Striding Across Reynolds Numbers: Representation Geometry in Neural PDE Generalisation

• URL: http://arxiv.org/abs/2605.30112v1
• Date: Thu, 28 May 2026 15:49:26 GMT
• Title: Striding Across Reynolds Numbers: Representation Geometry in Neural PDE Generalisation
• Abstract summary: Cross-Reynolds generalisation in neural PDE solvers remains poorly characterised.<n>We test this hypothesis through ConvAE-Relay, which matches states in a source-trained convolutional autoencoder latent space.<n>Experiments confirm that source-regime dynamics directions remain transferable (cosine similarity 0.84) when matching stays on-manifold.
• Score: 0.5451632647096261
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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