**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Publication# Physics-informed machine learning for reduced-order modeling of nonlinear problems

Abstract

A reduced basis method based on a physics-informed machine learning framework is developed for efficient reduced-order modeling of parametrized partial differential equations (PDEs). A feedforward neural network is used to approximate the mapping from the time-parameter to the reduced coefficients. During the offline stage, the network is trained by minimizing the weighted sum of the residual loss of the reduced-order equations, and the data loss of the labeled reduced coefficients that are obtained via the projection of high-fidelity snapshots onto the reduced space. Such a network is referred to as physics-reinforced neural network (PRNN). As the number of residual points in time-parameter space can be very large, an accurate network – referred to as physics-informed neural network (PINN) – can be trained by minimizing only the residual loss. However, for complex nonlinear problems, the solution of the reduced-order equation is less accurate than the projection of high-fidelity solution onto the reduced space. Therefore, the PRNN trained with the snapshot data is expected to have higher accuracy than the PINN. Numerical results demonstrate that the PRNN is more accurate than the PINN and a purely data-driven neural network for complex problems. During the reduced basis refinement, the PRNN may obtain higher accuracy than the direct reduced-order model based on a Galerkin projection. The online evaluation of PINN/PRNN is orders of magnitude faster than that of the Galerkin reduced-order model.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Loading

Related publications

Loading

Related MOOCs

Loading

Related publications (7)

Related MOOCs (11)

Related concepts (13)

Loading

Loading

Loading

Warm-up for EPFL

Warmup EPFL est destiné aux nouvelles étudiantes et étudiants de l'EPFL.

Neuronal Dynamics - Computational Neuroscience of Single Neurons

The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.

Neuronal Dynamics - Computational Neuroscience of Single Neurons

The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.

Neural network

A neural network can refer to a neural circuit of biological neurons (sometimes also called a biological neural network), a network of artificial neurons or nodes in the case of an artificial neural network. Artificial neural networks are used for solving artificial intelligence (AI) problems; they model connections of biological neurons as weights between nodes. A positive weight reflects an excitatory connection, while negative values mean inhibitory connections. All inputs are modified by a weight and summed.

Space

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.

Galerkin method

In mathematics, in the area of numerical analysis, Galerkin methods are named after the Soviet mathematician Boris Galerkin. They convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions.

Learning to embed data into a space where similar points are together and dissimilar points are far apart is a challenging machine learning problem. In this dissertation we study two learning scenario

Qian Wang, Jan Sickmann Hesthaven

A physics-informed machine learning framework is developed for the reduced-order modeling of parametrized steady-state partial differential equations (PDEs). During the offline stage, a reduced basis

2020Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility

2020