发表机构
Westlake University; University of Science and Technology of China（西湖大学; 中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出尺度不变神经算子SINO，通过双分支架构和低秩偏置在归一化尺度上学习物理场，在闭合问题上比基线误差降低1.5-38倍，参数效率提升2-23倍。
AI 中文摘要
在科学机器学习中，由偏微分方程控制的物理场表现出低秩结构和尺度不变性。在粗网格上求解方程时，信息缺失导致闭合问题：对未解析物理进行建模以恢复丢失的动力学。尽管闭合项依赖于网格分辨率，但它们代表尺度不变的物理定律。真正学习物理的模型应该通过低秩参数化捕获这些机制，而不是记忆特定于网格的模式。受此启发，我们提出了尺度不变神经算子（SINO），它通过谱域和空间域的双分支架构在归一化物理尺度上学习。SINO使用瓶颈多层感知机生成连续卷积核，嵌入显式低秩归纳偏置，该偏置将超过95%的方差集中在2-3个模式中，通过跨基准的PCA验证，同时大幅减少参数。这种原则性设计产生的缩放定律指数比FNO陡峭38倍，展示了卓越的参数效率。我们将SINO与传统模型（U-Net、DeepONet）、Transformer模型（Transolver、Oformer、GK-Transformer）以及频域模型（FNO、AMFNO、UFNO）在跨越外力驱动的Burgers湍流、衰减Burgers湍流、KS湍流、Kolmogorov驱动的NS湍流和衰减NS湍流的闭合问题上进行比较。实验表明，SINO在基线上实现了1.5-38倍的误差减少和2-23倍的参数效率，优越的缩放定律反映了来自原则性低秩设计的卓越数据效率。代码可在https://this URL获取。
英文摘要
In scientific machine learning, physical fields governed by partial differential equations exhibit low-rank structure and scale invariance. When solving equations on coarse grids, missing information leads to the closure problem: modeling unresolved physics to recover lost dynamics. Although closure terms depend on grid resolution, they represent scale-invariant physical laws. A model truly learning physics should capture these mechanisms with low-rank parameterization rather than memorizing grid-specific patterns. Inspired by this, we propose the Scale-Invariant Neural Operator (SINO), which learns on normalized physical scales via a dual-branch architecture operating in spectral and spatial domains. SINO uses bottleneck MLPs to generate continuous convolution kernels, embedding an explicit low-rank inductive bias that concentrates more than 95 percent of variance in 2-3 modes, as validated by PCA across benchmarks, while drastically reducing parameters. This principled design yields 38 times steeper scaling law exponents than FNO, demonstrating superior parameter efficiency. We compare SINO with traditional models (U-Net, DeepONet), Transformer models (Transolver, Oformer, GK-Transformer), and frequency-domain models (FNO, AMFNO, UFNO) on closure problems spanning externally forced Burgers turbulence, decaying Burgers turbulence, KS turbulence, Kolmogorov-forced NS turbulence, and decaying NS turbulence. Experiments show SINO achieves 1.5-38 times error reduction and 2-23 times parameter efficiency over baselines, with superior scaling laws reflecting exceptional data efficiency from principled low-rank design. Code is available at https://github.com/AI4Science-WestlakeU/SINO.
Comments38 pages, 9 figures
