In fact, this is some pretty cool stuff!

So what are PDEs? Well, they’re basically mathematical models that describe how things change over time and space. They come up all the time in physics (duh), but also in other fields like engineering, finance, and even biology. And solving them can be a real pain especially when you have to do it for complex systems with lots of variables.

That’s where neural networks come in! Neural nets are basically fancy algorithms that can learn patterns from data. They’re inspired by the structure of the human brain, but they don’t actually think like humans (sorry if that disappoints you). Instead, they use a bunch of mathematical operations to process information and make predictions based on what they’ve learned.

Now, here’s where things get interesting researchers have been trying to combine neural networks with PDEs for a while now. The idea is to create “physics-inspired” neural nets that can solve complex PDEs more efficiently than traditional methods. And guess what? It actually works!

One of the coolest examples of this is a technique called “deep learning for PDEs.” Basically, you feed your neural net some data (like temperature or pressure readings), and it learns to predict how those values will change over time based on the underlying physics. And because neural nets can handle complex systems with lots of variables, they’re great at solving PDEs that would be impossible for traditional methods.

But here’s where things get really interesting researchers have also been using “physics-inspired” neural nets to solve PDEs in reverse! That means instead of predicting how a system will change over time, you can use the neural net to figure out what initial conditions would lead to a certain outcome. This is called “backward solvers,” and it’s really useful for things like optimizing manufacturing processes or designing new drugs.

It might sound boring, but trust me this stuff is pretty cool. And who knows? Maybe someday we’ll be able to use these techniques to solve some of the biggest problems in science and engineering. Until then, keep learning and exploring!