PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network (1812.04426v2)

Abstract: Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work \cite{Long2018PDE} where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

• The paper demonstrates a hybrid numeric-symbolic network that accurately infers both differential operators and nonlinear response functions.
• The architecture integrates convolutional discretization with a symbolic neural network, enabling precise identification of PDE components in dynamic systems.
• Experimental results on Burgers', Diffusion, and Convection-Diffusion-Reaction models showcase robust predictions even in noisy environments.

Learning Partial Differential Equations with PDE-Net 2.0

The paper "PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network" presents a novel approach to identifying partial differential equations (PDEs) from data through a deep neural network. This method, termed PDE-Net 2.0, is inspired by recent advancements in data acquisition and computational power, offering a framework to infer PDEs governing complex dynamic systems.

Overview of PDE-Net 2.0

PDE-Net 2.0 elaborates on previous efforts by combining numerical approximations via convolutions with a symbolic neural network designed for model recovery. This hybrid approach allows for the learning of both differential operators and nonlinear response functions concurrently, providing a flexible and expressive tool for discovering underlying PDE models.

Architecture and Components

The architecture of PDE-Net 2.0 is composed of several key components:

• Discretization of Differential Operators: Using convolutional layers to approximate differential operators, ensuring transparency and interpretability of the network's learning.
• Symbolic Neural Network (SymNet): It is employed to approximate multivariate nonlinear response functions. SymNet operates by constructing a compact representation of possible response functions, allowing the system to infer potential analytic forms directly from data.
• Stacking of Multiple Layers: By organizing the formulated differential operator approximations and symbolic approximations into a layered architecture, PDE-Net 2.0 can accumulate predictions over multiple time steps, facilitating long-term dynamical forecasts.

Numerical Experiments and Results

The paper applies PDE-Net 2.0 to several PDEs, such as Burgers' Equation, the Diffusion Equation, and a Convection-Diffusion-Reaction model. The results underscore the model's proficiency in not only identifying the correct terms that constitute each PDE but also performing accurate long-term predictive simulations.

1. Burgers' Equation: The network successfully identifies both convection and diffusion terms, achieving close approximation coefficients. The predictive performance remains robust even in noisy environments.
2. Diffusion Equation: PDE-Net 2.0 accurately captures the parabolic nature of diffusion without significant deviations from expected coefficients.
3. Convection-Diffusion-Reaction System: The model shows its ability to handle complex interactions, accurately deducing both the convection terms and nonlinear reactive source terms.

Implications and Future Directions

The development of PDE-Net 2.0 presents a significant step forward in data-driven discovery of physical laws. By learning the governing equations from dynamic data, this framework can accelerate discoveries in complex systems where traditional mechanistic derivation of PDEs is challenging.

Possible extensions and improvements include:

• Enhanced Symbolic Network Designs: Incorporating additional operations in SymNet may extend its capacity to capture a broader scope of PDE forms.
• Temporal Scheme Advancements: Moving beyond forward Euler temporal discretization could yield improvements in model fidelity and stability.
• Application to Real-world Data: Implementing PDE-Net 2.0 on empirical datasets could potentially lead to novel insights, particularly in fields like neuroscience or biological sciences, where the governing dynamics are not fully understood.

In conclusion, PDE-Net 2.0 stands as a promising tool for bridging data science and mathematical modeling, paving the way for a deeper understanding of complex dynamical systems through computational methodologies.