Related papers: Generalization Error Bounds for Picard-Type Operat…

We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…

Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress in recent years, enabling efficient approximation of complex spatiotemporal dynamics. However, most existing methods rely on fixed time step…

We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE). We propose a novel probabilistic formulation of the operator learning problem based on…

In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case 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…

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…

We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…

Neural operators offer a powerful paradigm for solving partial differential equations (PDEs) that cannot be solved analytically by learning mappings between function spaces. However, there are two main bottlenecks in training neural…

In this paper, we address the issue of modeling and estimating changes in the state of the spatio-temporal dynamical systems based on a sequence of observations like video frames. Traditional numerical simulation systems depend largely on…

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control…

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…

Physics informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely…

The non-commutative strategy developed by Bagarello (see Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394) for the analysis of systems of ordinary differential equations (ODEs) is extended to a class of partial…

We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase…

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…

We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral…

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on…

We aim to develop physics foundation models for science and engineering that provide real-time solutions to Partial Differential Equations (PDEs) which preserve structure and accuracy under adaptation to unseen geometries. To this end, we…

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…