Inverse problems for physics-based process models
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 …
model in the context of uncertainty and random variability: the Bayesian inverse problem …
Machine learning of linear differential equations using Gaussian processes
M Raissi, P Perdikaris, GE Karniadakis - Journal of Computational Physics, 2017 - Elsevier
This work leverages recent advances in probabilistic machine learning to discover
governing equations expressed by parametric linear operators. Such equations involve, but …
governing equations expressed by parametric linear operators. Such equations involve, but …
Solving and learning nonlinear PDEs with Gaussian processes
We introduce a simple, rigorous, and unified framework for solving nonlinear partial
differential equations (PDEs), and for solving inverse problems (IPs) involving the …
differential equations (PDEs), and for solving inverse problems (IPs) involving the …
[HTML][HTML] Physics informed machine learning: Seismic wave equation
S Karimpouli, P Tahmasebi - Geoscience Frontiers, 2020 - Elsevier
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 …
extensively in geosciences for both small-and large-scale problems. However, the necessity …
Bayesian probabilistic numerical methods
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 …
as an alternative criterion with which to assess numerical methods. In contrast to worst-case …
[HTML][HTML] A modern retrospective on probabilistic numerics
CJ Oates, TJ Sullivan - Statistics and computing, 2019 - Springer
This article attempts to place the emergence of probabilistic numerics as a mathematical–
statistical research field within its historical context and to explore how its gradual …
statistical research field within its historical context and to explore how its gradual …
Error analysis of kernel/GP methods for nonlinear and parametric PDEs
We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear,
and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process …
and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process …
Gaussian process priors for systems of linear partial differential equations with constant coefficients
M Harkonen, M Lange-Hegermann… - … on machine learning, 2023 - proceedings.mlr.press
Partial differential equations (PDEs) are important tools to model physical systems and
including them into machine learning models is an important way of incorporating physical …
including them into machine learning models is an important way of incorporating physical …
Bayesian inverse problems are usually well-posed
J Latz - SIAM Review, 2023 - SIAM
Inverse problems describe the task of blending a mathematical model with observational
data---a fundamental task in many scientific and engineering disciplines. The solvability of …
data---a fundamental task in many scientific and engineering disciplines. The solvability of …
[图书][B] Probabilistic Numerics: Computation as Machine Learning
Probabilistic numerical computation formalises the connection between machine learning
and applied mathematics. Numerical algorithms approximate intractable quantities from …
and applied mathematics. Numerical algorithms approximate intractable quantities from …