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.

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.
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.

When considering simultaneously a finite number of tasks, multi-output learning enables one to account for the similarities of the tasks via appropriate regularizers. We propose a generalization of the classical setting to a continuum of tasks by using vector-valued RKHSs.
We propose an extension of multi-output learning to a continuum of tasks using operator-valued kernels.

There is no consensus yet on the question whether adaptive gradient methods like Adam are easier to use than non-adaptive optimization methods like SGD. In this work, we fill in the important, yet ambiguous concept of ‘ease-of-use’ by defining an optimizer’s tunability:  How easy is it to find good hyperparameter configurations using automatic random hyperparameter search? We propose a practical and universal quantitative measure for optimizer tunability that can form the basis for a fair optimizer benchmark.   Evaluating a variety of optimizers on an extensive set of standard datasets and architectures, we find  that Adam is the most tunable for the majority of problems, especially with a low budget for hyperparameter tuning.

We provide a method to benchmark optimizers that is cognizant to the hyperparameter tuning process.

