PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers
Time-dependent partial differential equations (PDEs) are ubiquitous in
science and engineering. Recently, mostly due to the high computational cost of
traditional solution techniques, deep neural network based surrogates have
gained increased interest. The practical utility of such neural PDE solvers
relies on their ability to provide accurate, stable predictions over long time
horizons, which is a notoriously hard problem. In this work, we present a
large-scale analysis of common temporal rollout strategies, identifying the
neglect of non-dominant spatial frequency information, often associated with
high frequencies in PDE solutions, as the primary pitfall limiting stable,
accurate rollout performance. Based on these insights, we draw inspiration from
recent advances in diffusion models to introduce PDE-Refiner; a novel model
class that enables more accurate modeling of all frequency components via a
multistep refinement process. We validate PDE-Refiner on challenging benchmarks
of complex fluid dynamics, demonstrating stable and accurate rollouts that
consistently outperform state-of-the-art models, including neural, numerical,
and hybrid neural-numerical architectures. We further demonstrate that
PDE-Refiner greatly enhances data efficiency, since the denoising objective
implicitly induces a novel form of spectral data augmentation. Finally,
PDE-Refiner's connection to diffusion models enables an accurate and efficient
assessment of the model's predictive uncertainty, allowing us to estimate when
the surrogate becomes inaccurate.
64d59d6c887f55fb6e055a63
Phillip Lippe
claimed_verified
2023-08-11T14:37:49.983Z
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/avatars/e6fb4143f4d2a1fe721a50167717803a.svg
Phillip Lippe
user
phlippe
64d59d6c887f55fb6e055a64
Bastiaan S. Veeling
64d59d6c887f55fb6e055a65
Paris Perdikaris
64d59d6c887f55fb6e055a66
Richard E. Turner
64d59d6c887f55fb6e055a67
Johannes Brandstetter
PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers
Time-dependent partial differential equations (PDEs) are ubiquitous in
science and engineering. Recently, mostly due to the high computational cost of
traditional solution techniques, deep neural network based surrogates have
gained increased interest. The practical utility of such neural PDE solvers
relies on their ability to provide accurate, stable predictions over long time
horizons, which is a notoriously hard problem. In this work, we present a
large-scale analysis of common temporal rollout strategies, identifying the
neglect of non-dominant spatial frequency information, often associated with
high frequencies in PDE solutions, as the primary pitfall limiting stable,
accurate rollout performance. Based on these insights, we draw inspiration from
recent advances in diffusion models to introduce PDE-Refiner; a novel model
class that enables more accurate modeling of all frequency components via a
multistep refinement process. We validate PDE-Refiner on challenging benchmarks
of complex fluid dynamics, demonstrating stable and accurate rollouts that
consistently outperform state-of-the-art models, including neural, numerical,
and hybrid neural-numerical architectures. We further demonstrate that
PDE-Refiner greatly enhances data efficiency, since the denoising objective
implicitly induces a novel form of spectral data augmentation. Finally,
PDE-Refiner's connection to diffusion models enables an accurate and efficient
assessment of the model's predictive uncertainty, allowing us to estimate when
the surrogate becomes inaccurate.
PDE-Refiner, inspired by diffusion models, provides accurate, stable, and data-efficient solutions for time-dependent PDEs by refining all frequency components.
partial differential equations deep neural networks neural PDE solvers temporal rollout strategies diffusion models PDE-Refiner multistep refinement fluid dynamics spectral data augmentation predictive uncertainty