Physics-informed learning of governing equations from scarce data

Abstract

Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel physics-informed deep learning framework to discover governing partial differential equations (PDEs) from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to (1) approximate the solution of system variables, (2) compute essential derivatives, as well as (3) identify the key derivative terms and parameters that form the structure and explicit expression of the PDEs. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of PDE systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.

46 pages; 1 table, 6 figures and 3 extended data figures in main text; 2 tables and 12 figures in supplementary information

References in corpus (3)

• SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics
• Neural Operator: Graph Kernel Network for Partial Differential Equations
• Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensors

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• Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression
• Derivative-based SINDy (DSINDy): Addressing the challenge of discovering governing equations from noisy data
• Discovering Sparse Interpretable Dynamics from Partial Observations
• Physics informed deep learning for computational elastodynamics without labeled data
• Noise-aware Physics-informed Machine Learning for Robust PDE Discovery
• Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion
• Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case
• Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations
• Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning
• Robust Data-Driven Discovery of Partial Differential Equations under Uncertainties
• One-shot learning for solution operators of partial differential equations
• Uncovering Closed-form Governing Equations of Nonlinear Dynamics from Videos
• Physics-informed Spline Learning for Nonlinear Dynamics Discovery
• Physics-informed Dyna-Style Model-Based Deep Reinforcement Learning for Dynamic Control
• Model discovery in the sparse sampling regime
• Discovering PDEs from Multiple Experiments