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diffphys-outlook.md

@@ -10,20 +10,20 @@ we can obtain _hybrid_ methods, that use the best numerical methods that we have
 
 ## Interaction
 
-One key component for these hybrids to work well is to let the NN _interact_ with the PDE solver at training time. Differentiable simulations allow a trained model to explore and experience the physical environment, and receive directed feedback regarding its interactions throughout the solver iterations. This combination nicely fits into the broader context of machine learning as _differentiable programming_. 
+One key component for these hybrids to work well is to let the ANN _interact_ with the PDE solver at training time. Differentiable simulations allow a trained model to explore and experience the physical environment, and receive directed feedback regarding its interactions throughout the solver iterations. This combination nicely fits into the broader context of machine learning as _differentiable programming_. 
 
 ## Generalization
 
-The hybrid approach also bears particular promise for simulators: it improves generalizing capabilities of the trained models by letting the PDE-solver handle large-scale changes to the data distribution such that the learned model can focus on localized structures not captured by the discretization. While physical models generalize very well, learned models often specialize in data distributions seen at training time. This was e.g. shown for the models reducing numerical errors of the previous chapter: the trained models can deal with solution manifolds with significant amounts of varying physical behavior, while simpler training variants quickly deteriorate over the course of recurrent time steps.
+The hybrid approach also bears particular promise for simulators: it improves generalizing capabilities of the trained models by letting the PDE-solver handle large-scale changes to the data distribution such that the learned model can focus on localized structures not captured by the discretization. While physical models generalize very well, learned models often specialize in data distributions seen at training time. This was, e.g., shown for the models reducing numerical errors of the previous chapter: the trained models can deal with solution manifolds with significant amounts of varying physical behavior, while simpler training variants quickly deteriorate over the course of recurrent time steps.
 
 ## Possibilities
 
 We've just scratched the surface regarding the possibilities of this combination. The examples with Burgers equation and Navier-Stokes solvers are non-trivial, and good examples for advection-diffusion-type PDEs. However, there's a wide variety of other potential combinations, to name just a few examples:
 
-* PDEs for chemical reactions often show complex behavior due to the interactions of mulltiple species. Here, and especially interesting direction is to train models that quickly learn to predict the evolution of an experiment or machine, and adjust control knobs to stabilize it, i.e., an online _control_ setting.
+* PDEs for chemical reactions often show complex behavior due to the interactions of multiple species. Here, and especially interesting direction is to train models that quickly learn to predict the evolution of an experiment or machine, and adjust control knobs to stabilize it, i.e., an online _control_ setting.
 
-* Plasma simulations share a lot with vorticity-based formulations for fluids, but additionally introduce terms to handle electric and magnetic interactions within the material. Likwise, controllers for plasma fusion experiments and generators are an excellent topic with plenty of potential for DL with differentiable physics.
+* Plasma simulations share a lot with vorticity-based formulations for fluids, but additionally introduce terms to handle electric and magnetic interactions within the material. Likewise, controllers for plasma fusion experiments and generators are an excellent topic with plenty of potential for DL with differentiable physics.
 
-* Finally, weather and climate are crucial topics for humanity, and highly complex systems of fluid flows interacting with a multitude of phenomena on the surface of our planet. Accurately modeling all these interacting systems and predicting their long-term behavior shows a lot of promise with benefitting from DL approaches that can interface with numerical simulations.
+* Finally, weather and climate are crucial topics for humanity, and highly complex systems of fluid flows interacting with a multitude of phenomena on the surface of our planet. Accurately modeling all these interacting systems and predicting their long-term behavior shows a lot of promise to benefit from DL approaches that can interface with numerical simulations.
 
 So overall, there's lots of exciting research work left to do - the next years and decades definitely won't be boring 👍