The previous sections have explained the differentiable physics approach for deep learning, and have given a range of examples: from a very basic gradient calculation, all the way to complex learning setups powered by advanced simulations. This is a good time to take a step back and evaluate: in the end, the differentiable physics components of these approaches are not too complicated. They are largely based on existing numerical methods, with a focus on efficiently using those methods not only to do a forward simulation, but also to compute gradient information. What is primarily exciting in this context are the implications that arise from the combination of these numerical methods with deep learning.



Most importantly, training via differentiable physics allows us to seamlessly bring the two fields together: we can obtain hybrid methods, that use the best numerical methods that we have at our disposal for the simulation itself, as well as for the training process. We can then use the trained model to improve forward or backward solves. Thus, in the end, we have a solver that combines a traditional solver and a learned component that in combination can improve the capabilities of numerical methods.


One key aspect that is important 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.


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.


To summarize, the pros and cons of training NNs via DP:

✅ Pro:

  • Uses physical model and numerical methods for discretization.

  • Efficiency and accuracy of selected methods carries over to training.

  • Very tight coupling of physical models and NNs possible.

  • Improved generalization via solver interactions.

❌ Con:

  • Not compatible with all simulators (need to provide gradients).

  • Require more heavy machinery (in terms of framework support) than previously discussed methods.

Outlook: the last negative point (regarding heavy machinery) is bound to strongly improve given the current pace of software and API developments in the DL area. However, for now it’s important to keep in mind that not every simulator is suitable for DP training out of the box. Hence, in this book we’ll focus on examples using phiflow, which was designed for interfacing with deep learning frameworks.

The training via differentiable physics (DP) allows us to integrate full numerical simulations into the training of deep neural networks. It is also a very generic approach that is applicable to a wide range of combinations of PDE-based models and deep learning.

In the next chapters, we will first compare DP training to model-free alternatives for control problems, and afterwards target the underlying learning process to obtain even better NN states.