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Poster
Solving High-Dimensional PDEs with Latent Spectral Models
Haixu Wu · Tengge Hu · huakun luo · Jianmin Wang · Mingsheng Long

Tue Jul 25 02:00 PM -- 04:30 PM (PDT) @ Exhibit Hall 1 #714
in Poster Session 1 »

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.

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Author Information

Haixu Wu (Tsinghua University)
Tengge Hu (Tsinghua University, Tsinghua University)
huakun luo (Tsinghua University)
Jianmin Wang (Tsinghua University)
Mingsheng Long (Tsinghua University)

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