Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Physics-Informed Neural Networks and Extensions

Maziar Raissi, Paris Perdikaris, Nazanin Ahmadi, George Em Karniadakis

TL;DR

The paper surveys Physics-Informed Neural Networks (PINNs) as a central paradigm in scientific machine learning for integrating data with governing PDEs. It covers core PINN methodology, several extensions (adaptive loss weighting, domain decomposition, long-time integration, and generalized PDE types), and a data-driven approach to discovering dynamical systems using multistep schemes and symbolic regression. A key example demonstrates learning the glycolytic oscillator dynamics and recovering explicit equations with PySR, illustrating data-efficient learning and equation discovery under partial physics. The discussion emphasizes the practical impact of PINNs in reducing mesh generation, enabling uncertainty quantification, and facilitating discovery of governing laws, while outlining challenges in accuracy, cost, and scalability that motivate ongoing research.

Abstract

In this paper, we review the new method Physics-Informed Neural Networks (PINNs) that has become the main pillar in scientific machine learning, we present recent practical extensions, and provide a specific example in data-driven discovery of governing differential equations.

Physics-Informed Neural Networks and Extensions

TL;DR

The paper surveys Physics-Informed Neural Networks (PINNs) as a central paradigm in scientific machine learning for integrating data with governing PDEs. It covers core PINN methodology, several extensions (adaptive loss weighting, domain decomposition, long-time integration, and generalized PDE types), and a data-driven approach to discovering dynamical systems using multistep schemes and symbolic regression. A key example demonstrates learning the glycolytic oscillator dynamics and recovering explicit equations with PySR, illustrating data-efficient learning and equation discovery under partial physics. The discussion emphasizes the practical impact of PINNs in reducing mesh generation, enabling uncertainty quantification, and facilitating discovery of governing laws, while outlining challenges in accuracy, cost, and scalability that motivate ongoing research.

Abstract

In this paper, we review the new method Physics-Informed Neural Networks (PINNs) that has become the main pillar in scientific machine learning, we present recent practical extensions, and provide a specific example in data-driven discovery of governing differential equations.
Paper Structure (5 sections, 1 equation, 2 figures, 1 table)

This paper contains 5 sections, 1 equation, 2 figures, 1 table.

Table of Contents

  1. Overview
  2. Physics-Informed Neural Networks - PINNs
  3. Extensions of PINNs
  4. Data-Driven Discovery of Dynamical Systems
  5. Outlook

Figures (2)

  • Figure 1: Schematic to illustrate three possible categories of physical problems and associated available data: Physics-informed neural networks can integrate seamlessly data and parameterized PDEs, including models with missing physics, in a unified way expressed compactly using automatic differentiation and PDE-induced neural networks.
  • Figure 2: Basic structure of PINN for conservation laws. The left (physics uninformed) network represents the PDE solution "U(x,t)" while the right (physics informed) network describes the PDE residual "F(x,t)". Currently, optimization is done by a human-in-the-loop empirically based on trial and error. Note that the "U" architecture is a fully-connected DNN here (or CNN, RNN, hybrid), while the "F" architecture is dictated by the PDE and is, in general, not possible to visualize explicitly. Its depth is proportional to the highest derivative order in the PDE times the depth of the uninformed "U" DNN.