IKNO for Navier–Stokes Stays Accurate Under Rotation, With 5.6x Fewer Parameters Than TFN

A neural surrogate for partial differential equations can fit its training data well and still give different answers for the same physical state when you rotate the coordinate system. A new arXiv paper proposes an architecture that avoids this problem by construction. The Invariant-Conditioned Isotropic Kernel Neural Operator, or IKNO, predicts three-dimensional Navier–Stokes dynamics on irregularly placed points. Its consistency under rotations holds to numerical precision. In accuracy, the model is comparable to the general rotation-aware Tensor Field Network, while having 5.6 times fewer parameters.

Where Conventional Surrogates Break

The task looks like this. You have a point cloud in 3D, with a velocity recorded at each point. The model must predict how the velocity will change on the next step. The physics is isotropic: if you rotate both the positions and the velocities, the velocity change must rotate in the same way. This is equivariance with respect to the SO(3) group.

A standard graph network knows nothing about this. It sees coordinates and vector components as a set of numbers, and it can learn a dependence on a specific orientation. The authors tested the standard remedy, which is training on randomly rotated examples. It reduces the dependence on the coordinate system but does not remove it completely. A residual error remains even with augmentation.

How IKNO Works

The idea is to separate roles. The model builds local interactions between neighboring points from two types of quantities:

• scalars that do not change under rotation (for example, distances and dot products);
• direction vectors that rotate together with the data.

Scalars determine the strength of an interaction, and vectors set the direction it points in. The kernel is isotropic, so the output rotates together with the input by construction. This property is built into the architecture rather than learned from data.

Because the model is specialized for vector PDEs, it does not need the heavy machinery of general equivariant networks. Tensor Field Network, which works with tensor representations of the rotation group, suits a wide range of problems but pays for that generality in size.

What the Measurements Showed

The evaluation used a held-out set that was frozen after the model design was finalized. This reduces the risk of tuning the architecture to the test.

• Rotation consistency of IKNO: to numerical precision.
• Unconstrained graph models trained with rotation augmentation: the dependence decreases but remains.
• Forecast accuracy: on par with Tensor Field Network.
• Size: 5.6 times fewer parameters than TFN.
• Against a graph simulator with the same parameter count: IKNO wins.

The last point matters for a fair comparison. The gain comes not from model size but from the built-in symmetry.

Where This Is Useful and Where the Limits Are

For anyone building surrogates in CFD, meteorology, particle simulation, or robotics, the takeaway is practical. If a problem has a known symmetry, it is better to build it into the architecture than to hope augmentation will teach it. A guarantee of exact equivariance also simplifies validation: you do not need to run the model over a grid of orientations to check that the answer does not drift.

The limits are also visible from the abstract. The results come from a single family of problems, three-dimensional Navier–Stokes dynamics. The claim about compact, PDE-specialized models has so far been confirmed only there. How the architecture will behave with boundary conditions, anisotropic media, or other equations cannot be determined from the description. Where the physics itself breaks rotational symmetry (for example, because of gravity or walls), pure SO(3) equivariance will need further work.

The general direction is clear. For scientific surrogates, a small, specialized model with the right symmetry can replace a universal, heavy one without a loss in quality.