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Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations
Maziar Raissi, Paris Perdikaris, George Em Karniadakis
TL;DR
The paper addresses how to discover parametrized nonlinear PDEs from limited observations when conventional machine-learning methods are unreliable in small-data settings. It uses physics-informed neural networks with continuous-time and discrete-time formulations, and demonstrates the approach across mathematical-physics benchmarks. The inferred models provide differentiable computational surrogates for forecasting, control, and optimization.
Problem
Data-driven discovery of nonlinear PDE parameters is needed from small, scattered, and potentially noisy observations, while standard machine-learning tools lack robustness and convergence guarantees in small-data regimes.
Method
Physics-informed neural networks encode the governing PDE while learning unknown parameters through continuous-time models or discrete-time models based on temporal snapshots.
Results
The approach demonstrates accurate and noise-robust parameter identification across benchmark problems including Burgers’ and Navier–Stokes equations.
Takeaways & Limitations
The inferred dynamic models yield computationally efficient, fully differentiable surrogates for predictive forecasting, control, and optimization.
Takeaways & Limitations
The authors characterize the treatise as presenting promising results while leaving further questions unanswered.
Abstract
from arXiv · showhide
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.
1. Introduction
The introduction frames data-driven PDE discovery as a small-data scientific computing problem where standard machine-learning methods lack robustness and convergence guarantees. It presents physics-informed neural networks and continuous-time and discrete-time approaches for identifying nonlinear PDE parameters from observations.
- Motivation: Standard deep, convolutional, and recurrent networks lack robustness and convergence guarantees when training examples are scarce.The paper identifies this small-data regime as one in which few training examples are available.
- Problem and contribution: This work shifts physics-informed neural networks toward data-driven discovery of parametrized nonlinear partial differential equations.The setup covers problems including conservation laws, diffusion, advection-diffusion-reaction systems, and kinetic equations.
- Problem and contribution: The discovery task is to infer parameters λ that best describe scattered, potentially noisy observations of a system’s hidden state u(t, x).The nonlinear operator is parametrized by λ, while u(t, x) denotes the latent solution.
- Approaches: The paper develops continuous-time models for scattered space-time measurements and discrete-time models that infer λ from two snapshots at distinct times.The two approaches differ in whether observations cover the spatio-temporal domain or consist of fixed temporal snapshots.
- Evaluation: The study reports systematic results across diverse mathematical-physics benchmarks while making its data and code publicly available.The introduction describes a series of benchmark studies and provides a GitHub repository for the manuscript’s data and code.
2. Continuous Time Models
Continuous-time physics-informed neural networks learn nonlinear PDE parameters while approximating solutions from scattered, potentially noisy spatio-temporal measurements. Burgers and Navier–Stokes examples show parameter identification, noise robustness, and pressure reconstruction from velocity data.
- Physics-informed formulation: The method approximates u(t, x) with a deep neural network and constructs f(t, x) by applying automatic differentiation to the governing PDE.The differential-operator parameters λ become trainable parameters of the physics-informed network.
- Physics-informed formulation: The loss combines data fit for u(t, x) with PDE residual enforcement at collocation points sharing the training data’s number and locations.MSEu measures agreement with observed solution data, while MSEf enforces the PDE structure.
- Burgers’ equation: 2,000 randomly sampled Burgers-equation points train a 9-layer network with 20 neurons per hidden layer to predict the full solution and identify unknown parameters.The experiment uses exact parameters λ1 = 1.0 and λ2 = 0.01/π and minimizes the mean squared error with L-BFGS.
- Burgers’ equation: The Burgers study reports reasonable identification accuracy under noise corruption up to 10%.The authors describe the methodology as robust to noise and compare it favorably with Gaussian-process and sparse-regression approaches.
3. Discrete Time Models
The discrete-time formulation identifies nonlinear PDE parameters from two temporal snapshots by embedding an implicit Runge–Kutta scheme in physics-informed neural networks. Burgers’ and KdV examples show accurate identification despite large temporal gaps, noise, and complex dynamics.
- Method: The method uses general q-stage Runge–Kutta schemes, covering both implicit and explicit time-stepping through the parameters {a_ij, b_j, c_j}.The stage values are represented by a multi-output neural network prior, producing physics-informed networks for the intermediate and next states.
- Method: Unknown PDE parameters and shared neural-network parameters are learned from noisy measurements at two distinct temporal snapshots by minimizing summed squared errors.The formulation trains on the two observed states while enforcing the differential-equation structure through the Runge–Kutta residuals.
- Burgers’ equation: Burgers’ equation recovered λ1 = 1.0 and λ2 = 0.01/π accurately for noise-free data and data with 1% Gaussian noise, despite ∆t = 0.8.The snapshots were taken at t_n = 0.1 and t_n+1 = 0.9, so they were far apart and could represent different dynamical regimes.
- Burgers’ equation: Sensitivity studies found the Burgers’ formulation robust to snapshot gaps, noise corruption, and neural-network architecture.The authors attribute this robustness mainly to implicit Runge–Kutta schemes admitting arbitrarily many stages without compromising nonlinear-dynamics resolution.
- Korteweg-de Vries equation: For KdV dynamics, parameter-estimation errors were 0.023% and 0.006% without noise, versus 0.057% and 0.017% with 1% noise, for λ1 and λ2 respectively.The two training snapshots differed dramatically because of complex nonlinear dynamics and were separated by a large temporal gap.
4. Summary and Discussion
The paper presents physics-informed neural networks for discovering dynamic models governed by parametrized nonlinear PDEs. It emphasizes that the inferred models provide efficient differentiable surrogates, while acknowledging that the broader treatise leaves substantial questions open.
- Contribution: The framework discovers dynamic models described by parametrized nonlinear partial differential equations using physics-informed neural networks.These networks encode underlying physical laws governing data sets when those laws are expressible as PDEs.
- Implications: The inferred models provide computationally efficient, fully differentiable surrogates for predictive forecasting, control, and optimization.These applications are presented as subsequent uses of the inferred models.
- Discussion: The two-part treatise presents promising results but creates more questions than it answers.The authors frame this open-ended status as part of a broader synergy between machine learning and classical computational physics.