Weakly-Supervised Deep Learning of Heat Transport via Physics Informed Loss

Abstract

In typical machine learning tasks and applications, it is necessary to obtain or create large labeled datasets in order to to achieve high performance. Unfortunately, large labeled datasets are not always available and can be expensive to source, creating a bottleneck towards more widely applicable machine learning. The paradigm of weak supervision offers an alternative that allows for integration of domain-specific knowledge by enforcing constraints that a correct solution to the learning problem will obey over the output space. In this work, we explore the application of this paradigm to 2-D physical systems governed by non-linear differential equations. We demonstrate that knowledge of the partial differential equations governing a system can be encoded into the loss function of a neural network via an appropriately chosen convolutional kernel. We demonstrate this by showing that the steady-state solution to the 2-D heat equation can be learned directly from initial conditions by a convolutional neural network, in the absence of labeled training data. We also extend recent work in the progressive growing of fully convolutional networks to achieve high accuracy (< 1.5% error) at multiple scales of the heat-flow problem, including at the very large scale (1024x1024). Finally, we demonstrate that this method can be used to speed up exact calculation of the solution to the differential equations via finite difference.

References in corpus (5)

• Adam: A Method for Stochastic Optimization
• Sequence to Sequence Learning with Neural Networks
• Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
• Deep Learning the Physics of Transport Phenomena
• From Language to Programs: Bridging Reinforcement Learning and Maximum Marginal Likelihood

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• A Combined Data-driven and Physics-driven Method for Steady Heat Conduction Prediction using Deep Convolutional Neural Networks
• MaxwellNet: Physics-driven deep neural network training based on Maxwell's equations
• Data-Driven Modeling of Geometry-Adaptive Steady Heat Transfer based on Convolutional Neural Networks: Heat Conduction
• DiffusionNet: Accelerating the solution of Time-Dependent partial differential equations using deep learning
• Variational Autoencoding the Lagrangian Trajectories of Particles in a Combustion System
• Deep-learning PDEs with unlabeled data and hardwiring physics laws
• Introducing a Generative Adversarial Network Model for Lagrangian Trajectory Simulation
• Solving the Reaction-Diffusion equation based on analytical methods and deep learning algorithm; the Case study of sulfate attack to concrete