Striding Across Reynolds Numbers: Representation Geometry in Neural PDE Generalisation
For researchers in neural PDE solvers, this work provides a systematic analysis of representation geometry's role in cross-Reynolds generalization, but claims are scoped to a single benchmark and are incremental.
The paper investigates cross-Reynolds generalization in neural PDE solvers, finding that representation geometry is a key factor. On the forced 2D Navier-Stokes benchmark, a retrieval method (ConvAE-Relay) achieves 38.34% relative L2 error under a 10x Reynolds shift, while a U-Net achieves 34.72%, with autoregressive drift identified as the main bottleneck.
Cross-Reynolds generalisation in neural PDE solvers remains poorly characterised. On the canonical forced 2D Navier-Stokes benchmark, a trained Fourier Neural Operator reaches 46.68% relative L2 error under a 10x Reynolds-number shift, yet zero-forward-model retrieval baselines already improve to 41-42%. This suggests representation geometry as a major organising variable among the tested methods. We test this hypothesis through ConvAE-Relay, which matches states in a source-trained convolutional autoencoder latent space and borrows dynamics from a source-regime database, achieving 38.34+/-0.07% using only a source-regime database and no target-regime fitting, labels, or database entries. A 2x2 ablation isolates matching quality as dominant over the update rule. Oracle experiments confirm that source-regime dynamics directions remain transferable (cosine similarity ~0.84) when matching stays on-manifold; autoregressive drift is the primary bottleneck (~12 percentage points). From the learned-prediction side, a U-Net with multi-scale skip connections achieves 34.72+/-0.60%, consistent with the retrieval-side finding that local, multi-scale representations organise cross-Reynolds transfer among tested methods. All claims are scoped to this benchmark.