相关论文: General Covariance Data Augmentation for Neural PD…

Pretraining for partial differential equation (PDE) modeling has recently shown promise in scaling neural operators across datasets to improve generalizability and performance. Despite these advances, our understanding of how pretraining…

PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) use solvers of evolution PDEs as substitutes for the conventional components in G-CNNs. PDE-G-CNNs can offer several benefits simultaneously: fewer parameters, inherent…

Deep neural networks suffer from over-fitting and catastrophic forgetting when trained with small data. One natural remedy for this problem is data augmentation, which has been recently shown to be effective. However, previous works either…

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…

There has recently been increasing attention towards developing foundational neural Partial Differential Equation (PDE) solvers and neural operators through large-scale pretraining. However, unlike vision and language models that make use…

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…

The problem of solving partial differential equations (PDEs) can be formulated into a least-squares minimization problem, where neural networks are used to parametrize PDE solutions. A global minimizer corresponds to a neural network that…

In this paper, we compute numerical approximations of the minimal surfaces, an essential type of Partial Differential Equation (PDE), in higher dimensions. Classical methods cannot handle it in this case because of the Curse of…

Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear…

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this…

Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in…

Data augmentation is a major component of many machine learning methods with state-of-the-art performance. Common augmentation strategies work by drawing random samples from a space of transformations. Unfortunately, such sampling…

Linear systems with large differences between coefficients ("discontinuous coefficients") arise in many cases in which partial differential equations(PDEs) model physical phenomena involving heterogeneous media. The standard approach to…

"AI for Science" aims to solve fundamental scientific problems using AI techniques. As most physical phenomena can be described as Partial Differential Equations (PDEs) , approximating their solutions using neural networks has evolved as a…

Data augmentation is an effective and universal technique for improving generalization performance of deep neural networks. It could enrich diversity of training samples that is essential in medical image segmentation tasks because 1) the…

In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…

Covariate adjustment is a ubiquitous method used to estimate the average treatment effect (ATE) from observational data. Assuming a known graphical structure of the data generating model, recent results give graphical criteria for optimal…

Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications. Neural PDE solvers offer a promising alternative to classical numerical methods. However, existing approaches typically…

This paper introduces PDEformer-1, a versatile neural solver capable of simultaneously addressing various partial differential equations (PDEs). With the PDE represented as a computational graph, we facilitate the seamless integration of…