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

TL;DR

The paper introduces physics-informed neural networks (PINNs) that integrate PDE-based physics as residual constraints into neural network training, enabling data-efficient solution of nonlinear PDEs via automatic differentiation. It develops two architectural families—continuous-time and discrete-time PINNs—to address data-driven solution and discovery tasks, and validates them on Burgers', Schrödinger, and Allen–Cahn equations. Continuous-time PINNs enforce PDE residuals at collocation points, while discrete-time PINNs leverage Runge–Kutta time-stepping to enable large, stable time steps with high-order accuracy. The results demonstrate accurate, differentiable surrogate solutions from limited data and set the stage for future work in uncertainty quantification and PDE discovery (Part II).

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 two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

Figures (4)

• Figure 1: Burgers' equation:Top: Predicted solution $u(t,x)$ along with the initial and boundary training data. In addition we are using 10,000 collocation points generated using a Latin Hypercube Sampling strategy. Bottom: Comparison of the predicted and exact solutions corresponding to the three temporal snapshots depicted by the white vertical lines in the top panel. The relative $\mathcal{L}_{2}$ error for this case is $6.7 \cdot 10^{-4}$. Model training took approximately 60 seconds on a single NVIDIA Titan X GPU card.
• Figure 2: Shrödinger equation:Top: Predicted solution $|h(t,x)|$ along with the initial and boundary training data. In addition we are using 20,000 collocation points generated using a Latin Hypercube Sampling strategy. Bottom: Comparison of the predicted and exact solutions corresponding to the three temporal snapshots depicted by the dashed vertical lines in the top panel. The relative $\mathcal{L}_{2}$ error for this case is $1.97 \cdot 10^{-3}$.
• Figure 3: Burgers equation:Top: Solution $u(t,x)$ along with the location of the initial training snapshot at $t=0.1$ and the final prediction snapshot at $t=0.9$. Bottom: Initial training data and final prediction at the snapshots depicted by the white vertical lines in the top panel. The relative $\mathcal{L}_{2}$ error for this case is $8.2 \cdot 10^{-4}$.
• Figure 4: Allen-Cahn equation:Top: Solution $u(t,x)$ along with the location of the initial training snapshot at $t=0.1$ and the final prediction snapshot at $t=0.9$. Bottom: Initial training data and final prediction at the snapshots depicted by the white vertical lines in the top panel. The relative $\mathcal{L}_{2}$ error for this case is $6.99\cdot 10^{-3}$.