  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 gain of 11.5% averaged on seven benchmarks covering both
solid and fluid physics. Code is available at
https://github.com/thuml/Latent-Spectral-Models.
