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 …
Leveraging viscous Hamilton–Jacobi PDEs for uncertainty quantification in scientific machine learning
Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful
predictive power of SciML with methods for quantifying the reliability of the learned models …
predictive power of SciML with methods for quantifying the reliability of the learned models …
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 …
Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots
Most problems within and beyond the scientific domain can be framed into one of the
following three levels of complexity of function approximation. Type 1: Approximate an …
following three levels of complexity of function approximation. Type 1: Approximate an …
A path-dependent PDE solver based on signature kernels
We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs).
Our numerical scheme leverages signature kernels, a recently introduced class of kernels …
Our numerical scheme leverages signature kernels, a recently introduced class of kernels …
Sparse Gaussian processes for solving nonlinear PDEs
This article proposes an efficient numerical method for solving nonlinear partial differential
equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) …
equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) …
[HTML][HTML] Operator learning with Gaussian processes
Operator learning focuses on approximating mappings G†: U→ V between infinite-
dimensional spaces of functions, such as u: Ω u→ R and v: Ω v→ R. This makes it …
dimensional spaces of functions, such as u: Ω u→ R and v: Ω v→ R. This makes it …
Towards Gaussian Process for operator learning: An uncertainty aware resolution independent operator learning algorithm for computational mechanics
S Kumar, R Nayek, S Chakraborty - Computer Methods in Applied …, 2025 - Elsevier
The growing demand for accurate, efficient, and scalable solutions in computational
mechanics highlights the need for advanced operator learning algorithms that can efficiently …
mechanics highlights the need for advanced operator learning algorithms that can efficiently …
Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations
S Kumar, R Nayek, S Chakraborty - arXiv preprint arXiv:2404.15618, 2024 - arxiv.org
The study of neural operators has paved the way for the development of efficient
approaches for solving partial differential equations (PDEs) compared with traditional …
approaches for solving partial differential equations (PDEs) compared with traditional …
Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
This paper introduces a novel kernel learning framework toward efficiently solving nonlinear
partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that …
partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that …