Deep Learning Tackles Complex PDEs: A breakthrough or Just More Math Hype?

The world of partial differential equations (PDEs) has long been a mathematical nightmare, especially when dealing with infinite dimensions. But now, researchers are throwing deep learning into the mix, aiming to simplify these complex beasts. Enter the Hilbert-Galerkin Neural Operators (HGNOs), a mouthful that's promising a lot.

The Universal Approximation Theorems

With these new methods, we're seeing the first Universal Approximation Theorems (UATs) that are allegedly up for the job. They're supposed to wrangle PDEs with non-sequential and non-metrizable topologies. In simpler terms, these equations are slippery and hard to pin down. Yet, HGNOs claim to approximate all the terms involved, even for control problems. But is this just more bullish hopium?

Deep Learning Meets Control Problems

On the control side of things, there's talk about optimal feedback controls. This should matter to anyone dealing with deterministic and stochastic systems, like the heat and Burgers' equations. They're claiming breakthroughs by minimizing the $L^2_\mu(H)$-norm of the PDE's residual across the entire Hilbert space. Not just a snippet, the whole space. Sounds promising, right? Or maybe it's just another funding rate lie?

Applications and Skepticism

Researchers are applying this to various fields, from physics to path-dependent systems. Functional differential equations, Kolmogorov and HJB PDEs, stochastic systems, the list goes on. But we've heard grand promises before in the AI space. Can deep learning truly revolutionize these areas, or is this another case of everyone having a plan until liquidation hits?

Let's not forget, the AI space loves to overextend itself. Are these models solid enough or are we just setting ourselves up for another round of disappointment? Zoom out on the history of AI and you'll see a pattern of bold claims followed by quiet retreats.

In the end, tackling these PDEs could be groundbreaking. But until we see real-world applications beyond controlled environments and academic papers, skepticism remains healthy. After all, the funding rate is lying to you again.