MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions

We introduce MetaColloc, which freezes a meta-learned basis dictionary and recovers each PDE solution by a single linear least-squares solve on a collocation system: no per-equation training, no solution data. Under the unregularised solve its accuracies on smooth two-dimensional benchmarks span $10^{-9}$ to $10^{-5}$. We also report the two quantities that bound what those numbers can mean; both are cheap to compute before any solve is attempted. The first is capacity. The dictionary is meta-trained on Gaussian Random Field samples, so it can represent that prior no better than $H$ functions can. The prior needs 594 Karhunen-Loeve modes to capture 90% of its energy against $H=512$, so the trained dictionary reproduces 16% of a fresh prior draw and fails an in-distribution manufactured problem under every backend we tried — worse than predicting zero. The second is numerical rank. The row-equilibrated collocation system has condition number $1.04\times 10^{17}$ and retains rank 309 of 512 at $10^{-12}{\sigma }_{\max }$, so the solve is rank-deficient beyond double precision and its answer is decided by the cutoff the library chooses: the same checkpoint solves Poisson to $4.2\times 10^{-5}$ under cuSOLVER and to $4.9\times 10^{-1}$ under LAPACK. Choosing a Tikhonov level by generalized cross-validation stays within an order of magnitude of the oracle cutoff, matching it on the equation that needs regularisation, while making the result a property of the data rather than of the backend. On a band-matched prior, where the dictionary is representable, the comparison does not resolve: across three task banks the meta-trained dictionary scores $1.08\times 10^{-1}$, $1.34\times 10^{-1}$ and $1.21\times 10^{-1}$, while a random basis from the same prior's spectrum ranges from $6.8\times 10^{-2}$ to 1.35.