Related papers: PDEformer: Towards a Foundation Model for One-Dime…

Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale…

This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of…

Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all…

Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…

In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of…

With the increases in computational power and advances in machine learning, data-driven learning-based methods have gained significant attention in solving PDEs. Physics-informed neural networks (PINNs) have recently emerged and succeeded…

Recent developments in the field of neural partial differential equation (PDE) solvers have placed a strong emphasis on neural operators. However, the paper "Message Passing Neural PDE Solver" by Brandstetter et al. published in ICLR 2022…

This paper explores the efficacy of diffusion-based generative models as neural operators for partial differential equations (PDEs). Neural operators are neural networks that learn a mapping from the parameter space to the solution space of…

The Transformer architecture has revolutionized artificial intelligence, yet a principled theoretical understanding of its internal mechanisms remains elusive. This paper introduces a novel analytical framework that reconceptualizes the…

The task of simultaneously reconstructing multiple physical coefficients in partial differential equations (PDEs) from observed data is ubiquitous in applications. In this work, we propose an integrated data-driven and model-based iterative…

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…

Time-independent Partial Differential Equations (PDEs) on large meshes pose significant challenges for data-driven neural PDE solvers. We introduce a novel graph rewiring technique to tackle some of these challenges, such as aggregating…

Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…

We present PDE-FM, a modular foundation model for physics-informed machine learning that unifies spatial, spectral, and temporal reasoning across heterogeneous partial differential equation (PDE) systems. PDE-FM combines spatial-spectral…

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained…

We propose a framework for solving nonlinear partial differential equations (PDEs) by combining perturbation theory with one-shot transfer learning in Physics-Informed Neural Networks (PINNs). Nonlinear PDEs with polynomial terms are…

Partial differential equations (PDEs) govern a wide range of physical systems, and recent multimodal foundation models have shown promise for learning PDE solution operators across diverse equation families. However, existing multi-operator…

We present a novel method for using Neural Networks (NNs) for finding solutions to a class of Partial Differential Equations (PDEs). Our method builds on recent advances in Neural Radiance Field research (NeRFs) and allows for a NN to…

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…

We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the…