用物理维度设计轻量级神经算子，提升PDE求解泛化能力
DimINO: Dimension-Informed Neural Operator Learning
- 基于物理维度分析，引入维度归一化与重维度操作
- 在多个PDE数据集上性能提升最高达76.3%
- 适用于需跨参数泛化的物理模拟场景
在计算物理中，求解偏微分方程（PDE）长期面临挑战。近年来，神经算子方法因其能逼近函数间映射而受到关注。尽管神经算子具备通用逼近性，但可靠误差界通常依赖庞大模型结构，如深层傅里叶层堆叠。这引发一个问题：能否设计轻量模型而不牺牲泛化能力？为此，我们提出DimINO（Dimension-Informed Neural Operators），受量纲分析启发。该框架包含两个核心组件：DimNorm与重维度操作，可无缝嵌入现有神经算子架构。二者增强模型在不同物理参数数据集间的泛化能力。理论上，我们建立了DimINO的通用逼近定理，并证明其满足关键性质——相似变换不变性（STI）。实验上，DimINO在多个PDE数据集上实现最高76.3%的性能提升，并展现出明确的STI证据。
原文摘要 · Abstract (English)
In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their ability to approximate operators-mappings between functions. Although neural operators benefit from a universal approximation theorem, achieving reliable error bounds often necessitates large model architectures, such as deep stacks of Fourier layers. This raises a natural question: Can we design lightweight models without sacrificing generalization? To address this, we introduce DimINO (Dimension-Informed Neural Operators), a framework inspired by dimensional analysis. DimINO incorporates two key components, DimNorm and a redimensionalization operation, which can be seamlessly integrated into existing neural operator architectures. These components enhance the model's ability to generalize across datasets with varying physical parameters. Theoretically, we establish a universal approximation theorem for DimINO and prove that it satisfies a critical property we term Similar Transformation Invariance (STI). Empirically, DimINO achieves up to 76.3% performance gain on PDE datasets while exhibiting clear evidence of the STI property.
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