Physics-informed learning of governing equations from scarce data (2005.03448v3)

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.

• The paper presents a physics-informed deep learning framework that fuses DNNs, automatic differentiation, and sparse regression to uncover governing PDEs from sparse, noisy data.
• The methodology employs a root-branch network to assimilate information from diverse initial and boundary conditions, outperforming traditional SINDy methods.
• Experimental validation on canonical PDEs and cell migration data demonstrates robust accuracy with errors under 5%, enhancing complex system modeling.

Physics-informed Learning of Governing Equations from Scarce Data: An Overview

The research presented in the paper focuses on unveiling governing equations of complex physical systems from minimal and noisy data by leveraging a physics-informed deep learning (PiDL) framework. The approach synergistically integrates deep neural networks (DNNs), physics embedding, automatic differentiation, and sparse regression to efficiently discover partial differential equations (PDEs) that describe the underlying dynamics of nonlinear spatiotemporal systems. This methodology stands as a robust solution to modeling challenges in scenarios where traditional methods struggle due to data scarcity or noise.

The contribution is significant in bridging data-driven methods with physics-based insights. These governing equations are not obtained through strict physical derivations but distilled from observational data, providing a novel angle to understand complex systems in fields like climate science and material engineering. The framework is particularly advantageous for real-world applications where capturing extensive, accurate datasets is challenging.

Methodology and Framework

The PiDL combines four main components:

1. Deep Neural Networks (DNNs): Utilized for rich representation learning of nonlinear functions. They approximate the solution of system variables.
2. Automatic Differentiation: Ensures accurate derivative computation, critical for constructing and validating candidate PDEs.
3. Sparse Regression: Facilitates the identification of key derivative terms and parameters that structure the PDEs, adhering to parsimony.
4. Physics Embedding: Embeds known physics into the learning process, enhancing generalization and interpretability.

These elements collectively form a cohesive framework that allows for the exploration of parsimonious models, even under significant data limitations and noise.

Results and Implications

The paper demonstrates the efficacy of the PiDL framework across a range of canonical PDEs, including Burgers', Kuramoto-Sivashinsky, nonlinear Schrödinger, Navier-Stokes, and reaction-diffusion equations. Utilizing datasets with varying levels of data scarcity and noise, the results exhibit significant accuracy, with averaged errors of non-zero coefficients remaining under 5% in most cases. This performance outpaces traditional methods like sparse identification of nonlinear dynamics (SINDy), especially for systems with chaotic behavior or high-dimensional datasets.

Pioneering in its approach, the framework leverages a "root-branch" network architecture to handle multiple independent datasets sampled under different initial/boundary conditions. This is crucial as it allows the network to assimilate information across varied datasets without explicit boundary conditions.

Experimental Validation

The practical application potential is illustrated through an experiment involving cell migration and proliferation data. The framework successfully discovers a PDE aligning with the Fisher-Kolmogorov model using only sparse and noisy data—a demonstration of its capability in experimental scenarios where other methods underperform.

Theoretical and Practical Implications

Practically, this research opens pathways to more efficient modeling of complex systems without relying on exhaustive datasets. Theoretically, it advances our understanding of how integrating physics with data-driven models enhances model interpretability and generalization.

Future developments could focus on extending the methodology to even higher dimensional systems and incorporating discrete DNNs to manage computational costs more effectively. Furthermore, the integration with multi-GPU architectures might alleviate scalability issues when handling numerous datasets.

In summary, the integration of machine learning and physics presented by this framework signifies a potent tool for scientific inquiry, promising advancements in both understanding and predicting complex dynamical systems in uncertain environments.