Key Takeaways

  • Classical physics-informed neural networks (PINNs) solve differential equations from scratch for each instance, often failing when equations describe turbulent or time-dependent physics.
  • Neural operators treat machine learning as a mapping across continuous, infinite-dimensional function spaces rather than fixed pixel grids or meshes.
  • By decoupling the model from fixed resolution, neural operators achieve zero-shot super-resolution, evaluating physics at any scale without retraining.
  • Anandkumar applied this architecture to build FourCastNet, generating global weather and climate forecasts thousands of times faster on a single GPU than numerical baselines.

Why PINNs Struggle on Real Physics

If you try to model fluid flow, aerodynamics, or weather using classical PINNs, you hit an immediate roadblock. PINNs optimize a neural network to fit a specific boundary condition and differential equation from scratch every single time.

Anandkumar points out the operational flaw in that design: “With the pinned, every instance of an equation you solve from scratch, at least in the classical sense. So you start, you take the specification of what equation you want to solve, and you hope that the optimization succeeds, which in many cases it doesn't.”

When fluid flows become turbulent or equations evolve over long time horizons, the loss surface gets messy. The network gets stuck in local minima. If you change the boundary conditions by two percent or advance the timeline by ten seconds, you have to throw away the weights and rerun the entire optimization from zero. PINNs act like slow numerical solvers with worse convergence guarantees.

Mapping Function Spaces Instead of Grids

Neural operators take a completely different architectural path. Instead of training a model on a single equation instance, you train it on thousands of equation variations. The model learns the underlying solution operator itself.

“Whereas with neural operators what we do is we have lots of great data,” Anandkumar explains. “We have a training phase. We teach it how to come up with solutions for different instances of equations.”

Standard computer vision models treat images as discrete pixel grids. Standard physical simulations use fixed finite-element meshes. But nature does not run on pixels.

As Anandkumar notes: “Our world is inherently multiscale. So you should not be deciding beforehand what the resolution is. Neural operators enable us to zoom in and out as we like.”

Because a neural operator maps between infinite-dimensional function spaces, it does not care what resolution you feed it at inference time. You can train the model on cheap, coarse grid simulations, then query the model at dense, continuous coordinates during deployment. It evaluates points off the training grid with zero-shot super-resolution.

That capability is what allowed Anandkumar and her team to create FourCastNet. Instead of running heavy supercomputer weather calculations on fixed spatial steps, FourCastNet maps atmospheric dynamics continuously, producing high-resolution global forecasts at speeds thousands of times faster than traditional numerical models on a single GPU.

What to Do With This

If you are building ML for physical systems, structural mechanics, or fluid dynamics, stop trying to train vanilla PINNs on individual simulation runs. Set up a synthetic data pipeline using your existing numerical solver to produce coarse simulations across variable boundary parameters, then train a Fourier neural operator to learn the operator mapping. Test the model by evaluating it directly on a finer mesh than the one used during training to verify resolution invariance before deploying.