Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

Inside the paper

The problem

Measurements of physical systems can be expensive and sparse, making it hard to infer solutions or governing parameters from data alone. Existing approaches to nonlinear equations also faced restrictions from approximation choices and time discretization.

The idea

The authors train neural networks using both observed data and constraints derived from a governing differential equation. They present continuous-time and discrete-time formulations for prediction and for identifying unknown equation parameters.

How it works

• A network represents the unknown solution as a function of its coordinates, and automatic differentiation supplies the derivatives needed by the equation.
• For continuous time, the loss combines mismatch to observed or boundary data with the equation residual at selected points.
• For discrete time, a Runge–Kutta construction connects network outputs at different stages to observations at selected times.
• In an inverse problem, unknown coefficients are trained alongside the network so that the inferred solution and equation fit the observations together.

What it showed

The paper reports experiments on several equations, including Schrödinger and Allen–Cahn examples for solution prediction and Navier–Stokes and Korteweg–de Vries examples for identification. These examples test the approach under particular data and noise settings rather than establishing a general guarantee.

Read it for

Read Section 2 for the problem setup and Section 3.1 for the continuous-time construction before inspecting Figure 1. Section 4 and Figure 3 show the inverse setting; the appendices contain further Burgers equation experiments and can be left for later.

Written in our own words from the paper itself. A draft the editors have not yet reviewed.