The main problem is that as soon as you try to describe a somewhat complicated system, it gets almost impossible to solve the associated differential equations with a pen and paper (as any student who's taken a physics class could tell you). That’s why I'm trying to combine two developping technologies to solve them:

• The first tool is something called Physics-Informed Neural Networks (PINNs). These are machine learning models that work similarly to regular neural networks - systems that learn patterns from data- but with one key difference: they’re trained to obey the laws of physics. Instead of learning blindly from examples, PINNs are guided by the equations that describe how things behave in the real world. This helps ensure that the results make physical sense and might actually reflect how FDM behaves. It also means we don’t need huge amounts of training data, since the model already has an "intuition" for the physical laws

• The second tool is quantum computing, which takes advantage of the strange rules of quantum physics - like superposition and entanglement - to process information in new ways. Unlike classical computers that work with bits (0s and 1s), quantum computers use quantum bits which, in my case are represented by photons (the particles that make up light). These can perform certain types of calculations far faster than even the best supercomputers, especially when simulating systems governed by quantum mechanics.

So how do we combine them? Instead of using a standard neural network with classical operations, we build a Quantum Physics-Informed Neural Network (QPINN). That means we replace the usual parts of the network with operations performed on a quantum circuit. The input is encoded into quantum states, which then evolve through a sequence of quantum gates (think of them as the quantum equivalent of the logic gates you'd find in a regular computer). The final output is measured and compared against what the physics tells us it should be. We adjust the quantum circuit parameters until the output matches our desired result, effectively training the network to learn the solution to a physical equation. This is just like what happens in PINNs, but powered by quantum hardware.