Rajdeep Singh

Rough notes. PINNs are the idea that if you already know the governing PDE, you shouldn't have to learn it from data — you can make the network obey it.

The move

Approximate the solution to a PDE $\mathcal{N}[u](t,\mathbf{x})=0$ with a neural network $u_{\theta }$, then train by minimizing a composite loss:

$$\mathcal{L}={\mathcal{L}}_{\text{data}}+\lambda {\mathcal{L}}_{\text{physics}}$$

The physics term is the residual of the PDE evaluated at sampled collocation points. Partial derivatives come from autodiff through the network. No mesh.

What I find compelling

• The PDE is inductive bias, not data. A well-posed physical constraint is stronger than even a very large dataset. With one, data requirements drop by orders of magnitude — in some problems to zero labels, just boundary conditions and the equation.
• Inverse problems fall out for free. Treat unknown parameters — viscosity, diffusivity, reaction rates — as learnable. The same objective that fits $u_{\theta }$ fits the coefficients.
• Failure modes are specific, not generic. Unlike black-box overfitting, when a PINN fails you can usually localize the failure to a region of the domain and a specific term of the residual.

Where the framing strains

• $\lambda$ is a hyperparameter, and it shouldn't be. The weight between data and physics is almost always hand-tuned. Adaptive weighting helps, but the underlying tension — two competing objectives on very different scales — doesn't go away cleanly.
• Stiff and multiscale PDEs. Autodiff through a deep network across sharp gradients is unreliable. Domain decomposition helps but feels like a workaround rather than a principle.
• No free lunch on high dimensions. The "no mesh" pitch sounds like a dimensionality win, but collocation points still have to cover the domain densely enough. You trade meshing for sampling.

Why I'm still interested

Symmetries and conservation laws should be structural — baked into the architecture or the loss — not statistical patterns a model has to rediscover from data. PINNs are one way to do that. Equivariant architectures are another. Neural operators are a third. The open question is how much of physics can move from learned to assumed, and at what cost.