通过表征几何分析神经微分方程模型跨雷诺数泛化机制
Striding Across Reynolds Numbers: Representation Geometry in Neural PDE Generalisation
- 用卷积自编码器隐空间匹配+源域动态借用实现零样本迁移
- 在10倍雷诺数变化下误差降至38.34%，优于已有基线
- 发现局部多尺度表征是跨雷诺数泛化的关键组织机制
神经微分方程求解器在跨雷诺数泛化方面仍缺乏清晰刻画。在经典的受迫二维纳维-斯托克斯基准测试中，训练好的傅里叶神经算子在雷诺数扩大10倍时相对L2误差为46.68%，而零样本检索基线已可提升至41-42%。这表明表征几何可能是方法间差异的主要组织变量。我们通过ConvAE-Relay验证该假设：仅利用源域训练的卷积自编码器隐空间状态匹配，并借用源域数据库中的动力学，即可在无需目标域拟合、标签或数据条目的情况下达到38.34±0.07%的误差。2×2消融实验表明，匹配质量优于更新规则。模拟实验确认，当匹配保持在流形上时，源域动力学方向仍具可转移性（余弦相似度~0.84），但自回归漂移是主要瓶颈（约12个百分点）。从学习预测角度看，带有跨尺度跳跃连接的U-Net实现34.72±0.60%误差，与检索侧结果一致，表明局部多尺度表征在所测方法中主导跨雷诺数迁移。所有结论均限定于该基准测试。
原文摘要 · Abstract (English)
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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