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Outlook
Despite the in-depth discussions and diverse examples we’ve explored, we’ve really only begun to tap into the vast potential of physics-based deep learning. The techniques covered in the previous chapters aren’t just useful -— they have the power to reshape computational methods for decades to come. As we’ve seen in the code examples, there’s no magic at play; rather, deep learning provides an incredibly powerful new tool to work with complex, non-linear functions.
Crucially, deep learning doesn’t replace traditional numerical methods. Instead, it enhances them. Together, they form a groundbreaking synergy, with a huge potential to unlock new frontiers in simulation and modeling. One aspect we haven’t yet touched upon is perhaps the most profound: at its core, our ultimate goal is to deepen human understanding of the world. The notion of neural networks as impenetrable “black boxes” is outdated. Instead, they should be seen as just another numerical tool—one that is as interpretable as traditional simulations when used correctly.
Looking ahead, one of the most exciting challenges is to refine our ability to analyze learned networks. By distilling the patterns and structures these networks uncover, we move closer to extracting fundamental, human-readable insights from their solution manifolds. The future of differentiable simulation isn’t just about better predictions -— it’s about revealing the hidden order of the physical world in ways we’ve never imagined.
Some specific directions
Beyond this long-term vision, there are plenty of exciting and immediate next steps. While our deep dives into Burgers’ equation and Navier-Stokes solvers have tackled non-trivial challenges, they represent just a fraction of the landscape of PDE models and operators that these techniques can improve. Here are just a few promising directions from other fields:
Chemical Reaction PDEs often exhibit intricate behaviors due to multi-species interactions. A particularly exciting avenue is training models that can rapidly predict experimental or industrial processes and dynamically adjust control parameters to stabilize them to enable real-time, intelligent control.
Plasma Simulations share similarities with vorticity-based fluid formulations but introduce additional complexities due to electric and magnetic interactions. This makes them a prime candidate for deep learning methods, especially for plasma fusion experiments and energy generators, where differentiable physics could be a game-changer.
Weather and Climate Modeling remain among the most critical scientific challenges for humanity. These highly complex, multi-scale systems involve fluid flows intertwined with countless environmental factors. Leveraging deep learning to enhance numerical simulations in this space holds immense potential. Not just for more accurate forecasts, but for unlocking deeper insights into the dynamics of our planet.
Closing remarks
These are just a few examples, but they illustrate the incredible breadth of opportunities where differentiable physics and deep learning can make an impact. There’s lots of exciting research work left to do - the next years and decades definitely won’t be boring. 🤗 👍
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