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.

翻译：摘要：本文提出"Courant"模型——一种基于感知器（Perceiver）的编码器-处理器-解码器代理模型，其潜变量特征在物理空间中展现出适应性特化与局部支撑能力，可实现类似自适应hp-细化方案的功能。这一属性在传统数值求解器及广义科学机器学习领域中具有高度需求性。所提架构融合了共享随机傅里叶特征坐标嵌入、状态自适应潜变量查询机制以及轻量化解码器。Courant采用稳态或瞬态仿真数据进行端到端训练，仅需物理空间中标准L_2预测损失即可在基准测试中达到竞争性精度。研究表明，Courant的归纳偏置使得潜变量天然具备可解释性：它们可在仿真域中发展多尺度几何特化，在时变情形下追踪相干结构，其行为类似于时变空间基函数，从而实现对仿真场的紧凑型、几何锚定、类单位分解型场分解解码。