We have developed a new data-driven paradigm for the rapid inference, modeling and simulation of the physics of transport phenomena by deep learning. Using conditional generative adversarial networks (cGAN), we train models for the direct generation of solutions to steady state heat conduction and incompressible fluid flow purely on observation without knowledge of the underlying governing equations. Rather than using iterative numerical methods to approximate the solution of the constitutive equations, cGANs learn to directly generate the solutions to these phenomena, given arbitrary boundary conditions and domain, with high test accuracy (MAE<1%) and state-of-the-art computational performance. The cGAN framework can be used to learn causal models directly from experimental observations where the underlying physical model is complex or unknown. Transport phenomena studies the exchange of energy, mass, momentum, and charge between systems, 1 encompassing fields as diverse as continuum mechanics and thermodynamics, and is used heavily throughout all engineering disciplines. Here, we show that modern deep learning models, such as generative adversarial networks, can be used for rapid simulation of transport phenomena without knowledge of the underlying constitutive equations, developing generative inference based models for steady state heat conduction and incompressible fluid flow problems with arbitrary geometric domains and boundary conditions. In contrast to conventional procedure, the deep learning models learn to generate realistic solutions in a data-driven approach and achieve state-of-the-art computational performance, while retaining high accuracy. Deep learning models for physical inference can be applied to any phenomena, given observed or simulated data, and can be used to learn and predict directly from experiments where the underlying causal model is complicated or unknown. In recent years, there have been several advances in the fields of computer vision and natural language processing applications brought on by deep learning. The convolutional neural network architecture, an example of a modern deep learning architecture, is a multilayer stack of optimizable convolution operations which compute non-linear input transformations. Each operation in the stack transforms its input in a manner that increases the selectivity and uniqueness of the output representation. The flexibility of deep neural networks allows models in principle to learn successively higher orders of features from raw data, making the application of deep learning models in physics highly attractive. Figure 1a . We construct models for steady state heat transfer and incompressible fluid flow that learn directly from observation. In the case of heat transfer, the observations are the temperature distribution given a set of constant temperature boundary conditions. In the case of incompressible fluid flow, the observations are the velocity fields and pressure distribution given a set of constant velocity boundary conditions. Details on the construction of train and test datasets are given in Methods (section B). The cases, the prediction is highly accurate for velocity and pressure fields. The flow patterns predicted by the model are nearly exact. One interesting point to note here is the accurate inference of the pressure field given only the velocity field boundary condition, highlighting the information transfer between velocity and pressure fields which occurs within our model. For all problems in field-based physics, there is an underlying physical relationship, described by di↵erential equations, between field values at discrete, adjacent nodes in space. Numerical solvers utilize this fact by discretization of the underlying constitutive equations in order to obtain the field solution self-consistently (for example, r 2 T=0 implies that T at each node is the average of four adjacent nodes). The cGAN model forms an estimate of the data distribution by treating the model underlying the observations as a Markov random field (MRF), 19-21 where each node is considered as a random variable. During training, the discriminator learns the local relationship between adjacent nodes in order to distinguish between real and generated samples, a conditional probability query of the data distribution. The generator learns to produce the most probable sample given known evidence (boundary conditions and domain), a maximum a posteriori query of the data distribution. The parameters of the generator are then optimized to maximize the probability that the discriminator assigns a real label to generated data. In Figure 4 , we demonstrate how the discriminator operates on an example selected from the heat transfer test set. The discriminator operates over patches of data in a given example, attempting to classify if each patch is real or fake. The discriminator correctly classifies most patches of the generated and ground truth solutions. Since the process of featurization and feature detection are unsupervised, any number of known or unknown physical phenomena can be learned from raw data by the discriminator. In the case of steadystate heat conduction, the cGAN discriminator learns to identify patches which locally satisfy the underlying partial di↵erential equation (Methods, Equation 4), i.e. the temperature at each node is the average of its four neighboring nodes. In the case of incompressible fluid flow, the cGAN discriminator scans the data within a particular field (e.g. u velocity field) as well as the coupling between fields (e.g. u and v velocity fields and pressure field). The cGAN discriminator learns to identify patches of multi-channel data that locally satisfy the coupled partial di↵erential equations in Methods, Equation 5 (conservation of mass and momentum). To demonstrate the accuracy of our framework, we compare the generated solutions from our cGAN model with the FD method solutions for the test set fraction of the heat transfer and fluid flow datasets. The generated solutions show an average per-sample mean absolute error of less than 1% (see SI). The evaluation time of the cGAN model is over an order of magnitude faster than the conventional finite di↵erence method (see SI). and their connectivity continue to grow exponentially, we can expect our framework will be used in predicting complex multiphysics phenomena in a faster, less compute-intensive manner. We expect the cGAN method to be broadly applicable to a wide range of scientific and engineering fields. METHODS We adapt the conditional Generative Adversarial Network (cGAN), used previously for image-to-image translation. We train D to maximize the probability of assigning the correct label to both training 5 examples and samples from G. The cGAN objective function is expressed as: Here, D and G participate in a two-player minimax game with value function L cGAN (G, D) , where G attempts to minimize this objective against an adversarial D that tries to maximize In addition, we apply an L1 distance loss function to the generator: The final objective is: where the hyperparameter is the L1 objective weight. The discriminator D (Figure 1c ) is a convolutional neural network that operates on either (c,x), the input and generated output, or (c,x 0 ), the input and ground truth solution, to produce the probability that a small patch of the discriminator input comes from the training data distribution. In this way, the discriminator treats each real or generated sample as a Markov random field (MRF), an undirected probabilistic graph which assumes statistical independence between nodes separated by more than a patch diameter. This operation is performed convolutionally across the entire solution, averaging all responses of all distinct patches to provide the output of D. By training the discriminator to correctly distinguish between real and generated samples (Methods, Equation 2), we build a joint probabilistic model over the desired field values at discrete grid points. In the image modeling community, the treatment of images as MRFs has been previously explored, and is commonly used in models to determine texture or style loss. For generator G (Figure 1b) , we adopt an encoder-decoder network, which has been used successfully in image and text translation. B. Models We consider steady-state heat conduction on an arbitrary two-dimensional domain with no heat generation. We wish to obtain a solution of the temperature field, T (x, y). Here, the Laplace equation applies: assuming constant thermal conductivity. The boundary conditions are expressed in the form and 512x512) was generated in a similar manner for timing purposes. 7 We consider steady-state fluid flow in a square two-dimensional domain. We wish to obtain a solution of the velocity fields, u(x, y) and v(x, y), and pressure field, p(x, y).
REF uses conditional generative models to find the equilibrium solution to a number of transport problems faster than traditional iterative methods.
Deep Learning the Physics of Transport Phenomena
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