We construct the first rigorously justified probabilistic algorithm for
recovering the solution operator of a hyperbolic partial differential equation
(PDE) in two variables from input-output training pairs. The primary challenge
of recovering the solution operator of hyperbolic PDEs is the presence of
characteristics, along which the associated Green's function is discontinuous.
Therefore, a central component of our algorithm is a rank detection scheme that
identifies the approximate location of the characteristics. By combining the
randomized singular value decomposition with an adaptive hierarchical partition
of the domain, we construct an approximant to the solution operator using
$O(\Psi_\epsilon^{-1}\epsilon^{-7}\log(\Xi_\epsilon^{-1}\epsilon^{-1}))$
input-output pairs with relative error $O(\Xi_\epsilon^{-1}\epsilon)$ in the
operator norm as $\epsilon\to0$, with high probability. Here, $\Psi_\epsilon$
represents the existence of degenerate singular values of the solution
operator, and $\Xi_\epsilon$ measures the quality of the training data. Our
assumptions on the regularity of the coefficients of the hyperbolic PDE are
relatively weak given that hyperbolic PDEs do not have the ``instantaneous
smoothing effect'' of elliptic and parabolic PDEs, and our recovery rate
improves as the regularity of the coefficients increases.

我们构建了第一个严格合理的概率算法，用于从输入 - 输出训练对中恢复二元双曲型偏微分方程的解算符。

双曲型偏微分方程的算子学习

Operator learning for hyperbolic partial differential equations

Neural operators have gained recognition as potent tools for learning
solutions of a family of partial differential equations. The state-of-the-art
neural operators excel at approximating the functional relationship between
input functions and the solution space, potentially reducing computational
costs and enabling real-time applications. However, they often fall short when
tackling time-dependent problems, particularly in delivering accurate long-term
predictions. In this work, we propose "waveformer", a novel operator learning
approach for learning solutions of dynamical systems. The proposed waveformer
exploits wavelet transform to capture the spatial multi-scale behavior of the
solution field and transformers for capturing the long horizon dynamics. We
present four numerical examples involving Burgers's equation, KS-equation,
Allen Cahn equation, and Navier Stokes equation to illustrate the efficacy of
the proposed approach. Results obtained indicate the capability of the proposed
waveformer in learning the solution operator and show that the proposed
Waveformer can learn the solution operator with high accuracy, outperforming
existing state-of-the-art operator learning algorithms by up to an order, with
its advantage particularly visible in the extrapolation region

通过波形函学习法学习动态系统解算符以提高准确性并超越现有的操作符学习算法的研究。

用于建模动态系统的 Waveformer

Waveformer for modelling dynamical systems

Spectral analysis provides one of the most effective paradigms for
information-preserving dimensionality reduction, as simple descriptions of
naturally occurring signals are often obtained via few terms of periodic basis
functions. In this work, we study deep neural networks designed to harness the
structure in frequency domain for efficient learning of long-range correlations
in space or time: frequency-domain models (FDMs). Existing FDMs are based on
complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform
computation on the spectrum and input data separately. This design introduces
considerable computational overhead: for each layer, a forward and inverse FT.
Instead, this work introduces a blueprint for frequency domain learning through
a single transform: transform once (T1). To enable efficient, direct learning
in the frequency domain we derive a variance-preserving weight initialization
scheme and investigate methods for frequency selection in reduced-order FDMs.
Our results noticeably streamline the design process of FDMs, pruning redundant
transforms, and leading to speedups of 3x to 10x that increase with data
resolution and model size. We perform extensive experiments on learning the
solution operator of spatio-temporal dynamics, including incompressible
Navier-Stokes, turbulent flows around airfoils and high-resolution video of
smoke. T1 models improve on the test performance of FDMs while requiring
significantly less computation (5 hours instead of 32 for our large-scale
experiment), with over 20% reduction in average predictive error across tasks.

本文介绍一种基于单个信号变换的频域学习模型，通过方差保持初始化方法和频率选择技术，可以简化模型设计，从而在各种动态系统的学习中获得更高的测试性能和更少的计算成本。