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

Maziar Raissi, Paris Perdikaris+1 more|arXiv (Cornell University)|Nov 28, 2017

Model Reduction and Neural Networks316 citations

TL;DR

The authors develop physics-informed neural networks (PINNs) for data-driven discovery of nonlinear PDEs, using continuous-time and discrete-time formulations to identify governing equations from scarce or noisy data across spatio-temporal domains.

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.

Motivation & Objective

• Motivate and formalize data-driven discovery of nonlinear PDEs under unknown parameters.
• Develop continuous-time and discrete-time PINN frameworks to infer PDE operators and parameters from data.
• Demonstrate robustness to noise and data sparsity across benchmarks (Burgers’, Navier–Stokes, KdV).
• Show that PINNs can recover full state and auxiliary quantities (e.g., pressure) from partial data.

Proposed method

• Represent the state u(t,x) with a neural network and enforce the PDE residual f := u_t + N[u; lambda] via automatic differentiation.
• Train by minimizing a loss that combines data misfit and PDE residuals at collocation points (MSE_u + MSE_f).
• For Navier–Stokes, jointly learn velocity components and pressure by enforcing continuity and solving for lambda in f and g residuals.
• In discrete-time, use Runge-Kutta time-stepping to relate snapshots at two times and train neural networks for Runge-Kutta stage solutions along with lambda by minimizing SSE over both snapshots.
• Apply to Burgers’, Navier–Stokes, and KdV equations with noisy and sparse data, improving robustness and accuracy of inferred parameters.

Experimental results

Research questions

• RQ1 Can unknown nonlinear PDE parameters lambda be accurately identified from limited, possibly noisy data?
• RQ2 How can continuous-time and discrete-time PINN formulations be designed to handle scattered data and data snapshots at two time instants, respectively?
• RQ3 How well do PINNs recover not only the governing equations but also auxiliary fields (e.g., pressure) from partial observations?
• RQ4 What is the impact of data density, noise, and network/ RK scheme choices on the accuracy and robustness of PDE discovery?
• RQ5 Can the framework handle higher-order derivatives (e.g., KdV) and still identify underlying dynamics from sparse data?

Key findings

• PINNs can identify Burgers’ equation parameters with high accuracy and remain robust up to 10% data noise.
• Navier–Stokes parameter recovery achieves sub-0.2% error for lambda1 and around 5–6% for lambda2 under noise.
• Pressure field can be qualitatively recovered in Navier–Stokes from velocity data alone, up to an additive constant.
• Discrete-time PINNs using Runge-Kutta formulations enable accurate parameter recovery from two distant snapshots, even with noise and large gaps.
• KdV example demonstrates accurate recovery of parameters for higher-order PDEs from sparse snapshots with noise.
• Overall, physics-informed learning yields accurate, differentiable surrogates for predictive modeling, control, and optimization.