解决物理神经网络高频细节收敛慢的问题
PILIR: Physics-Informed Local Implicit Representation
- 用可学习网格划分空间，局部编码高频率信息
- 在多个复杂方程上实现更快收敛和更高精度
- 适合需要精细结构重建的科学计算任务
物理信息神经网络已成为求解偏微分方程的强大无网格方法，但其性能常受谱偏差限制。标准 MLP 中的全局参数耦合使模型优先学习低频成分，导致高频细节收敛缓慢。为此，我们提出物理信息局部隐式表示（PILIR）。该方法将物理域离散化为潜在特征空间，并通过连续生成解码器进行重构。利用可学习网格显式编码空间局部性，PILIR 能在局部捕捉高频细节，避免被全局模式稀释。生成神经算子将这些局部潜在特征合成连续物理场，实现细粒度结构的精准重建。在一系列挑战性 PDE 上的实验表明，PILIR 有效缓解谱偏差，显著提升高频细节的收敛速度与精度，优于现有先进方法。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks have become a powerful mesh-free method for solving partial differential equations, but their performance is often limited by spectral bias. Specifically, in standard MLPs used in PINNs, the global parameter coupling causes the model to prioritize learning low-frequency components, resulting in slow convergence for high-frequency details. To overcome this limitation, we introduce the Physics-Informed Local Implicit Representation (PILIR). Our approach separates the global physical domain into a discrete latent feature space and a continuous generative decoder. By using a learnable grid to encode explicit spatial locality, PILIR can capture high-frequency details locally, preventing dilution by global patterns. A generative neural operator then synthesizes these local latent features into continuous physical fields, allowing accurate reconstruction of fine-scale structures. Experiments on a range of challenging PDEs show that PILIR effectively mitigates spectral bias, thereby boosting the convergence of high-frequency details and achieving superior accuracy compared to state-of-the-art methods.
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