We consider using deep neural networks to solve time-dependent partial
differential equations (PDEs), where multi-scale processing is crucial for
modeling complex, time-evolving dynamics. While the U-Net architecture with
skip connections is commonly used by prior studies to enable multi-scale
processing, our analysis shows that the need for features to evolve across
layers results in temporally misaligned features in skip connections, which
limits the model's performance. To address this limitation, we propose SineNet,
consisting of multiple sequentially connected U-shaped network blocks, referred
to as waves. In SineNet, high-resolution features are evolved progressively
through multiple stages, thereby reducing the amount of misalignment within
each stage. We furthermore analyze the role of skip connections in enabling
both parallel and sequential processing of multi-scale information. Our method
is rigorously tested on multiple PDE datasets, including the Navier-Stokes
equations and shallow water equations, showcasing the advantages of our
proposed approach over conventional U-Nets with a comparable parameter budget.
We further demonstrate that increasing the number of waves in SineNet while
maintaining the same number of parameters leads to a monotonically improved
performance. The results highlight the effectiveness of SineNet and the
potential of our approach in advancing the state-of-the-art in neural PDE
solver design. Our code is available as part of AIRS
(this https URL).

使用深度神经网络解决时间依赖的偏微分方程，提出了一种名为 SineNet 的方法，通过多个连续连接的网络块逐步演化高分辨率特征，降低每个阶段内的特征错位，以改进模型的性能，证明了 SineNet 的有效性及其在神经 PDE 求解器设计中的潜力。

SineNet: 学习时变偏微分方程中的时间动力学

SineNet: Learning Temporal Dynamics in Time-Dependent Partial  Differential Equations

Training neural networks sequentially in time to approximate solution fields
of time-dependent partial differential equations can be beneficial for
preserving causality and other physics properties; however, the
sequential-in-time training is numerically challenging because training errors
quickly accumulate and amplify over time. This work introduces Neural Galerkin
schemes that update randomized sparse subsets of network parameters at each
time step. The randomization avoids overfitting locally in time and so helps
prevent the error from accumulating quickly over the sequential-in-time
training, which is motivated by dropout that addresses a similar issue of
overfitting due to neuron co-adaptation. The sparsity of the update reduces the
computational costs of training without losing expressiveness because many of
the network parameters are redundant locally at each time step. In numerical
experiments with a wide range of evolution equations, the proposed scheme with
randomized sparse updates is up to two orders of magnitude more accurate at a
fixed computational budget and up to two orders of magnitude faster at a fixed
accuracy than schemes with dense updates.

通过在时序中顺序训练神经网络来逼近时变偏微分方程的解场，有助于保持因果性和其他物理特性；然而，由于训练误差随时间快速累积和放大，时序训练的数值挑战很大。本研究引入了 Neural Galerkin 方案，该方案在每个时间步骤更新网络参数的随机稀疏子集。随机化的操作避免了局部时间上的过拟合问题，因此有助于防止误差在时序训练中快速累积。该方案的更新稀疏性减少了训练的计算成本，而不会失去表达能力，因为在每个时间步骤中，很多网络参数是冗余的。通过与密集更新方案进行的大量数值实验表明，使用随机稀疏更新的建议方案在固定计算预算下精度高出两个数量级，而在固定精度下速度快两个数量级。

随机稀疏神经 Galerkin 方案用于使用深度网络求解演化方程

Randomized Sparse Neural Galerkin Schemes for Solving Evolution  Equations with Deep Networks

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.

基于时间依赖的偏微分方程，神经 PDE 求解器，时域展开策略，扩散模型和 PDE-Refiner 是该论文的五个关键词，通过多步细化过程实现对所有频率分量的准确建模，提供稳定准确的结果，优于现有模型，并通过去噪目标实现新颖的频谱数据增强。