[Paper Review] Data-driven discovery of partial differential equations

Samuel Rudy|arXiv (Cornell University)|Sep 21, 2016

Time Series Analysis and Forecasting3 citations

TL;DR

This paper introduces PDE-FIND, a sparse regression method that discovers governing partial differential equations (PDEs) from time-series data collected at fixed spatial locations (Eulerian) or moving with the dynamics (Lagrangian). By leveraging sparsity-promoting regression and Pareto analysis, the method accurately identifies sparse, physically meaningful PDEs—such as Navier-Stokes, Korteweg-de Vries, and Schrödinger equations—directly from data, even under noise and subsampling, enabling robust discovery of physical laws in complex systems where first-principles derivation is intractable.

ABSTRACT

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity promoting techniques to select the nonlinear and partial derivative terms terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework where the sensors are fixed spatially, or in a Lagrangian framework where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems of mathematical physics including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially non-unique dynamical terms by using multiple time series taken with different initial data. Thus for a traveling wave, the method can distinguish between a linear wave equation or the Korteweg-deVries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parametrized spatio-temporal systems where first-principles derivations are intractable.

Motivation & Objective

• To develop a data-driven method for discovering governing partial differential equations (PDEs) from time-series measurements in space and time.
• To overcome the challenge of identifying PDEs when first-principles derivations are infeasible, especially in complex systems like neuroscience, epidemiology, and fluid dynamics.
• To enable robust discovery of PDEs using sparse regression that balances model accuracy and complexity via Pareto analysis.
• To demonstrate the method's capability on canonical models of mathematical physics, including nonlinear and dissipative systems.
• To disambiguate between non-unique dynamical terms by using multiple time series with different initial conditions.

Proposed method

• The method constructs a large library of candidate terms, including nonlinear functions and spatial derivatives of the state variable, from time-series data.
• It applies sparse regression to identify the minimal set of terms that best represent the data, promoting parsimonious models through L0-norm minimization.
• The algorithm uses compressed sensing techniques to reduce computational cost by subsampling data while preserving regression accuracy.
• It supports both Eulerian (fixed sensors) and Lagrangian (moving sensors) data collection frameworks to accommodate different measurement scenarios.
• Pareto analysis is used to balance model complexity and regression error, selecting the optimal trade-off between sparsity and accuracy.
• The method is implemented as PDE-FIND (PDE Functional Identification of Nonlinear Dynamics), with open-source code available.

Experimental results

Research questions

• RQ1 Can sparse regression be extended to discover partial differential equations from time-series data, rather than just ordinary differential equations?
• RQ2 How can spatial derivatives and nonlinear terms be robustly identified from noisy, sampled data in both Eulerian and Lagrangian frameworks?
• RQ3 Can the method disambiguate between non-unique dynamical models, such as distinguishing a linear wave equation from the Korteweg-de Vries equation?
• RQ4 What level of data subsampling and noise can the method tolerate while still accurately recovering the true governing PDE?
• RQ5 Can the method generalize to predict dynamics in parameter regimes not present in the training data, such as higher Reynolds numbers in Navier-Stokes?

Key findings

• The method successfully discovered the Navier-Stokes equation at Re = 100 with a reconstruction error of 1% ± 0.2% under no noise and 7% ± 6% under 1% noise.
• For the Korteweg-de Vries (KdV) equation, the method achieved 1% ± 0.2% error without noise and 7% ± 5% with 1% noise, even when using only 2.22% of the full data for subsampling.
• The method accurately recovered the Schrödinger equation with 0.25% ± 0.01% error in the absence of noise and 10% ± 7% with 1% noise.
• In the reaction-diffusion system, the method achieved 0.02% ± 0.01% error without noise and 3.8% ± 2.4% with 0.5% noise, demonstrating robustness to moderate noise.
• The method enabled accurate prediction of turbulent Navier-Stokes dynamics at Re = 10,000 using a model trained only at Re = 100, showcasing generalization beyond training regimes.
• The use of multiple initial conditions allowed disambiguation between competing PDEs, such as distinguishing the KdV equation from a linear wave equation in soliton dynamics.