Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

TL;DR

The paper tackles data-driven discovery of nonlinear PDEs using physics-informed neural networks (PINNs) in two regimes: continuous-time models with a PDE residual and discrete-time models based on Runge-Kutta time stepping. It demonstrates accurate identification of governing parameters in Burgers', Navier–Stokes, and KdV equations, even from scarce or noisy data and from two time-snapshot pairs, while also reconstructing unobserved fields like pressure. The results show robustness to noise, large temporal gaps, and variations in network architecture, underscoring the method's potential for efficient, differentiable surrogates in forecasting, control, and optimization. The work highlights a productive synergy between machine learning and classical computational physics, with public code and data available for reproducibility.

Abstract

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this second part of our two-part treatise, we focus on the problem of data-driven discovery of partial differential equations. Depending on whether the available data is scattered in space-time or arranged in fixed temporal snapshots, we introduce two main classes of algorithms, namely continuous time and discrete time models. The effectiveness of our approach is demonstrated using a wide range of benchmark problems in mathematical physics, including conservation laws, incompressible fluid flow, and the propagation of nonlinear shallow-water waves.

Figures (5)

• Figure 1: Burgers equation:Top: Predicted solution $u(t,x)$ along with the training data. Middle: Comparison of the predicted and exact solutions corresponding to the three temporal snapshots depicted by the dashed vertical lines in the top panel. Bottom: Correct partial differential equation along with the identified one obtained by learning $\lambda_1$ and $\lambda_2$.
• Figure 2: Navier-Stokes equation:Top: Incompressible flow and dynamic vortex shedding past a circular cylinder at $Re=100$. The spatio-temporal training data correspond to the depicted rectangular region in the cylinder wake. Bottom: Locations of training data-points for the the stream-wise and transverse velocity components, $u(t,x,y)$ and $v(t,x,t)$, respectively.
• Figure 3: Navier-Stokes equation:Top: Predicted versus exact instantaneous pressure field $p(t,x,y)$ at a representative time instant. By definition, the pressure can be recovered up to a constant, hence justifying the different magnitude between the two plots. This remarkable qualitative agreement highlights the ability of physics-informed neural networks to identify the entire pressure field, despite the fact that no data on the pressure are used during model training. Bottom: Correct partial differential equation along with the identified one obtained by learning $\lambda_1, \lambda_2$ and $p(t,x,y)$.
• Figure 4: Burgers equation:Top: Solution $u(t,x)$ along with the temporal locations of the two training snapshots. Middle: Training data and exact solution corresponding to the two temporal snapshots depicted by the dashed vertical lines in the top panel. Bottom: Correct partial differential equation along with the identified one obtained by learning $\lambda_1, \lambda_2$.
• Figure 5: KdV equation:Top: Solution $u(t,x)$ along with the temporal locations of the two training snapshots. Middle: Training data and exact solution corresponding to the two temporal snapshots depicted by the dashed vertical lines in the top panel. Bottom: Correct partial differential equation along with the identified one obtained by learning $\lambda_1, \lambda_2$.