Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.
07/10/2019 ∙ by Lu Lu, et al. ∙ 33 ∙ share
Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a flock, and use a fractional PDE to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics, we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in 1D and 2D nonlocal flocking dynamics. In particular, a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization. We show that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system. The proposed method offers new insights on how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.
10/27/2018 ∙ by Zhiping Mao, et al. ∙ 0 ∙ share
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