Related papers: Data-Driven Theory-guided Learning of Partial Diff…

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial…

Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that…

We investigate methods for learning partial differential equation (PDE) models from spatiotemporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select…

Stochastic partial differential equations (SPDEs) represent a very active research field with numerous recent developments and breakthrough results. There are several well-established approaches and methods used to construct solutions for…

We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central…

A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear SPDEs (stochastic partial differential equations) driven by multiplicative noise is considered. The discretization of the Lyapunov…

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…

Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but…

In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to…

Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…

We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of pattern-forming physics. The motivation comes from developmental…

Physics-informed neural networks (PINNs) were recently proposed in [1] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution while a PDE-induced NN is coupled to the solution NN,…

Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations…

(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…

We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven…

Scientific machine learning has been successfully applied to inverse problems and PDE discovery in computational physics. One caveat concerning current methods is the need for large amounts of ("clean") data, in order to characterize the…

The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to…

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction…