Related papers: Mesh-clustered Gaussian process emulator for parti…
We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…
Machine learning solvers for partial differential equations (PDEs) have attracted growing interest. However, most existing approaches, such as neural network solvers, rely on stochastic training, which is inefficient and typically requires…
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial…
Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
Inverse problems for Partial Differential Equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm…
We introduce and study a new class of partial differential equations (PDEs) with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared to purely stochastic PDEs or purely fuzzy PDEs, fuzzy-stochastic PDEs offer powerful…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
We propose a framework for training neural networks that are coupled with partial differential equations (PDEs) in a parallel computing environment. Unlike most distributed computing frameworks for deep neural networks, our focus is to…
Gaussian processes provide a flexible framework for spatial prediction, but their computational cost limits applicability to large-scale data with large sample size $n$. Predictive processes (PPs), a popular low-rank approximation, mitigate…
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
As further progress in the accurate and efficient computation of coupled partial differential equations (PDEs) becomes increasingly difficult, it has become highly desired to develop new methods for such computation. In deviation from…
The study presents a general framework for discovering underlying Partial Differential Equations (PDEs) using measured spatiotemporal data. The method, called Sparse Spatiotemporal System Discovery ($\text{S}^3\text{d}$), decides which…
Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities…