Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative error reduction of 11.5% averaged on seven benchmarks covering both solid and fluid physics.

翻译：深度学习模型在求解偏微分方程方面取得了显著进展。一种新兴范式是学习神经算子来逼近偏微分方程的输入-输出映射。尽管先前的深度学习模型探索了多尺度架构和多种算子设计，但它们局限于在坐标空间中将算子作为一个整体进行学习。在实际物理科学问题中，偏微分方程是复杂的耦合方程，其数值求解依赖于将高维坐标空间离散化，这使得单一算子无法精确逼近，同时由于维度灾难也难以高效学习。我们提出了潜在谱模型，旨在成为高维偏微分方程的高效精确求解器。超越坐标空间，LSM 采用基于注意力机制的层次化投影网络，在线性时间内将高维数据压缩至紧凑的潜在空间。受数值分析中经典谱方法的启发，我们在潜在空间中设计了一个神经谱模块，通过学习多个基算子来逼近复杂的输入-输出映射，并具有收敛性和逼近性的良好理论保证。实验表明，LSM 持续达到最先进水平，在涵盖固体和流体物理的七个基准测试中，平均相对误差降低了 11.5%。