相关论文: Neural SPDE solver for uncertainty quantification …

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form $L^{\beta}u = \mathcal{W}$, where $\mathcal{W}$ is…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Accurate quantification of uncertainty in neural network predictions remains a central challenge for scientific applications involving high-dimensional, correlated data. While existing methods capture either aleatoric or epistemic…

Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the…

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit…

Bayesian inference was once a gold standard for learning with neural networks, providing accurate full predictive distributions and well calibrated uncertainty. However, scaling Bayesian inference techniques to deep neural networks is…

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…

Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer…

Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…

The spatiotemporal resolution of Partial Differential Equations (PDEs) plays important roles in the mathematical description of the world's physical phenomena. In general, scientists and engineers solve PDEs numerically by the use of…

In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest.…

The rapid growth of earth observation systems calls for a scalable approach to interpolate remote-sensing observations. These methods in principle, should acquire more information about the observed field as data grows. Gaussian processes…

We present the first deep-learning solver for backward stochastic Volterra integral equations (BSVIEs) and their fully-coupled forward-backward variants. The method trains a neural network to approximate the two solution fields in a single…

Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms.…

We introduce a novel framework for uncertainty quantification of solution operators associated with stochastic partial differential equations (SPDEs). Although SPDEs play a central role in modeling complex physical systems under…

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

Deep Gaussian process models typically employ discrete hierarchies, but recent advancements in differential Gaussian processes (DiffGPs) have extended these models to infinite depths. However, existing DiffGP approaches often overlook the…

The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather…

In this paper a new approach for constructing \emph{multivariate} Gaussian random fields (GRFs) using systems of stochastic partial differential equations (SPDEs) has been introduced and applied to simulated data and real data. By solving a…

Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental for modeling stochastic dynamics across the natural sciences and modern machine learning. Learning their solution operators with…