechanics and computational science require not only accurate predictions but also reliable uncertainty quantification (UQ). Herein we investigate the UQ capabilities of tabular FMs in regression tasks through a comprehensive empirical study comparing Tabular Prior-Data Fitted Networks (TabPFN) against Gaussian processes (GPs). We systematically evaluate these two methods across a host of regression problems with varying complexity, dataset sizes, and input dimensionalities. We use a default setting to build all the GPs and for a fair comparison against TabPFN v2.5. Our findings highlight an important trade-off between explicit and learned priors: while TabPFN achieves highly competitive performance for complex, high-dimensional problems with sufficient data, GPs often provide superior predictive accuracy and UQ in data-scarce settings. Moreover, when the chosen kernel constitutes a good prior for the underlying function, GP performance can substantially exceed that of TabPFN. Our results can be reproduced from https://github.com/kianswarehouse/GPvsPFN.

A preliminary data fusion study to assess the feasibility of Foundation Process-Property Models in Laser Powder Bed Fusion

Foundation models are at the forefront of an increasing number of critical applications. In regards to technologies such as additive manufacturing (AM), these models have the potential to dramatically accelerate process optimization and, in turn, design of next generation materials. A major challenge that impedes the construction of foundation process-property models is data scarcity. To understand the impact of this challenge, and since foundation models rely on data fusion, in this work we conduct controlled experiments where we focus on the transferability of information across different material systems and properties. More specifically, we generate experimental datasets from 17-4 PH and 316L stainless steels (SSs) in Laser Powder Bed Fusion (LPBF) where we measure the effect of five process parameters on porosity and hardness. We then leverage Gaussian processes (GPs) for process-property modeling in various configurations to test if knowledge about one material system or property can be leveraged to build more accurate machine learning models for other material systems or properties. Through extensive cross-validation studies and probing the GPs' interpretable hyperparameters, we study the intricate relation among data size and dimensionality, complexity of the process-property relations, noise, and characteristics of machine learning models. Our findings highlight the need for structured learning approaches that incorporate domain knowledge in building foundation process-property models rather than relying on uninformed data fusion in data-limited applications.

SEEK: Self-adaptive Explainable Kernel For Nonstationary Gaussian Processes

Gaussian processes (GPs) are powerful probabilistic models that define flexible priors over functions, offering strong interpretability and uncertainty quantification. However, GP models often rely on simple, stationary kernels which can lead to suboptimal predictions and miscalibrated uncertainty estimates, especially in nonstationary real-world applications. In this paper, we introduce SEEK, a novel class of learnable kernels to model complex, nonstationary functions via GPs. Inspired by artificial neurons, SEEK is derived from first principles to ensure symmetry and positive semi-definiteness, key properties of valid kernels. The proposed method achieves flexible and adaptive nonstationarity by learning a mapping from a set of base kernels. Compared to existing techniques, our approach is more interpretable and much less prone to overfitting. We conduct comprehensive sensitivity analyses and comparative studies to demonstrate that our approach is not robust to only many of its design choices, but also outperforms existing stationary/nonstationary kernels in both mean prediction accuracy and uncertainty quantification.