Separable Tensor-Product Spline Trial Layer

1. Integration point: replace the first coordinate-mixing layer of a KAN, coordinate MLP, or implicit neural representation. For each rank \(k\) and coordinate \(d\), store a vector of spline coefficients \(c_{k,d,m}\); evaluate one-dimensional B-spline bases locally and combine their outputs multiplicatively. Use \(R\) separable products so expressivity is controlled by rank instead of a full tensor grid. 2. Pseudocode: u=zeros(batch); for k in range(R): z=ones(batch); for d in range(D): B=bspline_basis(x[:,d], knots[d], order=r); s=B @ coeff[k,d]; z=z*s; u=u+z; optionally add a residual additive KAN or MLP path. For two-dimensional PDEs this is u=sum_k (Nx @ a[k])*(Ny @ b[k]). Enforce known boundary conditions by multiplying the output by a boundary mask or retain a boundary penalty. 3. Compute spline bases and derivatives analytically or with a differentiable B-spline routine. The tensor-product representation is exact for rank-one separable fields; approximation quality for general fields is controlled empirically by rank \(R\). Products may cause exploding or vanishing activations, so initialize factor coefficients with scale \(R^{-1/(2D)}\), normalize each factor output, and use gradient clipping. 4. First experiment: fit a 2D manufactured PDE solution and compare a dense tensor-grid spline layer, additive KAN, and rank-\(R\) multiplicative KAN at matched parameter counts. Sweep \(R\in\{1,2,4,8\}\) and grid sizes, measuring relative field error, parameter memory, forward latency, and weak-residual error. The expected signal is lower memory and faster evaluation than a dense tensor-product basis for moderate rank, with accuracy improving as rank increases. Reject the method if the rank required to match additive-KAN accuracy removes the memory or speed advantage.