When you look at the actual implementation, you realize that the "Physics-Informed" part isn't just a fancy label. You are literally embedding physical laws—like Navier-Stokes or Schrödinger's equation—into the loss function of your neural network. This means your loss function isn't just measuring the error between a prediction and a label; it's measuring how much your model violates the laws of physics.
So, where should you actually put your focus during your undergraduate or graduate studies?
The Math route #
If you choose Mathematics, you are going to be the person who actually understands why the convergence works (or fails). PINNs rely heavily on functional analysis, optimization theory, and numerical analysis. You’ll spend your time dealing with Sobolev spaces and how to approximate solutions to complex differential equations. The downside? You might find yourself incredibly capable of proving that a solution exists, but you might struggle to actually deploy a model on a GPU or optimize a PyTorch training loop. You’ll be the theorist who understands the "why" but might get stuck on the "how" of modern AI workflows.
The Physics route #
A Physics major is the most direct way to understand the "what." You will intuitively understand the constraints you are trying to impose on the network. You'll know why a fluid flow shouldn't have certain discontinuities or why energy conservation must be maintained.
However, there is a massive technical gap here. Most physics departments don't teach the heavy-duty computational side of things—things like backpropagation through complex solvers, stochastic gradient descent, or distributed training. You might understand the physics perfectly, but you'll be starting from scratch when it comes to the actual machine learning engineering.
The Computer Science route #
This is the most "practical" path if your goal is to build tools, libraries, or scalable AI agents that handle physical simulations. You will learn the deployment, the data structures, and the hardware acceleration necessary to make these models run in real-time.
The risk is that you might treat the physics as a "black box." If you don't have a solid grasp of the underlying calculus and differential equations, you won't be able to design the custom loss functions that make PINNs unique. You'll just be a standard ML engineer trying to force-fit a model into a problem it wasn't designed for.
My take on the ideal path #
If I were starting from scratch to master this specific niche, I wouldn't pick just one. The most effective way to approach this is a "double threat" strategy. If you can only pick one major, go for Computer Science or Math, but you absolutely must take elective courses in Computational Physics or Fluid Dynamics. The real magic happens in the deep dive where you understand the PDE, translate it into a residual loss term, and then write the efficient CUDA code to solve it. It’s a specialized field, and being a generalist in any of these three areas will likely leave you frustrated when you hit the actual research frontier.
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