Related papers: Tensor Gaussian Processes: Efficient Solvers for N…

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is…

Decoders built on Gaussian processes (GPs) are enticing due to the marginalisation over the non-linear function space. Such models (also known as GP-LVMs) are often expensive and notoriously difficult to train in practice, but can be scaled…

Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…

Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion…

In recent years, there has been a notable surge in research on machine learning techniques for combinatorial optimization. It has been shown that learning-based methods outperform traditional heuristics and mathematical solvers on the…

Partial Differential Equations (PDEs) are the bedrock for modern computational sciences and engineering, and inherently computationally expensive. While PDE foundation models have shown much promise for simulating such complex…

This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…

Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed…

We propose a nonparametric density estimator based on the Gaussian process (GP) and derive three novel closed form learning algorithms based on Fisher divergence (FD) score matching. The density estimator is formed by multiplying a base…

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical…

We propose in this paper a Proper Generalized Decomposition (PGD) solver for reduced-order modeling of linear elastodynamic problems. It primarily focuses on enhancing the computational efficiency of a previously introduced PGD solver based…

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…

Scientific and engineering problems often require the use of artificial intelligence to aid understanding and the search for promising designs. While Gaussian processes (GP) stand out as easy-to-use and interpretable learners, they have…

In image reconstruction, an accurate quantification of uncertainty is of great importance for informed decision making. Here, the Bayesian approach to inverse problems can be used: the image is represented through a random function that…

The Gaussian process (GP) is a popular statistical technique for stochastic function approximation and uncertainty quantification from data. GPs have been adopted into the realm of machine learning in the last two decades because of their…

High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…

Memory bound applications such as solvers for large sparse systems of equations remain a challenge for GPUs. Fast solvers should be based on numerically efficient algorithms and implemented such that global memory access is minimised. To…

Stiff ordinary differential equations (ODEs) play an important role in many scientific and engineering applications. Often, the dependence of the solution of the ODE on additional parameters is of interest, e.g.\ when dealing with…

We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any…

This work introduces the Efficient Transformed Gaussian Process (ETGP), a new way of creating C stochastic processes characterized by: 1) the C processes are non-stationary, 2) the C processes are dependent by construction without needing a…