Courant用自适应潜变量实现物理场的局部精准建模，像智能网格加密。
Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition
- 基于Perceiver架构，用随机傅里叶特征嵌入坐标，潜变量随状态自适应变化。
- 在稳态与瞬态数据上训练，仅用L2损失即达基准线精度，潜变量具可解释性。
- 适合需高精度局部建模的科学计算场景，如流体模拟、多尺度物理系统建模。
我们提出「Courant」，一种基于Perceiver的编码器-处理器-解码器代理模型，其潜变量在物理空间中表现出自适应专化和局部支持特性，功能类似自适应hp-加密方案，这在传统数值求解器和科学机器学习中极为重要。该架构结合共享的随机傅里叶特征坐标嵌入、状态自适应潜变量查询及轻量解码器。Courant以稳态或瞬态模拟数据端到端训练，仅使用物理空间中的标准L₂预测损失，便在基准测试中达到竞争性精度。我们证明，其归纳偏置使潜变量天生可解释：在仿真域中发展出多尺度几何专化，在时变情况下追踪相干结构，类比于随时间演化的空间基函数，支持解码出紧凑、几何锚定、类似单位分解的场分解。
原文摘要 · Abstract (English)
We introduce "Courant", a Perceiver-based encoder-processor-decoder surrogate model that has latent features exhibiting adaptive specialization and local support in the physical space, enabling functionality akin to an adaptive hp-refinement scheme, an attribute that is highly desirable in traditional numerical solvers and scientific machine learning broadly. The proposed architecture combines a shared random Fourier feature coordinate embedding, state-adapted latent queries, and a light-weight decoder. Courant is trained end-to-end with steady or transient simulation data and only a standard L_2 prediction loss in the physical space, achieving competitive accuracy on benchmarks. We demonstrate that Courant's inductive biases yield latents that are interpretable by design: they develop multiscale geometric specialization in the simulation domain and track coherent structures in the time-dependent case, acting analogously to time-evolving spatial basis functions and allowing for decoding a compact, geometry-anchored, partition-of-unity-like decomposition of the simulated field.
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