Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

Abstract

In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model transient systems with non-linear dynamics at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.

Transfer learning enhanced physics informed neural network for phase-field modeling of fracture

“…Coming to our rescue is the physics informed/constrained neural networks (PINN). Over the past two years, a number of studies on PINN can be found in the literature [1][2][3][4][5][6]. In these methods, the neural networks are trained to solve supervised learning problems while respecting any given law of physics described by general non-linear partial differential equations.…”

Multi-fidelity generative deep learning turbulent flows

“…, x N . In our past work [9], we formulated a deep convolutional auto-regressive model for modeling the evolution of a transient PDE as a Markov chain. To increase the predictive capability of our model and integrate the low-fidelity observations, in this work we will use a deep recurrent neural network (RNN) formulation which is a standard approach for time-series predictions in deep learning [13].…”

“…Examples of the encoding and dense blocks are illustrated in Fig. 9 which have been used successfully for modeling many physical systems in the past [9,41,75,76]. The encoding blocks down-scale the feature maps forcing the model to learn low-dimensional representations while the densely connected blocks increase predictive accuracy of the model and have better performance than standard residual connections [22].…”

“…1b, we wish to build a deep generative model to infer from a low-fidelity flow field the corresponding high-fidelity realizations. Due to their past success for modeling physical systems [9,75,76], we will choose to use a convolution based generative model with learnable parameters θ. The use of convolutions implies that the data is placed onto a structured Euclidean grid, akin to that of pixels in images.…”

“…This simplifies the learning task significantly by providing a physical coarse estimate of the flow, which is important for the prediction of the solution of the highly non-linear N-S equations at high Reynolds numbers. While we have shown in our past work that deep learning surrogates can successfully model many complex physical systems independently [9,75,76], the systems of interest in these works are far less complex than the turbulent N-S equations.…”