
Train Once and Use Forever: Solving Boundary Value Problems in Unseen Domains with Pretrained Deep Learning Models
Physicsinformed neural networks (PINNs) are increasingly employed to re...
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Sobolev Training for the Neural Network Solutions of PDEs
Approximating the numerical solutions of Partial Differential Equations ...
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A DerivativeFree Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks
We introduce a deep neural network based method for solving a class of e...
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PhysicsInformed Neural Network Method for Solving OneDimensional Advection Equation Using PyTorch
Numerical solutions to the equation for advection are determined using d...
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Acceleration of the NVTflash calculation for multicomponent mixtures using deep neural network models
Phase equilibrium calculation, also known as flash calculation, has been...
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Predicting atmospheric optical properties for radiative transfer computations using neural networks
The radiative transfer equations are wellknown, but radiation parametri...
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WeaklySupervised Deep Learning of Heat Transport via Physics Informed Loss
In typical machine learning tasks and applications, it is necessary to o...
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Deep learning approaches to surrogates for solving the diffusion equation for mechanistic realworld simulations
In many mechanistic medical, biological, physical and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs) can make simulations impractically slow. Biological models require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speedups of several orders of magnitude compared to direct calculation. PDE surrogates enable use of larger models than are possible with direct calculation and can make including such simulations in realtime or nearreal time workflows practical. Creating a surrogate requires running the direct calculation tens of thousands of times to generate training data and then training the neural network, both of which are computationally expensive. We use a Convolutional Neural Network to approximate the stationary solution to the diffusion equation in the case of two equaldiameter, circular, constantvalue sources located at random positions in a twodimensional square domain with absorbing boundary conditions. To improve convergence during training, we apply a training approach that uses rollback to reject stochastic changes to the network that increase the loss function. The trained neural network approximation is about 1e3 times faster than the direct calculation for individual replicas. Because different applications will have different criteria for acceptable approximation accuracy, we discuss a variety of loss functions and accuracy estimators that can help select the best network for a particular application.
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