With the availability of data and computational technologies in the modern world, machine
learning (ML) has emerged as a preferred methodology for data analysis and prediction …

Surrogate modeling and uncertainty quantification tasks for PDE systems are most often
considered as supervised learning problems where input and output data pairs are used for …

We introduce a simple, rigorous, and unified framework for solving nonlinear partial
differential equations (PDEs), and for solving inverse problems (IPs) involving the …

We present a deep learning framework for quantifying and propagating uncertainty in
systems governed by non-linear differential equations using physics-informed neural …

This work leverages recent advances in probabilistic machine learning to discover
governing equations expressed by parametric linear operators. Such equations involve, but …

We describe and compare two formulations of inverse problems for a physics-based process
model in the context of uncertainty and random variability: the Bayesian inverse problem …

For more than two centuries, solutions of differential equations have been obtained either
analytically or numerically based on typically well-behaved forcing and boundary conditions …

Similar to many fields of sciences, recent deep learning advances have been applied
extensively in geosciences for both small-and large-scale problems. However, the necessity …

Over forty years ago average-case error was proposed in the applied mathematics literature
as an alternative criterion with which to assess numerical methods. In contrast to worst-case …

Linear partial differential equations (PDEs) are an important, widely applied class of
mechanistic models, describing physical processes such as heat transfer, electromagnetism …