Related papers: A Geometry-Aware Operator Learning Framework for I…

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…

Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving…

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…

We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The proposed framework employs a loss function…

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning.…

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically…

The neural operator (NO) framework has emerged as a powerful tool for solving partial differential equations (PDEs). Recent NOs are dominated by the Transformer architecture, which offers NOs the capability to capture long-range…

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…

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform…

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets…

We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics…

We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator…

The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast…

Koopman analysis of a general dynamics system provides a linear Koopman operator and an embedded eigenfunction space, enabling the application of standard techniques from linear analysis. However, in practice, deriving exact operators and…

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural…

In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first…

Initial boundary value problems arise commonly in applications with engineering and natural systems governed by nonlinear partial differential equations (PDEs). Operator learning is an emerging field for solving these equations by using a…

Learning solution operators for partial differential equations (PDEs) has become a foundational task in scientific machine learning. However, existing neural operator methods require abundant training data for each specific PDE and lack the…

Physics-informed neural operators have emerged as a powerful paradigm for solving parametric partial differential equations (PDEs), particularly in the aerospace field, enabling the learning of solution operators that generalize across…

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…