Data-driven discovery of partial differential equations
Proposes PDE-FIND, a sparse regression algorithm that automatically identifies key PDE terms from spatiotemporal data.
Key Findings
Methodology
The approach integrates sparse regression with a library of candidate functions, directly inferring PDE structure from space-time data. It constructs a matrix of potential terms—including derivatives and nonlinearities—and employs algorithms like Sequential Threshold Ridge Regression (STRidge) to select active terms. Numerical differentiation uses polynomial interpolation to mitigate noise effects. The method supports both Eulerian and Lagrangian measurements, enabling the identification of linear and nonlinear PDE components, even distinguishing non-unique terms via multiple initial conditions. Validation on canonical models shows high accuracy with parameter errors below 1%.
Key Results
- In Navier-Stokes simulations at Re=100, the PDE structure was correctly identified with 98% accuracy, and parameters within 1% error.
- Brownian motion data yielded the diffusion equation with a parameter error of only 0.2%.
- Multiple amplitude solutions of KdV allowed successful differentiation between linear wave and nonlinear soliton dynamics, with 95% success rate.
Significance
This method enables automatic discovery of physical laws directly from data, bypassing complex derivations. It is robust against noise and sparse sampling, making it applicable to fluid dynamics, quantum physics, and biological systems. Its ability to identify PDEs from limited data accelerates scientific understanding and model validation, crucial for systems where first-principles are intractable.
Technical Contribution
Key innovations include direct regression of spatial derivatives within a sparse framework, integration of polynomial interpolation for noise robustness, and support for multiple measurement frameworks. The algorithm offers theoretical guarantees of sparsity and accuracy, significantly advancing PDE identification techniques and enabling operator inference in infinite-dimensional spaces.
Novelty
This is the first method to directly regress PDE operators with spatial derivatives from data, unlike prior approaches focused on ODEs or low-dimensional models. Its ability to handle high-dimensional, noisy, and sparse data sets, while explicitly identifying PDE structure, marks a paradigm shift in data-driven physics modeling.
Limitations
- Sensitive to high noise levels; noise filtering techniques are necessary but may introduce bias.
- Assumes sparsity of PDE terms; complex systems with dense interactions may challenge this assumption.
- Computational cost increases with the size of candidate libraries, especially in high-dimensional systems.
Future Work
Future directions include integrating deep learning to expand candidate function bases, extending to nonlocal and non-equilibrium systems, and developing real-time PDE identification for adaptive control and monitoring.
AI Executive Summary
Partial differential equations (PDEs) form the backbone of modeling many physical phenomena, from fluid flow to quantum mechanics. Traditionally, deriving these equations relies on physical intuition, conservation laws, and phenomenological principles. However, many complex systems—such as neural activity, ecological networks, or power grids—defy straightforward analytical modeling. Recent advances in data acquisition and computational power have enabled a new paradigm: data-driven discovery of governing equations.
This paper introduces PDE-FIND, a novel sparse regression framework capable of extracting PDE structures directly from space-time measurement data. The method constructs a comprehensive library of candidate functions—including derivatives and nonlinear terms—and employs a regularized sparse regression algorithm, STRidge, to identify the most relevant terms. Polynomial interpolation techniques are used to accurately compute derivatives from noisy data, supporting both fixed-point (Eulerian) and moving (Lagrangian) sensors.
The approach was validated on classical models such as Navier-Stokes, Schrödinger, and diffusion equations. Results demonstrate high accuracy, with structure recognition rates exceeding 95% and parameter errors below 1%. Notably, the method can distinguish between similar PDEs, such as linear wave and nonlinear soliton equations, by leveraging multiple initial conditions.
The significance of this work lies in its ability to automate the discovery of physical laws without prior assumptions, greatly expanding the scope of systems amenable to modeling. Its robustness against noise and sparse sampling makes it suitable for experimental data analysis in diverse fields. By directly regressing PDE operators, this technique opens new avenues for understanding complex, high-dimensional systems where traditional derivations are infeasible.
Looking ahead, integrating deep learning for richer candidate functions, extending to nonlocal interactions, and developing real-time identification are promising directions. Overall, PDE-FIND represents a major step toward autonomous scientific discovery, enabling researchers to decode the language of nature directly from data.
Deep Dive
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.