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Partial differential equations (PDEs) govern a wide range of physical systems, but solving them efficiently remains a major challenge. The idea of a scientific foundation model (SciFM) is emerging as a promising tool for learning…

Uncertainty quantification (UQ) provides a resource-efficient solution for on-device monitoring of tinyML models deployed without access to true labels. However, existing UQ methods impose significant memory and compute demands, making them…

The intrusive (sample-free) spectral stochastic finite element method (SSFEM) is a powerful numerical tool for solving stochastic partial differential equations (PDEs). However, it is not widely adopted in academic and industrial…

(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…

The goals of this chapter are twofold. First, we wish to introduce molecular dynamics (MD) and uncertainty quantification (UQ) in a common setting in order to demonstrate how the latter can increase confidence in the former. In some cases,…

We consider the problem of providing optimal uncertainty quantification (UQ) --- and hence rigorous certification --- for partially-observed functions. We present a UQ framework within which the observations may be small or large in number,…

Predicting the future frames of a video is a challenging task, in part due to the underlying stochastic real-world phenomena. Prior approaches to solve this task typically estimate a latent prior characterizing this stochasticity, however…

Uncertainty quantification (UQ) methods play an important role in reducing errors in weather forecasting. Conventional approaches in UQ for weather forecasting rely on generating an ensemble of forecasts from physics-based simulations to…

Optimal control problems are crucial in various domains, including path planning, robotics, and humanoid control, demonstrating their broad applicability. The connection between optimal control and Hamilton-Jacobi (HJ) partial differential…

Accurate uncertainty quantification is a critical challenge in machine learning. While neural networks are highly versatile and capable of learning complex patterns, they often lack interpretability due to their ``black box'' nature. On the…

Current methods commonly used for uncertainty quantification (UQ) in deep learning (DL) models utilize Bayesian methods which are computationally expensive and time-consuming. In this paper, we provide a detailed study of UQ based on…

Solving high-dimensional partial differential equations (PDEs) is a critical challenge where modern data-driven solvers often lack reliability and rigorous error guarantees. We introduce Simulation-Calibrated Scientific Machine Learning…

We propose new and original mathematical connections between Hamilton-Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks…

Inverse uncertainty quantification (UQ) tasks such as parameter estimation are computationally demanding whenever dealing with physics-based models, and typically require repeated evaluations of complex numerical solvers. When partial…

Uncertainty quantification is an important and challenging problem in deep learning. Previous methods rely on dropout layers which are not present in modern deep architectures or batch normalization which is sensitive to batch sizes. In…

Hamilton-Jacobi (HJ) reachability analysis is a widely used method for ensuring the safety of robotic systems. Traditional approaches compute reachable sets by numerically solving an HJ Partial Differential Equation (PDE) over a grid, which…

Inverse Uncertainty Quantification (IUQ) method has been widely used to quantify the uncertainty of Physical Model Parameters (PMPs) in nuclear Thermal Hydraulics (TH) systems. This paper introduces a novel hierarchical Bayesian model which…

Researchers have proposed several approaches for neural network (NN) based uncertainty quantification (UQ). However, most of the approaches are developed considering strong assumptions. Uncertainty quantification algorithms often perform…

%Large vision-language models (LVLMs) have shown substantial advances in multimodal understanding and generation. However, when presented with incompetent or adversarial inputs, they frequently produce unreliable or even harmful content,…

Accurate representations of unknown and sub-grid physical processes through parameterizations (or closure) in numerical simulations with quantified uncertainty are critical for resolving the coarse-grained partial differential equations…