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In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by…

This work introduces a paradigm for constructing parametric neural operators that are derived from finite-dimensional representations of Green's operators for linear partial differential equations (PDEs). We refer to such neural operators…

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…

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically…

In this work, we describe a novel approach to building a neural PDE solver leveraging recent advances in transformer based neural network architectures. Our model can provide solutions for different values of PDE parameters without any need…

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…

Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the…

In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces,…

Transfer learning (TL) enables the transfer of knowledge gained in learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labeling, potential computational power…

In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their…

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each…

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…

In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as…

Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network…

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…

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…

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data…

Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…

Neural networks can be used to learn the solution of partial differential equations (PDEs) on arbitrary domains without requiring a computational mesh. Common approaches integrate differential operators in training neural networks using a…

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel…