In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Input: Consider Input: Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, the uses have been restricted by the fact the training data are obtained using PDE solvers. Thereby, the uses were limited to domains, where the PDE solver was applicable, but no further.

We present methods for training on small domains, while applying the trained models on larger domains, with consistency constraints ensuring the solutions are physically meaningful even at the boundary of the small domains. We demonstrate the results on an air-pollution forecasting model for Dublin, Ireland.

Output: We present RNNs for training surrogate models of PDEs, wherein consistency constraints ensure the solutions are physically meaningful, even when the training uses much smaller domains than the trained model is applied to.

Input: Consider Input: The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Schoenholz et al. (2017) who showed that for deep feedforward neural networks only a specific choice of hyperparameters known as the 'edge of chaos' can lead to good performance.
We complete this analysis by providing quantitative results showing that, for a class of ReLU-like activation functions, the information propagates indeed deeper for an initialization at the edge of chaos. By further extending this analysis, we identify a class of activation functions that improve the information propagation over ReLU-like functions. This class includes the Swish activation, $\phi_{swish}(x) = x \cdot ext{sigmoid}(x)$, used in Hendrycks & Gimpel (2016),
Elfwing et al. (2017) and Ramachandran et al. (2017). This provides a theoretical grounding for the excellent empirical performance of $\phi_{swish}$ observed in these contributions. We complement those previous results by illustrating the benefit of using a random initialization on the edge of chaos in this context.

Output: How to effectively choose Initialization and Activation function for deep neural networks

Input: Consider Input: We improve previous end-to-end differentiable neural networks (NNs) with fast
weight memories. A gate mechanism updates fast weights at every time step of
a sequence through two separate outer-product-based matrices generated by slow
parts of the net. The system is trained on a complex sequence to sequence variation
of the Associative Retrieval Problem with roughly 70 times more temporal
memory (i.e. time-varying variables) than similar-sized standard recurrent NNs
(RNNs). In terms of accuracy and number of parameters, our architecture outperforms
a variety of RNNs, including Long Short-Term Memory, Hypernetworks,
and related fast weight architectures.
Output: An improved Fast Weight network which shows better results on a general toy task.